The Navier–Stokes Problem
in the 21st Century
Second Edition
The complete resolution of the Navier–Stokes equation—one of the Clay Millennium Prize Problems—remains an important open challenge in partial differential equations (PDEs) research
despite substantial studies on turbulence and three-dimensional fluids. The Navier–Stokes Problem in the 21st Century, Second Edition continues to provide a self-contained guide to the role
of harmonic analysis in the PDEs of fluid mechanics, now revised to include fresh examples,
theorems, results, and references that have become relevant since the first edition published in
2016.
Pierre Gilles Lemarié-Rieusset is a professor at the University of Evry Val d’Essonne. Dr.
Lemarié-Rieusset has constructed many widely used bases, such as the Meyer-Lemarié wavelet
basis and the Battle-Lemarié spline wavelet basis. His current research focuses on the application of harmonic analysis to the study of nonlinear PDEs in fluid mechanics. He is the author or
co-author of several books, including Recent Developments in the Navier-Stokes Problem.
Taylor & Francis
Taylor & Francis Group
http://taylorandfrancis.com
The Navier–Stokes Problem
in the 21st Century
Second Edition
Pierre Gilles Lemarié-Rieusset
University of Evry Val d’Essonne
Designed cover image: Bibliothèque nationale de France
Second edition published 2024
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© 2024 Taylor & Francis Group, LLC
First edition published by CRC Press 2016
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ISBN: 978-0-367-48726-3 (hbk)
ISBN: 978-1-032-62373-3 (pbk)
ISBN: 978-1-003-04259-4 (ebk)
DOI: 10.1201/ 9781003042594
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Contents
Preface to the First Edition
xi
Preface to the Second Edition
xxi
1 Presentation of the Clay Millennium Prizes
1.1
Regularity of the Three-Dimensional Fluid Flows: A Mathematical
Challenge for the 21st Century . . . . . . . . . . . . . . . . . . . .
1.2
The Clay Millennium Prizes . . . . . . . . . . . . . . . . . . . . . .
1.3
The Clay Millennium Prize for the Navier–Stokes Equations . . . .
1.4
Boundaries and the Navier–Stokes Clay Millennium Problem . . .
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2 The
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
Physical Meaning of the Navier–Stokes Equations
Frames of References . . . . . . . . . . . . . . . . . . . .
The Convection Theorem . . . . . . . . . . . . . . . . .
Conservation of Mass . . . . . . . . . . . . . . . . . . .
Newton’s Second Law . . . . . . . . . . . . . . . . . . .
Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . .
Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Equations of Hydrodynamics . . . . . . . . . . . . .
The Navier–Stokes Equations . . . . . . . . . . . . . . .
Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary Terms . . . . . . . . . . . . . . . . . . . . . .
Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . .
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . .
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13
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3 History of the Equation
3.1
Mechanics in the Scientific Revolution Era . . . .
3.2
Bernoulli’s Hydrodymica . . . . . . . . . . . . . .
3.3
D’Alembert . . . . . . . . . . . . . . . . . . . . .
3.4
Euler . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Laplacian Physics . . . . . . . . . . . . . . . . .
3.6
Navier, Cauchy, Poisson, Saint-Venant and Stokes
3.7
Reynolds . . . . . . . . . . . . . . . . . . . . . .
3.8
Oseen, Leray, Hopf and Ladyzhenskaya . . . . .
3.9
Turbulence Models . . . . . . . . . . . . . . . . .
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25
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4 Classical Solutions
4.1
The Heat Kernel . . . . . . . . . . .
4.2
The Poisson Equation . . . . . . . .
4.3
The Helmholtz Decomposition . . .
4.4
The Stokes Equation . . . . . . . . .
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42
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v
vi
Contents
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
The Oseen Tensor . . . . . . . . . . . . . . . . . . . . .
Classical Solutions for the Navier–Stokes Problem . . .
Maximal Classical Solutions and Estimates in L∞ Norms
Small Data . . . . . . . . . . . . . . . . . . . . . . . . .
Spatial Asymptotics . . . . . . . . . . . . . . . . . . . .
Spatial Asymptotics for the Vorticity . . . . . . . . . . .
Maximal Classical Solutions and Estimates in L2 Norms
Intermediate Conclusion . . . . . . . . . . . . . . . . . .
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5 A Capacitary Approach of the Navier–Stokes Integral Equations
5.1
The Integral Navier–Stokes Problem . . . . . . . . . . . . . . . . .
5.2
Quadratic Equations in Banach Spaces . . . . . . . . . . . . . . . .
5.3
A Capacitary Approach of Quadratic Integral Equations . . . . . .
5.4
Generalized Riesz Potentials on Spaces of Homogeneous Type . . .
5.5
Dominating Functions for the Navier–Stokes Integral Equations . .
5.6
Oseen’s Theorem and Dominating Functions . . . . . . . . . . . .
5.7
Functional Spaces and Multipliers . . . . . . . . . . . . . . . . . .
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6 The
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Differential and the Integral Navier–Stokes Equations
Very Weak Solutions for the Navier–Stokes Equations . . .
Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . .
The Leray Projection Operator . . . . . . . . . . . . . . . .
Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . .
Oseen Equations . . . . . . . . . . . . . . . . . . . . . . . .
Mild Solutions for the Navier–Stokes Equations . . . . . . .
Suitable Solutions for the Navier–Stokes Equations . . . . .
7 Mild
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Solutions in Lebesgue or Sobolev Spaces
Kato’s Mild Solutions . . . . . . . . . . . . .
Local Solutions in the Hilbertian Setting . . .
Global Solutions in the Hilbertian Setting . .
Sobolev Spaces . . . . . . . . . . . . . . . . .
A Commutator Estimate . . . . . . . . . . . .
Lebesgue Spaces . . . . . . . . . . . . . . . .
Maximal Functions . . . . . . . . . . . . . . .
Basic Lemmas on Real Interpolation Spaces .
Uniqueness of L3 Solutions . . . . . . . . . .
8 Mild
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
Solutions in Besov or Morrey Spaces
Morrey Spaces . . . . . . . . . . . . . . . . . .
Morrey Spaces and Maximal Functions . . . . .
Uniqueness of Morrey Solutions . . . . . . . . .
Besov Spaces . . . . . . . . . . . . . . . . . . .
Regular Besov Spaces . . . . . . . . . . . . . .
Triebel–Lizorkin Spaces . . . . . . . . . . . . .
Fourier Transform and Navier–Stokes Equations
The Cheap Navier–Stokes Equation . . . . . .
Plane Waves . . . . . . . . . . . . . . . . . . .
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Contents
9 The
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
vii
Space BMO−1 and the Koch and Tataru Theorem
The Koch and Tataru Theorem . . . . . . . . . . . . . .
A Variation on the Koch and Tataru Theorem . . . . .
Q-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Special Subclass of BM O−1 . . . . . . . . . . . . . .
Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . .
Further Results on Ill-posedness . . . . . . . . . . . . .
Large Data for Mild Solutions . . . . . . . . . . . . . . .
Stability of Global Solutions . . . . . . . . . . . . . . . .
Analyticity . . . . . . . . . . . . . . . . . . . . . . . . .
Small Data . . . . . . . . . . . . . . . . . . . . . . . . .
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192
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309
309
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350
12 Leray’s Weak Solutions
12.1 The Rellich Lemma . . . . . . . . . . . . . . . . . . . . . .
12.2 Leray’s Weak Solutions . . . . . . . . . . . . . . . . . . . .
12.3 Weak-Strong Uniqueness: The Prodi–Serrin Criterion . . .
12.4 Weak-Strong Uniqueness and Morrey Spaces on the Product
12.5 Almost Strong Solutions . . . . . . . . . . . . . . . . . . . .
12.6 Weak Perturbations of Mild Solutions . . . . . . . . . . . .
12.7 Non-uniqueness of Weak Solutions . . . . . . . . . . . . . .
12.8 The Inviscid Limit . . . . . . . . . . . . . . . . . . . . . . .
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Space R × R3
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13 Partial Regularity Results for Weak Solutions
13.1 Interior Regularity . . . . . . . . . . . . . . .
13.2 Serrin’s Theorem on Interior Regularity . . .
13.3 O’Leary’s Theorem on Interior Regularity . .
13.4 Further Results on Parabolic Morrey Spaces .
13.5 Hausdorff Measures . . . . . . . . . . . . . .
13.6 Singular Times . . . . . . . . . . . . . . . . .
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10 Special Examples of Solutions
10.1 Symmetries for the Navier–Stokes Equations . . .
10.2 Two-and-a-Half Dimensional Flows . . . . . . . . .
10.3 Axisymmetrical Solutions . . . . . . . . . . . . . .
10.4 Helical Solutions . . . . . . . . . . . . . . . . . . .
10.5 Brandolese’s Symmetrical Solutions . . . . . . . .
10.6 Self-similar Solutions . . . . . . . . . . . . . . . . .
10.7 Stationary Solutions . . . . . . . . . . . . . . . . .
10.8 Landau’s Solutions of the Navier–Stokes Equations
10.9 Time-Periodic Solutions . . . . . . . . . . . . . . .
10.10 Beltrami Flows . . . . . . . . . . . . . . . . . . . .
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11 Blow-up?
11.1 First Criteria . . . . . . . . . . . . . . . . . . . . . .
11.2 Blow-up for the Cheap Navier–Stokes Equation . . .
11.3 Serrin’s Criterion . . . . . . . . . . . . . . . . . . . .
11.4 A Remark on Serrin’s Criterion and Leray’s Criterion
11.5 Some Further Generalizations of Serrin’s Criterion .
11.6 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . .
11.7 Squirts . . . . . . . . . . . . . . . . . . . . . . . . . .
11.8 Eigenvalues of the Strain Matrix . . . . . . . . . . .
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viii
Contents
13.7
13.8
13.9
13.10
13.11
The Local Energy Inequality . . . . . . . . . . . . . . . . . . . . .
The Caffarelli-Kohn-Nirenberg Theorem on Partial Regularity . . .
Proof of the Caffarelli–Kohn–Nirenberg Criterion . . . . . . . . . .
Parabolic Hausdorff Dimension of the Set of Singular Points . . . .
On the Role of the Pressure in the Caffarelli, Kohn, and Nirenberg
Regularity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
14 A Theory of Uniformly Locally L2 Solutions
14.1 Uniformly Locally Square Integrable Solutions . . . . . . . . .
14.2 Local Inequalities for Local Leray Solutions . . . . . . . . . . .
14.3 The Caffarelli, Kohn and Nirenberg ϵ–Regularity Criterion . .
14.4 A Weak-Strong Uniqueness Result . . . . . . . . . . . . . . . .
14.5 Global Existence for Local Leray Solutions . . . . . . . . . . .
14.6 Weighted Estimates . . . . . . . . . . . . . . . . . . . . . . . .
14.7 A Stability Estimate . . . . . . . . . . . . . . . . . . . . . . . .
14.8 Barker’s Theorem on Weak-Strong Uniqueness . . . . . . . . .
14.9 Further Results on Global Existence of Suitable Weak Solutions
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419
424
431
443
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444
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452
452
462
468
480
483
491
500
504
511
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L3 Theory of Suitable Solutions
520
Local Leray Solutions with an Initial Value in L3 . . . . . . . . . . . . . . 520
Blow up in Finite Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Backward Uniqueness for Local Leray Solutions . . . . . . . . . . . . . . . 528
Seregin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
Further Comments on Seregin’s Theorem . . . . . . . . . . . . . . . . . . 533
Critical Elements for the Blow-up of the Cauchy Problem in L3 . . . . . . 537
Known Results on the Cauchy Problem for the Navier–Stokes Equations in
Presence of a Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
15.8 Local Estimates for Suitable Solutions . . . . . . . . . . . . . . . . . . . . 543
15.9 Uniqueness for Suitable Solutions . . . . . . . . . . . . . . . . . . . . . . . 545
15.10 A Quantitative One-scale Estimate for the Caffarelli–Kohn–Nirenberg
Regularity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
15.11 The Topological Structure of the Set of Suitable Solutions . . . . . . . . . 551
15.12 Escauriaza, Seregin and Šverák’s Theorem . . . . . . . . . . . . . . . . . . 555
15 The
15.1
15.2
15.3
15.4
15.5
15.6
15.7
16 Self-similarity and the Leray–Schauder Principle
16.1 The Leray–Schauder Principle . . . . . . . . . . .
16.2 Steady-state Solutions . . . . . . . . . . . . . . .
16.3 The Liouville Problem for Steady Solutions . . .
16.4 Self-similarity . . . . . . . . . . . . . . . . . . . .
16.5 Statement of Jia and Šverák’s Theorem . . . . .
16.6 The Case of Locally Bounded Initial Data . . . .
16.7 The Case of Rough Data . . . . . . . . . . . . .
16.8 Non-existence of Backward Self-similar Solutions
16.9 Discretely Self-similar Solutions . . . . . . . . . .
16.10 Time-periodic Weak Solutions . . . . . . . . . . .
17 α-Models
17.1 Global Existence, Uniqueness and Convergence
Equations . . . . . . . . . . . . . . . . . . . . .
17.2 Leray’s Mollification and the Leray-α Model . .
17.3 The Navier–Stokes α-Model . . . . . . . . . . .
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557
557
557
559
573
575
578
587
589
598
608
619
Issues
. . . .
. . . .
. . . .
for Approximated
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
619
620
625
Contents
17.4
17.5
17.6
ix
The Clark-α Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Simplified Bardina Model . . . . . . . . . . . . . . . . . . . . . . . .
Reynolds Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 Other Approximations of the Navier–Stokes Equations
18.1 Faedo–Galerkin Approximations . . . . . . . . . . . . .
18.2 Frequency Cut-off . . . . . . . . . . . . . . . . . . . . .
18.3 Hyperviscosity . . . . . . . . . . . . . . . . . . . . . . .
18.4 Ladyzhenskaya’s Model . . . . . . . . . . . . . . . . . .
18.5 Damped Navier–Stokes Equations . . . . . . . . . . . .
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635
642
648
650
650
658
661
666
674
19 Artificial Compressibility
679
19.1 Temam’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
19.2 Višik and Fursikov’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . 687
19.3 Hyperbolic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 694
20 Conclusion
20.1 Energy Inequalities . . . . . . . . .
20.2 Critical Spaces for Mild Solutions .
20.3 Models for the (Potential) Blow-up
20.4 The Method of Critical Elements .
20.5 Some Open Questions . . . . . . .
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706
706
707
710
713
714
Notations and Glossary
715
Bibliography
717
Index
747
Taylor & Francis
Taylor & Francis Group
http://taylorandfrancis.com
Preface to the First Edition
In a provocative paper published in 1974, Irigaray [240] muses on fluid mechanics with,
according to Hayles [230], “elliptical prose and incendiary reasoning.” Her thesis has been
turned into a kind of “post-modern myth” by the French-Theory bashers Sokal and Bricmont, relayed by Dawkins [147] in his survey in Nature of Sokal and Bricmont’s book
Fashionable Nonsense [67].
Dawkins’s irony focused on Hayles’s 1992 paper where “Katherine Hayles made the
mistake of re-expressing Irigaray’s thoughts in (comparatively) clear language. For once, we
get a reasonably unobstructed look at the emperor and, yes, he has no clothes.” According
to Hayles, the meaning of Irigaray’s paper is the following one:
The privileging of solid over fluid mechanics, and indeed the inability of science
to deal with turbulent flow at all, she attributes to the association of fluidity
with femininity. Whereas men have sex organs that protrude and become rigid,
women have openings that leak menstrual blood and vaginal fluids... From this
perspective it is no wonder that science has not been able to arrive at a successful
model for turbulence. The problem of turbulent flow cannot be solved because the
conceptions of fluids (and of women) have been formulated so as necessarily to
leave unarticulated remainders.
And Dawkins adds:
It helps to have Sokal and Bricmont on hand to tell us the real reason why
turbulent flow is a hard problem: the Navier–Stokes equations are difficult to
solve.
There is at least one point when one should disagree with Dawkins: whom can it help
to have Sokal’s libel on hand? Irigaray’s essay is a seven-page paper in the journal L’Arc,
in a special issue dedicated to the psychoanalyst Lacan; it is clearly not a treatise on the
meaning of fluid mechanics but a variation on Lacanian themes. The stake of the paper is the
confrontation between the symbolic order and reality: in Irigaray’s views, rationality deals
with universality and reality with singularity; she opposes the (masculine) “tout” [every] to
the (feminine) “pas-toute” [not every] in light of the theses Lacan developed in the early
seventies. Hayles’s exposition of Irigaray’s theses is purely provocative: she completely omits
the psychoanalytic context of enunciation and tries to give the most dramatic presentation
to an “outrageous” thesis that would uprise engineers and hydraulicians.
There is at least one point where one may agree with Dawkins: “the Navier–Stokes
equations are difficult to solve.” In a parody of Churchill’s analysis of the Soviet policy in
the thirties, Constantin writes [126]:
The Reynolds equations are still a riddle. They are based on the Navier–Stokes
equations, which are still a mystery. The Navier–Stokes equations are a viscous
regularization of the Euler equations, which are still an enigma. Turbulence is a
riddle wrapped in a mistery inside an enigma.
xi
xii
Preface to the First Edition
In this book, we are going to address the mystery part of the riddle-mistery-enigma
trilogy. Obviously, with no great hopes of success: Tao expressed in 2007 his views on “Why
global regularity for Navier-Stokes is hard” [460]. He insists that there is no hope to find an
explicit formula to solve the equations, or even to re-write explicitly the equations in simpler
ones. It may seem an obvious point in the light of centuries of studies in mathematical
physics; however, one may find every now and then authors who try to find such a miraculous
formula. Another point is that we lack controlled quantities that would impede any eventual
blow-up; for instance, the vectorial structure of the equations does not allow arguments
based on the control of the signs of associated quantities. The only control we have is
Leray’s energy inequality which provides the control of the L2 norm of the velocity of the
fluid, while we should control the vorticity of the fluid, i.e., the curl of the velocity. In
dimension 2, the evolution equation of the vorticity allows such a control, and we know
global existence of regular solutions; in dimension 3, however, we have no control on the
vorticity (to the exception of the case of some symmetrical fluids, as the axisymmetrical
fluids with no swirl studied by Ladyzhenskaya [295]) unless the data are very small.
The fact is that we have barely any control on the non-linear terms of the equations.
The Navier–Stokes equations combine a diffusive term and a convective term. Diffusion will
damp the high gradients, but convection may transfer energy from small gradients to high
gradients at a rate faster than the damping induced by diffusion. We still do not know,
eighty years after the seminal work of Leray on the Navier–Stokes equations [328], whether
this mechanism of energy transfer between different scales may open the way to the blow-up
of solutions or whether the damping effect of the diffusion will prevail and block the way to
the blow-up.
When in the end of 2011, my editor Sunil Nair invited me to write another book on
the Navier–Stokes equations, ten years after Recent Developments in the Navier–Stokes
Problem [313], I felt that I had included in my first book all I knew about the Navier–
Stokes equations; thus, I had the project of writing a book on everything I did not know
about Navier–Stokes. Such a project turned out to be unrealistic, as the topic is immense. I
then came back to my field of expertise, the sole study of the incompressible deterministic
Navier–Stokes equations in the case of a fluid filling the whole space. Even within such a
restricted frame, I discovered that I had a lot to learn, so the content of the book is much
larger than foreseen, and the resulting book has very few parts in common with the old one.
The references in the Bibliography are essentially either very old (historical references to
the (pre)history of the Navier–Stokes equations) or extremely recent, testifying that those
equations are a very active field in contemporary mathematical research.
It might be interesting to list what is not included in the book, so that a reader whose
interest into fluid mechanics is piqued would be tempted to get further information elsewhere
on further topics:
ˆ the book is not a treatise on hydraulics, nor on whatever application of fluid mechanics.
There will be no pipes nor vessels, as the fluid is assumed to fill the whole space; thus,
no physically reasonable modelization of the real world is proposed hereby. In the
same way, it is not a treatise on aerodynamics; no drag forces are investigated, and
no body is immersed in the fluid at any page of the book.
ˆ the book is not a treatise on turbulence. No modelization is commented, no statistical
study is presented and no stochastical theory is introduced. Stochastic fluid mechanics
is a very active research field, as there is some hope that the possible deterministic
singular solutions are rare and unstable enough to be ignored by a random description
of the equations.
Preface to the First Edition
xiii
ˆ the book does not study general fluids that may have various behaviors (compressibility, inhomogeneity) and may be subject to various forces, whether external (as the
Coriolis force) or internal (due to conductivity or thermal effects). We shall stick to
the very simplified frame of a Newtonian, incompressible, homogeneous isotropic fluid
subject only to the internal forces of friction.
ˆ the book does not study fluids in a bounded domain, or an exterior domain. Thus,
we do not have to consider many delicate problems in handling the pressure at the
boundary, or to deal with the vorticity generated at the boundary.
ˆ the book does not address issues from the computational fluid dynamics. No Galerkin
bases are constructed in any place to provide the basis of algorithms. Some notions
derived from the Large Eddy Simulation (as the α-models) will be studied but with
no aim at practical computations.
Now that we know what this book is not, we may serenely comment on what this book
is actually about, and in which way it differs from the 2002 book:
ˆ One of the main differences between the two books is the systematic inclusion of forces
in the theorems. Sometimes, it is a mere adaptation of the same theorems on equations
without forcing terms. But very often, the choice of the hypotheses on the forces is
not obvious. This is especially clear for the chapters on mild solutions: if you want to
get a solution in L4t L6 for instance, then a natural choice for the space where to pick
−1/2
the initial value is the Besov space Ḃ6,4 ; the choice of the space where to pick the
force f⃗ is more complex.
ˆ An obvious benefit of this inclusion of forcing terms in the equations is, of course,
that one may consider problems that are linked to the behavior of the forces: if the
force is stationary or time-periodic, will the solutions be asymptotically stationary or
time-periodic?
ˆ Another difference between the two books is the stress put in this new book on Morrey
spaces, whereas the old book insisted on Besov spaces. In both cases, the idea is
to deal with critical hypotheses on the data, where the criticality is intended with
respect to the scaling properties of the equations. Littlewood–Paley analysis which
was extensively used in the old book is very often replaced by Hedberg’s inequality
in the new book. In order to see the usefulness of such an approach, the reader may
compare the proofs of Theorems 10.2 and 10.3 in the book with Gallagher’s original
proofs [195, 197], which used the Littlewood–Paley decomposition, or the proof of
global existence of helical solutions (Theorem 10.7) with the same proof by Titi and
co-workers in [347].
ˆ The stress on Morrey spaces and on the related theory of singular multipliers has
given rise to a wholly new chapter on parabolic Morrey spaces and capacitary theory
applied to the existence of mild solutions (Chapter 5). (Let us remark, however, that
parabolic Morrey spaces have already been used by several authors in the study of
partial regularity results for the Navier–Stokes equations; see for instance, the papers
by O’Leary [379], Ladyzhenskaya and Seregin [297] or Kukavica [287])
ˆ Younger readers are often puzzled by the way the Navier–Stokes equations are stated
and solved. I found interesting to include chapters that would introduce the equations in a simple context. Thus, a chapter is devoted to the physical meaning of each
term involved in the equation (viscosity, density, velocity, vorticity, pressure, stress
xiv
Preface to the First Edition
tensor, . . . ). Another chapter is devoted to the history of the equations up to Leray;
I hope those historical indications will help the reader to better understand the inner
logic of this theory. A chapter is devoted to a classical resolution of the equations,
with tools pertaining to the nineteenth century, before the birth of functional analysis. This chapter thus introduces basic notions, as the Green function, the Leray
projection operator, the heat kernel, the Oseen tensor and so on.
ˆ In relation with Morrey spaces, special emphasis will be put on scaling properties
of the Navier–Stokes equations. A related important result will be Jia and Šverak’s
theorem on the existence of self-similar solutions for large homogeneous initial values
[245].
ˆ Another important result related to scaling properties is the partial regularity result of
Caffarelli, Kohn and Nirenberg for suitable solutions [74]. We shall discuss some recent
variants of this theorem, and explore in a systematical way the various approximating
processes of weak solutions, and show how they lead to those suitable solutions.
Summary of the Book
The book is divided into 19 chapters. We give here a brief presentation of those chapters.
1. Presentation of the Clay Millenium Prizes.
This chapter gives a loose presentation of the Clay Millenium Prizes, based on the
book published by the Clay Mathematics Institute [91]. We present more especially
the formulation of the Clay Millennium Problem on Navier–Stokes equations, as given
by Fefferman [171].
2. The physical meaning of the Navier–Stokes equations.
This chapter gives a short presentation of each term in the Navier–Stokes equations
in order to explain how and why they are introduced in fluid mechanics. In the case
of an isotropic Newtonian fluid, and in the absence of other internal forces than the
forces exerted by the hydrostatic pressure or the friction due to viscosity, we find the
equations of hydrodynamics:
D
ρ + ρ div ⃗u = 0
(0.1)
Dt
and
D
⃗ + µ∆⃗u + λ∇(div
⃗
ρ ⃗u = −∇p
⃗u) + f⃗ext
(0.2)
Dt
Equation (0.1) describes the mass conservation: ρ is the density of the fluid, ⃗u = ⃗u(t, x)
D
the velocity of the parcel of fluid that occupies at time t the position x and Dt
is the
material derivative
3
X
D
h=h+
ui ∂i h.
Dt
i=1
Equation (0.2) expresses Newton’s second law on the momentum balance in the presence of forces. The forces are induced by the pressure p (the force density is given
⃗
by −∇p),
by viscosity (the force density is given by the divergence of the viscous
stress tensor div T; in the case of a Newtonian fluid, the tensor T depends linearly on
the strain tensor ϵ described by Cauchy – ϵi,j = 21 (∂i uj + ∂j ui ) – ; more precisely,
Preface to the First Edition
xv
T = 2µϵ + η tr(ϵ) I3 , where µ is the dynamical viscosity of the fluid and η [= λ − µ]
is the volume viscosity of the fluid), and by external forces whose density f⃗ext is
assumed to be independent of ⃗u.
In the case of a Newtonian, isotropic, homogeneous and incompressible fluid, those
equations of hydrodynamics are transformed into the Navier–Stokes equations.
ρ is then constant (it does not depend neither on time t nor on position x); one
then divides the equations by ρ, and replaces the force density f⃗ext with a reduced
density f⃗r = ρ1 f⃗ext , the pressure p with a reduced pressure pr = ρ1 p (the kinematic
pressure), and the dynamical viscosity µ by the kinematic viscosity. We then
have:
The Navier–Stokes equations
⃗ u = −∇p
⃗ r + ν∆⃗u + f⃗r
∂t ⃗u + (⃗u.∇)⃗
(0.3)
div ⃗u = 0
(0.4)
3. History of the equation.
In this chapter, we give a short history of the Navier–Stokes equations, based on
Darrigol’s recent book Worlds of flow [145] and on the classical papers of Truesdell [479, 480, 481]. We recall the first works on hydrodynamics by Bernoulli [37],
D’Alembert [137, 138, 139] and Euler [167]. Then we describe how the Navier–Stokes
equations were introduced by Navier [373], Cauchy [96], Poisson [403], Saint-Venant
[418] and Stokes [451].
Then we show how Lorentz computed the Green function for the steady Stokes problem [342], paving the way to modern Navier–Stokes theory: Oseen [384, 385] extended
the work of Lorentz to the case of evolutionary Stokes equations and then to the
Navier–Stokes equations, and proved local-in-time existence of classical solutions;
Leray [328] extended Oseen’s work to the existence of global-in-time weak solutions.
The formulas derived by Lorentz in 1896 and Oseen in 1911 for hydrodynamic potentials were explicitly known only for very simple domains and not available for more
complex domains. Hopf [238] in 1951 and Ladyzhenskaya in 1957 [262] then used a
Faedo–Galerkin method to deal with the case of a general domain, where no explicit
formula could be used.
4. Classical solutions.
In this chapter, we solve the Navier–Stokes equations, using only classical tools of
differential calculus, as they were used in the end of the 19th century or the beginning
of the 20th century. More precisely, we stick to the spirit of Oseen’s paper, which was
published in 1911 [384] (a similar treatment can be found in a 1966 paper of Knightly
[263]).
This chapter introduces the main equations and fundamental solutions used in the
book: the heat equation and the heat kernel, the Poisson equation and the Green
function, the Helmholtz decomposition and the Leray projection operator, the Stokes
problem and the Oseen tensor.
xvi
Preface to the First Edition
5. A capacitary approach of the Navier–Stokes integral equations.
In this chapter, we use a new method for solving the Navier–Stokes equations: we
re-write the problem as a quadratic integral equation, and we solve it by the classical
Picard iterative scheme. The novelty is the fact that we prove convergence by use
of a dominating function that solves a quadratic integral problem with a positive
symmetric kernel. We may then use a 1999 result of Kalton and Verbitsky [250] to
describe those functions.
This chapter introduces important tools for the study of parabolic equations: parabolic
Morrey spaces, parabolic Riesz potentials and Hedberg’s inequality. New functional
spaces are introduced for the study of the Navier–Stokes equations, such as the space
1/2,1
1/2
of pointwise multipliers between the parabolic Sobolev space Ḣt,x = L2t Ḣx1 ∩L2x Ḣt
2
and Lt,x , or the Triebel–Lizorkin–Morrey–Campanato spaces.
6. The differential and the integral Navier–Stokes equations.
In this chapter, we discuss the relations between the differential version and the integral version of the Navier–Stokes equations and the way to get rid of the pressure
through the Leray projection operator.
Thus, we discuss various definitions of a solution of the Cauchy initial value problem
for the Navier–Stokes equations:
ˆ very weak solution:
– div ⃗u = 0
– ⃗u is locally square integrable on (0, T ) × R3
– the map t ∈ (0, T ) 7→ ⃗u(t, .) is continuous from (0, T ) to D′ (R3 ) and
limt→0+ ⃗u(t, .) = ⃗u0
– for all φ
⃗ ∈ D((0, T ) × R3 ) with div φ
⃗ = 0, we have
⟨∂t ⃗u − ν∆⃗u + div(⃗u ⊗ ⃗u) − f⃗|⃗
φ⟩D′ ,D = 0
(0.5)
No other regularity is assumed on ⃗u than the continuity of t ∈ [0, T ) 7→ ⃗u ∈
D′ , and no regularity is required on the distribution f⃗. The pressure p is only
defined implicitly by the property (0.5).
ˆ Oseen solution:
under appropriate assumptions on f⃗ and ⃗u, we may get rid of the pressure with
the help of the Leray projection operator P and write
⃗ = (Id − P)(f⃗ − div(⃗u ⊗ ⃗u)) = ∇
⃗ 1 div(f⃗ − div(⃗u ⊗ ⃗u)).
∇p
∆
An Oseen solution ⃗u of the Navier–Stokes equations on (0, T )×R3 , for initial value
⃗u0 and forcing term f⃗ is then a distribution vector field ⃗u(t, x) ∈ D′ ((0, T ) × R3 )
is a very weak solution such that moreover:
– ⃗u ∈ (L2 L2 )uloc
⃗ − f⃗ = P(div(⃗u ⊗ ⃗u) − f⃗)
– div(⃗u ⊗ ⃗u) + ∇p
ˆ mild solutions:
when the Oseen solution ⃗u may be computed by Picard’s iteration method, we
shall speak of mild solution.
ˆ weak solutions:
when Picard’s iterative scheme does not work, the existence of solutions is provided by energy estimates involving the (local) L2 norm of the gradient of ⃗u.
Thus, one is led to consider weak solutions, i.e., Oseen solutions ⃗u such that
Preface to the First Edition
xvii
2
– ⃗u ∈ (L∞
t Lx )uloc
⃗ ⊗ ⃗u ∈ (L2t L2x )uloc
– ∇
associated to an initial value ⃗u0 and to a forcing term f⃗ such that
– ⃗u0 ∈ L2uloc with div ⃗u0 = 0
– f⃗ = div F , where the tensor F is such that F ∈ (L2t L2x )uloc
ˆ suitable solutions:
a suitable solution is a weak solution that satisfies in D′ the local energy inequality
|⃗u|2
|⃗u|2
|⃗u|2
2
⃗
) ≤ ν∆(
) − ν|∇ ⊗ ⃗u| − div (p +
)⃗u + ⃗u.f⃗
(0.6)
∂t (
2
2
2
ˆ Leray weak solution:
a weak solution ⃗u of the Navier–Stokes equations is called a Leray weak solution
if it satisfies the Leray energy inequality:
2
2 1
– ⃗u ∈ L∞
t Lx ∩ Lt Ḣx
– f⃗ ∈ L2 H −1
t
x
– for every t ∈ (0, T ),
∥⃗u(t, .)∥2 ≤ ∥⃗u0 ∥22 − 2ν
Z
0
t
⃗ ⊗ ⃗u∥22 ds + 2
∥∇
Z
t
⟨⃗u|f⃗⟩H 1 ,H −1 ds.
0
7. Mild solutions in Lebesgue or Sobolev spaces.
This chapter is devoted to the classical results of Kato on mild solutions: solutions in
Sobolev spaces H s for s ≥ 1/2 (Fujita and Kato [185]) and in Lebesgue spaces Lp
for p ≥ 3 (Kato [255]).
We also give the proof of uniqueness in C([0, T ), (L3 )3 ) (Furioli, Lemarié-Rieusset, and
Terraneo [187]).
8. Mild solutions in Besov or Morrey spaces.
This chapter is devoted to the study of mild solutions in Besov or Morrey spaces, in
the spirit of the books of Cannone [81] and Lemarié-Rieusset [313].
At the end of the chapter, one also considers the case of Fourier-Herz spaces (as in
the results of Le Jan and Sznitman [305], of Lei and Lin [306] or of Cannone and Wu
[86]).
9. The space BM O−1 and the Koch and Tataru theorem.
This chapter deals with initial values in the largest critical spaces associated to the
−1
Navier–Stokes equations: the space BM O−1 or the Besov space Ḃ∞,∞
Koch and Tataru’s theorem [266] is proved by following the strategy recently given
by Auscher and Frey [11].
Then we consider the Navier–Stokes problem with a null force (f⃗ = 0), and we present
many important results:
−1
ˆ We prove ill-posedness in Ḃ∞,∞
(Bourgain and Pavlović [52]).
ˆ We develop an example of global mild solutions associated to large initial value
given by Chemin and Gallagher [108].
ˆ We prove the stability theorem of Auscher, Dubois and Tchamitchian [10] for
global solutions in BM O−1 .
xviii
Preface to the First Edition
ˆ We present the persistence theory of Furioli, Lemarié-Rieusset, Zahrouni and
Zhioua in [188] for the propagation of initial regularity for mild solutions in
BM O−1 .
ˆ We give a simple proof of time and space analyticity, following Cannon and
Knightly [80].
10. Special examples of solutions.
The symmetries for the Navier–Stokes equations were discussed one century ago by
Wilczynski [502]. In this chapter, we study the solutions that are invariant with respect
to those symmetries:
ˆ Two-and-a-half dimensional flows: ⃗u is invariant under the action of space
translations parallel to the x3 axis. Global existence and regularity are similar
to the case of the 2D Navier–Stokes equations, a case well understood since the
works of Leray [327, 328, 329], and fully developed by Ladyzhenskaya, Lions and
Prodi [293, 339].
ˆ Axisymmetrical solutions: ⃗u is invariant under the action of rotations around
the x3 axis. In the case of axisymmetric flows with no swirl, Ladyzhenskaya
[295], Uchovskii and Yudovich [486] proved global existence under regularity
assumptions on ⃗u0 and f⃗ but without any size requirements on the data.
ˆ Helical solutions: ⃗u is invariant under the action of a one-parameter group of
screw motions
Rθ (x1 , x2 , x3 ) = (x1 cos θ − x2 sin θ, x1 sin θ + x2 cos θ, x3 + αθ)
(where α =
̸ 0 is fixed). Global existence of helical flows has been studied by
Mahalov, Titi and Leibovich [347].
ˆ Brandolese’s symmetrical solutions: ⃗u is invariant under the action of a
finite (non-trivial) group of isometries of R3 (Brandolese [60]).
ˆ Self–similar solutions: ⃗u is invariant under the action of time-space rescalings,
i.e., we consider self–similar solutions:
for every λ > 0,
λ⃗u(λ2 t, λx) = ⃗u(t, x).
Backward self–similar solutions were first considered by Leray [328], but ruled
out by Nečas, Růžička and Šverák [375] and by Tsai [482]. Forward self-similar
mild solutions have been studied by many authors (see, for instance, Cannone,
Meyer and Planchon [83, 84]) in the case of small data; the case of forward selfsimilar weak solutions associated with large data has recently been solved by Jia
and Šverák [245].
ˆ Stationary solutions: ⃗u is invariant under the action of time translations, i.e.,
we consider steady solutions. We present the results of Kozono and Yamazaki
[280], Bjorland, Brandolese, Iftimie and Schonbek [44] and Phan and Phuc [395].
ˆ Landau’s solutions: those special solutions were described first (quite implicitly) by Slezkin [439], then independently by Landau [301] and Squire [447]. They
are self-similar, axisymmetrical with no swirl and steady.
ˆ Time-periodic solutions: ⃗u is invariant under the action of a discrete group
of time translations, i.e., ⃗u is time-periodic. Such solutions have been considered
by Maremonti [349], Kozono and Nakao [272], Yamazaki [509] and Kyed [290].
Preface to the First Edition
xix
11. Blow-up?
This chapter discusses various refinements of Serrin’s criterion [435] for blow-up of the
solutions, including the classical criterion of Beale, Kato and Majda [27]. We present
the extension of the criterion to the setting of Besov spaces (Kozono and Shimada
[274], Chen and Zhang [116], May [354] and Kozono, Ogawa and Taniuchi [273]).
12. Leray’s weak solutions.
Classical theory on existence and weak-strong uniqueness of Leray solutions are presented (Leray [328], Prodi [406], Serrin [435]). Extensions of the Prodi–Serrin criterion to larger classes of solutions (Kozono and Taniuchi [277], Kozono and Sohr [275],
Lemarié-Rieusset [313, 316], Chen, Miao and Zhang [115]) are described. Uniqueness
for “almost strong” solutions is proved (Chemin [106], Lemarié-Rieusset [317], May
[355], Chen, Miao and Zhang [115]). Results on stability of mild solutions through L2
perturbation will be discussed (Karch, Pilarczyk and Schonbek [251]).
13. Partial regularity results for weak solutions.
This chapter is devoted to Serrin’s theory of interior regularity (Serrin [434], Struwe
[455] and Takahashi [459]), and to the celebrated theorem of Caffarelli, Kohn and
Nirenberg [74]. For this theorem, new proofs are provided, including a new result
where the pressure is submitted to (quite) no assumptions at all; this extension is
based on the notion of dissipative solutions introduced by Duchon and Robert [159].
14. A theory of uniformly locally L2 solutions.
This chapter recalls the theory developed in Recent Developments in the Navier–Stokes
Problem about suitable weak solutions with infinite energy.
15. The L3 theory of suitable solutions.
This chapter applies the theory of uniformly locally L2 solutions to the case of solutions
with values in L3 . We prove the recent results of Jia, Rusin and Šverák on minimal data
for blowing-up solutions [417, 244] on the (potential) existence of a minimal-norm
initial value for a blowing-up mild solution to the Navier–Stokes Cauchy problem. We
3
prove as well the L∞
t Lx regularity result of Escauriaza, Seregin and Šverák [163] for
suitable solutions of the Navier–Stokes equations.
16. Self-similarity and the Leray–Schauder principle.
The theory of self-similar solutions has known an impressive advance with the publication in 2014 of a paper of Jia and Šverák [245] establishing the existence of such
solutions for any large homogeneous initial value. This chapter presents this result
with some slight extensions.
17. α-models.
α-models were developed (mainly by Holm) in recent years to provide efficient solvers
for the Reynolds equations associated to turbulent flows. In this chapter, we discuss
the existence of solutions of various α-models and their convergence to weak solutions
of the Navier–Stokes equations when α goes to 0. Those α-models are:
ˆ the Leray–α model (discussed by Cheskidov, Holm, Olson and Titi [119])
ˆ the Navier–Stokes α-model, also known as viscous Camassa–Holm equations
(studied by Chen, Foias, Holm, Olson, Titi and Wynne [117])
ˆ the Clark-α model (studied by Cao, Holm and Titi [87])
ˆ the simplified Bardina model (studied by Cao, Lunasin and Titi [89])
xx
Preface to the First Edition
18. Other approximations of the Navier–Stokes equations.
In this chapter, we discuss various approximations of the Navier–Stokes equations,
including frequency cut-off, hyperviscosity (Beirão da Vega [28], and damping (Cai and
Jiu [75]). We present an important example of Ladyzhenskaya’s model of a Stokesian
fluid with a non-linear damping of the high frequencies through the friction tensor
(Ladyzhenskaya [294]).
19. Artificial compressibility.
In order to simplify the estimation of the pressure in the Navier–Stokes equations,
some authors have presented an approximation of the equations by introducing a
small amount of compressibility on ⃗u in order to turn the Navier–Stokes equations,
⃗ (given by the Leray projection operator, thus
which contains a non-local term ∇p
by a singular integral), into a system of partial differential operators that contain no
non-local terms. We present in this chapter two classical models (given by Temam
[469, 470] and by Višik and Fursikov [189]) and Hachicha’s recent model [225] of a
hyperbolic approximation with finite speed of propagation.
Preface to the Second Edition
Navier–Stokes equations is a very difficult problem and a highly competitive field of research.
This book is the second edition of a second book I wrote on the topic. Between the first
book, released in 2002 [313], and the first edition of this book, released in 2016 [319], the
number of references grew from 116 to 441 (including 180 references published after the
first book was written) and the number of pages grew from approximately 400 pages to 740.
This second edition contains more than 900 pages1 and more than 500 references (including
50 references published after the first edition).
Beyond those quantitative data, we must underline that the field has known major
breakthroughs that we could only allude to (the book would have been very much heavier
if we detailed the lengthy and technical proofs involved). Concerning uniqueness of weak
solutions, Buckmaster and Vicol [71] proved in 2019 non-uniqueness of very weak solutions
in C([0, T ], L2 ) and in 2021, Albritton, Brué and Colombo [5] gave an example of nonuniqueness of suitable Leray solutions to the Navier–Stokes equations on (0, T ) × R3 with
body force f⃗ ∈ L1 ((0, T ), L2 (R3 )) and initial condition ⃗u0 = 0. Another important result was
Tao’s result in 2019 [463] giving an explicit a priori bound for the L3 norm of a blowing up
mild solution, based on Fourier analysis of the solution, while Barker and Prange [19, 20, 21]
developed an analysis of the blow up in spatially localized estimates.
The new (or not so new) results we chose to include in the second edition (besides
correcting many typos and some serious mistakes, such as in pages 361 or 438) are the
following ones:
ˆ In Chapter 4, section 4.11, we added Swann’s beautiful theorem [458] on the existence
time for very regular solutions. We deal again with Swann’s theorem in Chapter 12,
where we added Section 12.8 on Kato’s theorem on the inviscid limit of the Navier–
Stokes equations [254].
ˆ In Chapter 6, we modified the presentation of general weak solutions, replacing the
role of uniform estimates by weighted estimates. In particular, we added a section
(Section 6.3) on Leray’s projection operator in order to take into account the recent
results of by Bradshaw and Tsai [58] and Fernández-Dalgo and Lemarié-Rieusset [174].
ˆ In Chapter 8, we added a small section (Section 8.9) on solutions expressed as a
countable superposition of plane waves, as discussed by Dinaburg and Sinai [153].
ˆ In Chapter 9, we added a variation on the Koch and Tataru theorem (Section 9.2)
based on recent results of Lemarié-Rieusset [323]. We included a proof of Wang’s result
−1
on norm inflation in the critical Besov space Ḃ∞,2
[495] .
ˆ In Chapter 10, Section 10.3, we corrected the statement of Theorem 10.3 on Muckenhoupt weights and we enriched the section with 15 pages devoted to axisymmetric
solutions in Morrey spaces (including the theorem of Gallay and Šverák [203]).
1 The amount of pages given in this preface is estimated in the trim size of the first edition. This size has
been changed for the second edition, so that the final result contains “only” 800 pages, and not 930 pages.
xxi
xxii
Preface to the Second Edition
ˆ In Chapter 11, we added a section (Section 11.8) devoted to the role of the second
eigenvalue of the strain matrix, as discussed by Miller [361].
ˆ In Chapter 12, we corrected the proof of Proposition 12.1 on the strong Leray energy
inequality and we added two small sections, Section 12.7 on non-uniqueness of weak
solutions and Section 12.8 on inviscid limits.
ˆ In Chapter 13, we corrected the proof of Lemma 13.4.
ˆ With 40 new pages, Chapter 14 has been largely extended, with new sections on
weighted Leray solutions (Section 14.6, based on the results of Fernández-Dalgo and
Lemarié-Rieusset [173] and of Bradshaw, Kukavica and Tsai [56]), global existence
for local Leray solution (Section 14.5, which generalizes results presented in [313]),
Barker’s theorem on weak-strong uniqueness (Section 14.8, based on the papers of
Barker [18], and Lemarié-Rieusset [324]) and a final section (Section 14.9) where
we present a theorem (Theorem 19.2) loosely based on the theory of homogeneous
statistical solutions (Višik and Fursikov [190], Basson [23]).
ˆ In Chapter 15, we added a small section (Section 15.5) on the recent results on the
L3 norm of blowing up solutions (Tao [463], Barker and Prange [19, 20, 21])
ˆ With 40 new pages, Chapter 16 has been largely extended. The section on existence
of steady solution has been completed with a section (Section 16.3) on the Liouville
problem for steady solutions (based mainly on Seregin’s work [430]). New examples of
application of the Leray–Schauder principle have been given: existence of discretely
self-similar solutions for large data (Section 16.9, following Chae and Wolf [100], Bradshaw and Tsai [57], Fernández-Dalgo and Lemarié-Rieusset [173]), existence of timeperiodic weak solutions for large data (Section 16.10, following Kyed [290]).
ˆ In Chapter 17, we completed Theorem 17.2 on the Navier–Stokes-α model.
ˆ In Chapter 19, we corrected the proof of Theorem 19.1.
ˆ In conclusion (Chapter 20), we added a final section (Section 20.5) with a small list
of open questions.
Chapter 1
Presentation of the Clay Millennium Prizes
1.1
Regularity of the Three-Dimensional Fluid Flows: A
Mathematical Challenge for the 21st Century
Modern mathematical hydrodynamics was born in the 18th century. In 1750, Euler [166]
expressed the conviction that the mechanics of continuous media could be reduced to the
application of Newton’s law to the infinitely small elements constituting the continuum. In
1755, Euler [167] presented a memoir (published in 1757) entitled Principes généraux du
mouvement des fluides [General principles concerning the motion of fluids], where he could
derive the equations for a general fluid, compressible or not, in the presence of arbitrary
external forces. The Euler equations use Newton’s law when the fluid element is submitted
only to the external forces and to the pressure exerted by the other elements.
However successful Euler had been in applying his program, his results suffered from
two severe limitations. The first one was underlined by Euler himself in his conclusion:
Cependant tout ce que la Théorie des Fluides renferme est contenu dans ces
deux équations, de sorte que ce ne sont pas les principes de Méchanique qui
nous manquent dans la poursuite de ces recherches, mais uniquement l’Analyse,
qui n’est pas encore assés cultivée, pour ce dessein1
As a matter of fact, the complete resolution of the Euler equations is still an open problem
nowadays.
The second limitation is even more severe. In Euler’s equations, the internal forces (i.e.,
the forces exerted on parts of the fluid by the other parts of the fluid) are described only
in terms of the pressure. If we consider a fluid element as a little cube, the pressure exerts
a force on the faces of the cube in the normal direction to the faces. But, due to the
fact that the other elements of fluids have a different velocity, there is another force (the
friction) exerted on the fluid element, in directions that are tangential to the faces. This
shear stress has been described in the 19th century as the effect of viscosity. Viscous fluids
behave drastically differently from the inviscid ones. Von Neumann coined the term “dry
water”2 to underline the inefficiency of modelization that would neglect viscosity forces, as
commented by Feynman [175]:
When we drop the viscosity term, we will be making an approximation which
describes some ideal stuff rather than real water. John von Neumann was well
aware of the tremendous difference between what happens when you don’t have
1 Everything that is held within the Theory of Fluids is contained in those two equations, so that it is
not the principles of Mechanics that are lacking for the continuation of our research, but only the Analysis,
which is still not developed enough for that purpose.
2 Ironically enough, dry water exists. It was patented in 1968. Dry water, in this acceptance of the term,
is a water-air emulsion in which tiny water droplets are surrounded by a sandy silica coating. The silica
coating prevents the water droplets from combining and turning back into a bulk liquid.
DOI: 10.1201/9781003042594-1
1
2
The Navier–Stokes Problem in the 21st Century (2nd edition)
the viscous terms and when you do; and he was also aware that, during most of
the development of hydrodynamics until about 1900, almost the main interest was
in solving beautiful mathematical problems with this approximation which had
almost nothing to do with real fluids. He characterized the theorist who made
such analyses as a man who studied “dry water.” Such analyses leave out an
essential property of the fluid.
Taking into account the viscosity led to the Navier–Stokes equations. Those equations were first introduced by Navier in 1822 [373]. Though they have been rediscovered by
many authors, such as Cauchy, Poisson or de Saint–Venant, they remained quite controversial until they were settled on a firmer basis by Stokes in 1845 [451] (see the paper of
Darrigol [144] on the “five births” of the Navier–Stokes equations). They still had to wait
dozens of years before being definitely adopted by physicists, after that they were proven
to be in accordance with Maxwell’s kinetic theory of gases. Again, the complete resolution
of the Navier–Stokes equations is still an open problem nowadays. Given some initial value
that is smooth and well localized, we are not able to prove the existence of a global-in-time
solution (except when the initial value is small enough). Local existence was rigorously established by Oseen [385] and his co-workers at the beginning of the 20th century. Then, in
1934, Leray [328] proved that those local-in-time solutions could be prolongated in globalin-time weak solutions that might be no longer smooth, so that the derivatives are to be
taken in some weak sense. Thereafter, very few further results could be obtained for the
3D fluids, and the question of global existence of classical solutions remained an important
challenge.
A major issue in the theory of fluid mechanics is the understanding of turbulence.
Turbulence occurs when the motion of the fluid becomes disordered. The flow then turns
out to be highly irregular and quite unpredictable. Reynolds [410] studied the instability
of steady flows and gave experimental evidence of the transition from laminar flows (i.e.,
regular flows) to turbulence through the increase in the velocities. The Navier–Stokes equations are believed to be a good frame to establish transition to turbulence in a rigorous
mathematical setting.
The study of three-dimensional fluids remains an important issue nowadays. It is considered as an important challenge for the 21st century. Such challenges have been presented
by the International Mathematical Union (IMU) at the occasion of the World Mathematical
Year 2000. More precisely, in 1992 in Rio de Janeiro, IMU, with support of the UNESCO,
declared the year 2000 to be the World Mathematical Year. The purpose was to highlight
mathematics for a larger audience, in an effort of world-wide promotion. The Declaration
of Rio set three aims:
ˆ The great challenges of the 21st century
ˆ Mathematics, as a key for development
ˆ The image of mathematics
In August 2000, the American Mathematical Society held an extraordinary meeting on
the UCLA campus under the title “Mathematical Challenges of the 21st Century.” In the
editorial of the Notices of the AMS [69], Browder, the President of the AMS, explained the
aims of the meeting:
1. To exhibit the vitality of mathematical research and to indicate some of its potential
major growing points: these include some of the major classical problems (the Riemann
Hypothesis, the Poincaré conjecture, the regularity of three-dimensional fluid flows) as
well as some of the recently developed major research programs like those associated
with the names of Langlands and Thurston.
Presentation of the Clay Millennium Prizes
3
2. To point up the growing connections between the frontiers of research in the mathematical sciences and cutting-edge developments in such areas as physics, biology,
computational science, and finance.
Browder was not the only one to promote the issue of the regularity of three-dimensional
fluid flows to such a prestigious neighborhood as the one of the Riemann Hypothesis and
of the Poincaré conjecture. In 1997 in a conference at the Fields Institute at Toronto, the
1966 Fields medalist Smale [442] gave a list of problems he selected as “likely to have great
importance for mathematics and its development in the next century.” That list was an
answer to an invitation of Arnold, on behalf of the International Mathematical Union, to
describe some great problems for the 21st century, in a reminiscent way of Hilbert who
described in the 1900 meeting of the IMU in Paris a list of twenty-three great problems for
the 20th century. Smale listed eighteen problems, including what he considered as the three
greatest open problems of mathematics: the Riemann Hypothesis, the Poincaré conjecture
and the “Does P=NP?” problem. Smale’s fifteenth problem is the question about global
existence and regularity for the three-dimensional Navier–Stokes equations, “perhaps the
most celebrated problem in partial differential equations,” whose solution “might well be a
fundamental step toward the very big problem of understanding turbulence.”
The most spectacular effort to promote mathematics, however, had been the establishment of the Millennium Prizes by the Clay Mathematics Institute. Seven $1 million prizes
were established to reward the solution of seven classical mathematical problems that have
resisted solutions for many years. Once more, the Navier–Stokes equations were selected, as
well as other great problems such as the Riemann Hypothesis, the Poincaré conjecture and
the “Does P=NP?” problem.
1.2
The Clay Millennium Prizes
The Clay Mathematics Institute is a non-profit foundation. It was established in
1998 by the American businessman Landon T. Clay. As indicated on its Web site
(http://www.claymath.org/), the primary objectives and purposes of The Clay Mathematics Institute are:
ˆ to increase and disseminate mathematical knowledge,
ˆ to educate mathematicians and other scientists about new discoveries in the field of
mathematics,
ˆ to encourage gifted students to pursue mathematical careers,
ˆ and to recognize extraordinary achievements and advances in mathematical research.
According to this mission, the Institute offers postdoctoral grants, funds summer schools
and conferences, co-publishes with the American Mathematical Society monographs devoted
to the “exposition of recent developments, both in emerging areas and in older subjects
transformed by new insights or unifying ideas,” and has a program of providing to a large
readership digital facsimiles of major mathematical works from the past.
The Institute is best known for establishing the Millennium Prize Problems in 2000. “The
Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second Millennium; to elevate in the consciousness
of the general public the fact that in mathematics, the frontier is still open and abounds
4
The Navier–Stokes Problem in the 21st Century (2nd edition)
in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of
historical magnitude.”
A committee of experts (Michael Atiyah, Andrew Wiles, John Tate, Arthur Jaffe, Alain
Connes, Edward Witten) selected seven problems that were officially presented at a conference in the Collège de France (Paris) in May 2000. For each problem, the first person to
solve it would be awarded $1,000,000 by the CMI. The problems to be solved are:
P versus NP
The Hodge Conjecture
The Poincaré Conjecture
The Riemann Hypothesis
Yang–Mills Existence and Mass Gap
Navier–Stokes Existence and Smoothness
The Birch and Swinnerton-Dyer Conjecture
The announcement of the prizes drew the attention of media to this area of science. As
early as 2003, Perelman solved the Poincaré conjecture; he was awarded the Fields Medal in
2006 and the Clay Millennium Prize in 2010, and declined both. This had a rather large echo
in the media. If most journals focused on Perelman’s personality and his alleged eccentricity,
many papers tried and explained the Poincaré conjecture to their laymen readers.
In 2006, Science [345] labeled Perelman’s achievements “Breakthrough of the year,” a
distinction that Science had never given to any mathematical result. In the same year,
Smith posted a paper on arXiv which was supposed to have solved the question about the
Navier–Stokes equations. Smith withdrew her paper within two weeks, after a serious flaw
was found in her proof, but Nature [235] had already given a large echo to Smith’s paper.
This buzz in the media has been severely criticized by mathematicians. Vershik [489]
expressed his doubts about the utility of the Millennium Prizes.
Around the year 2000 /. . . / I met my old friend Arthur Jaffe, who was then
president of the Clay Mathematics Institute. I asked him: “What is this being
done for?” At the time I felt that the assignment of huge (million dollar) prizes
was more in keeping with the style of show business, aiming at drawing attention
to something or somebody at any price, whereas scientific life should avoid cheap
popularization.
/. . . / Arthur answered me decidedly and professionally: ‘You understand nothing about the American way of life. If a politician, a businessman, a housewife
will see that one can earn a million by doing mathematics, they will not discourage their children from choosing that profession, will not insist on their doing
medicine, law, or going in for some lucrative activity. And other rich philanthropists will be more likely to give money to mathematics, which is in such need
of it’.
One interesting remark of Vershik about the Prizes states:
I would also like to note that the stir created around the seven “Millennium
prizes” creates the wrong impression in society about the work of mathematicians, supporting the hackneyed notion that it consists only in solving concrete
problems. You don’t have to be an expert to understand how misleading that
notion is. The discovery of new domains and relationships between different
branches of mathematics, the setting of new problems, the development and perfection of the mathematical apparatus, and so on, are no less important and
difficult parts of our science, without which it cannot exist.
Presentation of the Clay Millennium Prizes
5
In The Millennium Prize Problems [91], a collective book edited by the Clay Mathematics Institute, Gray [216] makes a presentation of the history of prizes and challenges
in mathematics, with a special emphasis on the 18th and the 19th centuries and with a
section devoted to the Hilbert problems. The aim of the chapter is, of course, to illustrate
the “tradition of stimulating problems that the Clay Mathematics Institute has also sought
to promote,” but it indicates as well that financial motivation could lead to some disaster,
as in the case of the Wolfskehl prize, offered for a solution of Fermat’s Last Theorem:
From some perspectives, such as generating enthusiasm for mathematics, the
prize was a great success; from others, such as the advancement of knowledge, it
was a complete disaster. In the first year [1907] no fewer than 621 solutions were
submitted, and over the years more than 5,000 came in. These had to be read,
the errors spotted, and the authors informed, who often replied with attempts to
fix their ‘proofs’.
This Millennium Prize Problems book [91] presents the official description of each of the
seven problems, given by eminent specialists of the field (while a book by Devlin [152] gives
a loose description of the problems at stake, just sketching the general background of each
problem in an effort of popularization toward the largest audience):
ˆ The Birch and Swinnerton-Dyer Conjecture: the official description has been given
by Wiles [503], the famous number theorist who proved Fermat’s last theorem. This
conjecture is a problem in Diophantine analysis. The question is to determine whether
an algebraic equation f (x, y) = 0, where f is a polynomial with rational coefficients
has rational solutions and how many. The answer depends on the genus of the curve
defined by the equation f (x, y) = 0. When this genus is equal to 0, Hilbert and
Hurwitz proved in 1890 that either there are no solutions or there are infinitely many.
When this genus is no less than 2, then Faltings proved in 1980 that there are only
finitely many solutions. The question is more complex when the genus is equal to 1.
In that case, one can take a model of the curve of the form y 2 = x3 + ax + b with
integer coefficients a, b ∈ Z and a non-null discriminant ∆ = 4a3 + 27b2 . The Birch
and Swinnerton-Dyer Conjecture relates the number of rational points to the numbers
Np of solutions of the same equation in the field Z/p for every prime p, which is not
a divisor of 2∆.
In 1965, Birch and Swinnerton-Dyer based their conjecture on computer simulations.
If the curve C has a rational point, then there is a natural law group on the rational
points of the curve that makes the set of rational points an Abelian group C(Q). In
1922, Mordell proved that the group is finitely generated, thus isomorphic to some
Zr ⊗ F , where F is finite; r is called the rank of the group C(Q). The Hasse L-function
of the curve C is defined for ℜs > 3/2 as
L(C, s) =
p
Y
1
prime ;
p−Np
ps
2∆∈pN
/
(1 −
+
1
p2s−1 )
A by-product of the proof by Wiles and Taylor of Fermat’s Last Theorem gives that
the Hasse function may be continued as a holomorphic function on the plane. The
official Clay Millennium Problem is then to solve the following conjecture:
6
The Navier–Stokes Problem in the 21st Century (2nd edition)
Conjecture (Birch and Swinnerton-Dyer)
The Taylor expansion of L(C, s) at s = 1 has the form
L(C, s) = c(s − 1)r + higher order terms
with c ̸= 0 and r = rank (C(Q)).
Note that, if this conjecture is true, the equation has infinitely many rational solutions
if and only if L(C, 1) = 0.
ˆ The Hodge Conjecture: the official description has been given by the 1978 Fields
medalist Deligne [151]. The official Clay Millennium Problem is to solve the following
conjecture:
Hodge Conjecture
On a projective non-singular algebraic variety over C, any Hodge class
is a rational linear combination of classes cl(Z) of algebraic cycles.
The conjecture was presented by Hodge at the International Congress of Mathematicians in 1950. This conjecture concerns harmonic differential forms on a projective
non-singular complex algebraic variety. It states that every rational harmonic (p, p)form on the variety is (modulo exact forms) a rational linear combination of algebraic
cycles, i.e., of classes induced by algebraic subvarieties of complex co-dimension p.
ˆ Navier–Stokes Existence and Smoothness: the official description has been given by
the 1978 Fields medalist Fefferman [171]. The official Clay Millennium Problem is
then the following one:
Navier–Stokes existence and smoothness
We ask for a proof of one of the four following statements:
A) Existence and smoothness of Navier–Stokes solutions on R3
B) Existence and smoothness of Navier–Stokes solutions on R3 /Z3
C) Breakdown of Navier–Stokes solutions on R3
D) Breakdown of Navier–Stokes solutions on R3 /Z3
The problem concerns the initial value problem for a fluid that fills the whole space
(so that there is no boundary problem) and which is viscous, homogeneous and incompressible. The question raised is whether, for a smooth initial value, the Navier–Stokes
problem has a (unique) global smooth solution or whether one can exhibit an example
of initial value for which the solution blows up in finite time. This question appears in
the work of Leray who proved in 1934 global existence of weak solutions which may
be non-unique and irregular. We shall discuss in Section 1.3 the terms of this problem
to a greater extent.
Presentation of the Clay Millennium Prizes
7
ˆ P versus NP: the official description has been given by the 1982 Turing Award winner
Cook [130]. The official Clay Millennium Problem has a very simple statement:
Problem statement
Does P=NP?
The “Does P= NP?” problem appeared in 1971–1973 in the independent works of
Cook, Karp and Levin in complexity theory. The class P is the class of decision
problems solvable by some algorithm within a number of steps bounded by some fixed
polynomial in the length of the input, while the class NP is the class of problems
whose proposed solutions can be checked in polynomial time.
ˆ The Poincaré Conjecture: the official description has been given by the 1962 Fields
medalist and 2011 Abel Prize winner Milnor [362]. While the classification of all
possible orientable compact two-dimensional surfaces has been well understood in the
19th century, the problem turned out to be much more complex in higher dimensions.
In 1904, Poincaré formulated a conjecture that remained unsolved all along the 20th
century. The Clay Millennium Problem was to prove the Poincaré Conjecture:
Question [the Poincaré Conjecture]
If a compact three-dimensional manifold M 3 has the property that every simple
closed curve within the manifold can be deformed continuously to a point, does it
follow that M 3 is homeomorphic to the sphere S 3 ? [The manifold M 3 is assumed
to be connected and with no border.]
The analogue of this conjecture had been proved in higher dimension, by Smale in 1961
for dimensions greater than four, then by Freedman in 1982 for the four-dimensional
case. The conjecture was finally proven by Perelman in 2002–2003. On March 18, 2010,
Carlson, on behalf of the Clay Mathematics Institute, announced that the conjecture
was proved and the prize awarded [90].
ˆ The Riemann Hypothesis: the official description has been given by the 1974 Fields
medalist Bombieri [50]. This is a famous problem in mathematics history. It deals with
the Riemann zeta function ζ(s); this function is defined for ℜ(s) > 1 as the series
ζ(s) =
∞
X
1
s
n
n=1
and then is prolongated by analytic continuation to a holomorphic function of s ̸= 1
(with a simple pole at s = 1). It is easy to see that the negative even numbers
−2, −4, −6, . . . are zeroes of the function ζ. They are called trivial zeroes. Riemann
conjectured in 1859 that all the other zeroes should satisfy ℜ(s) = 1/2. The Clay
Millennium Problem is to prove the Riemann Hypothesis:
8
The Navier–Stokes Problem in the 21st Century (2nd edition)
Riemann Hypothesis
The non-trivial zeroes of ζ(s) have real part equal to 12 .
This conjecture is important to our knowledge of prime numbers but has other farreaching consequences as evoked by Bombieri in his presentation of the problem. To
prove the Riemann Hypothesis was already one of the twenty-three problems Hilbert
had listed for the 20th century.
ˆ Quantum Yang–Mills theory: the official description has been given by the specialist
of constructive quantum field theory and founding President of the Clay Mathematics Institute Jaffe and the 1990 Fields medalist Witten [243]. The problem concerns
quantum field theory. In 1954, Yang and Mills introduced a non-Abelian gauge theory to modelize quantum electrodynamics and obtained a non-linear generalization
of Maxwell’s equations. The problem at stake now is to develop a gauge theory for
the modelization of weak interactions and strong interactions. Those forces involve
massive particles and require new tools, since the model of Yang and Mills dealt
with long-range fields describing massless particles. Nowadays, we still are lacking
a mathematically complete example of a quantum gauge theory in four-dimensional
space-time. The official Clay Millennium Problem is thus the following one:
Yang–Mills Existence and Mass Gap
Prove that for any compact simple gauge group G, a non-trivial quantum Yang–
Mills theory exists on R4 and has a positive mass gap ∆ > 0. Existence includes
establishing axiomatic properties at least as strong as those cited in [387, 388, 453].
1.3
The Clay Millennium Prize for the Navier–Stokes Equations
We now turn to the precise formulation of the Clay Millennium Problem on Navier–
Stokes equations. The Navier–Stokes equations considered in this formulation are the
following partial differential equations:
∂t ⃗u(t, x) = ν∆⃗u −
3
X
⃗ + f⃗
ui ∂i ⃗u − ∇p
for t > 0 and x ∈ R3
(1.1)
i=1
div ⃗u =
3
X
∂i ui = 0
(1.2)
i=1
with initial condition
⃗u(0, x) = ⃗u0 (x)
where ∂i stands for
∂
∂xi ,
∂t for
∂
∂t
for x ∈ R3
 
∂1
⃗ for the gradient operator ∇
⃗ = ∂2 .
and ∇
∂3
(1.3)
Presentation of the Clay Millennium Prizes
9


u1 (t, x)
The unknown are ⃗u(t, x) = u2 (t, x) and p(t, x). The equations describe the motion
u3 (t, x)
of a fluid filling the whole space R3 . The vector ⃗u is the velocity of the fluid element that,
at time t, occupies the position x. The scalar quantity p measures the pressure exerted on
the fluid element.
The fluid is assumed to be homogeneous and incompressible. Incompressibility is expressed by Equation (1.2). The constant density ρ is taken equal to 1. The fluid is assumed
to be viscous and Newtonian, i.e., the friction of fluid elements of different velocities generates a force of the form ν∆⃗u, where ν is a positive constant (the viscosity) and ∆ is the
P3
Laplacian ∆ = i=1 ∂i2 . Finally, the fluid may be submitted to external forces; the force
density is expressed by the vector f⃗(t, x). The forces expressed by f⃗ are assumed to be
independent from the velocity field ⃗u (for instance, the problem does not concern fluids in
⃗ ∧ ⃗u).
a rotating frame, submitted to the Coriolis force Ω
Solving Equations (1.1), (1.2) and (1.3) is a Cauchy initial value problem. Given the
initial state ⃗u0 at time t = 0 and the force f⃗ for t > 0, one wants to determine the evolution
of the system for t > 0. For the Clay Millennium Problem, one assumes that the initial data
⃗u0 and the force f⃗ are given by smooth and well-localized functions: ⃗u0 is a C ∞ divergencefree vector field on R3 such that, for all α ∈ N3 and all K > 0,
|∂ α ⃗u0 (x)| ≤ Cα,K (1 + |x|)−K
on R3
(1.4)
 
x1
p
(where | x2  | = x21 + x22 + x23 ); similarly, f⃗ is C ∞ on [0, +∞) × R3 and satisfies for all
x3
α ∈ N3 , all m ∈ N and all K > 0
|∂xα ∂tm f⃗(t, x)| ≤ Cα,m,K (1 + t + |x|)−K
on [0, +∞) × R3
(1.5)
Admissible solutions are smooth functions ⃗u and p with bounded energy:
p, ⃗u ∈ C ∞ ([0, +∞) × R3 )
Z
|⃗u(t, x)|2 dx < C
for all t > 0
(bounded energy)
(1.6)
(1.7)
R3
There is no need to specify the value of p at time t = 0. Indeed, we have:
⃗
∆⃗u = − curl(curl ⃗u) + ∇(div
⃗u) = − curl(curl ⃗u)
and, since the size of ⃗u at x = ∞ is limited by the condition of integrability (1.7), ⃗u is
uniquely determined through its curl. Let


∂2 u3 − ∂3 u2
⃗ ∧ ⃗u = ∂3 u1 − ∂1 u3  ;
ω
⃗ = curl ⃗u = ∇
∂1 u2 − ∂2 u1
ω is called the vorticity of the fluid.
10
The Navier–Stokes Problem in the 21st Century (2nd edition)
Taking the curl of Equation (1.1) gives
3
X
∂t ω
⃗ (t, x) = ν∆⃗
ω − curl(
ui ∂i ⃗u) + curl f⃗
for t > 0 and x ∈ R3
(1.8)
i=1
with initial condition
ω
⃗ (0, x) = curl ⃗u0 (x)
for x ∈ R3 .
(1.9)
Thus, we have a Cauchy initial value problem for ω
⃗ with no dependence on p. When ω
⃗,
and thus ⃗u, is known, p is determined by Equation (1.1).
Now, we can state precisely the Clay Millennium Problem for a viscous fluid (ν > 0):
Navier–Stokes existence and smoothness (whole space)
We ask for a proof of one of the two following statements:
ˆ A) Existence and smoothness of Navier–Stokes solutions on R3 : Let ⃗u0 be
any smooth, divergence-free vector field satisfying (1.4). Take f⃗(t, x) to be
identically zero. Then there exist smooth functions p(t, x), ui (t, x) on R3 ×
[0, ∞) that satisfy (1.1), (1.2), (1.3), (1.6) and (1.7).
ˆ C) Breakdown of Navier–Stokes solutions on R3 : There exist a smooth,
divergence-free vector field ⃗u0 on R3 and a smooth f⃗ on [0, +∞) × R3 satisfying (1.4) and (1.5) for which there exists no solution (p, ⃗u) of (1.1), (1.2),
(1.3), (1.6) and (1.7) on [0, +∞) × R3 .
The Clay Millennium Problem may also be solved on a compact domain instead of the
whole space. In order to avoid boundary terms, the domain is assumed to be the torus
R3 /Z3 , i.e., one deals with periodical functions. Hypotheses (1.4) and (1.5) are replaced
with:
ˆ ⃗u0 and f⃗ are smooth and satisfy
⃗u0 (x + k) = ⃗u0 (x) and f⃗(t, x + k) = f⃗(t, x)
for all k ∈ R3
(1.10)
ˆ for all α ∈ N3 , all m ∈ N and all K > 0
|∂xα ∂tm f⃗(t, x)| ≤ Cα,m,K (1 + t)−K
on [0, +∞) × R3
(1.11)
Admissible solutions are smooth functions ⃗u and p such that:
p, ⃗u ∈ C ∞ ([0, +∞) × R3 )
⃗u(t, x + k) = ⃗u(t, x)
for all k ∈ R3
(1.12)
(1.13)
The statement of the Clay Millennium Problem in the periodical case is then the following one:
Presentation of the Clay Millennium Prizes
11
Navier–Stokes existence and smoothness (torus)
We ask for a proof of one of the two following statements:
ˆ B) Existence and smoothness of Navier–Stokes solutions on R3 /Z3 : Let ⃗u0 be
any smooth, divergence-free vector field satisfying (1.10). Take f⃗(t, x) to be
identically zero. Then there exist smooth functions p(t, x), ui (t, x) on R3 ×
[0, ∞) that satisfy (1.1), (1.2), (1.3), (1.12) and (1.13).
ˆ D) Breakdown of Navier–Stokes solutions on R3 /Z3 : There exist a smooth,
divergence-free vector field ⃗u0 on R3 and a smooth f⃗ on [0, +∞)×R3 satisfying
(1.10) and (1.11) for which there exists no solution (p, ⃗u) of (1.1), (1.2), (1.3),
(1.12) and (1.13) on [0, +∞) × R3 .
Remark: If we want to get rid of the pressure, we may take the equations (1.8) on the
vorticity; but ⃗u has to be uniquely determined from ω
⃗ . In the setting of the whole space,
this is ensured by the spatial decay hypothesis onR⃗u at infinity. In the setting of periodic
solutions, this is ensured by the hypothesis that ⃗u(t, x) dx = 0 (or, equivalently when
f⃗ = 0, that p is periodical). Otherwise, we have the trivial example of non-uniqueness for
⃗u0 = 0 given by ⃗u(t, x) = ⃗v (t) with no dependence on x, associated with the pressure
d
p(t, x) = −⃗x. dt
⃗v (t), where ⃗v is any smooth function with ⃗v (0) = 0. This trivial example can
be turned into an example of blow-up by choosing a blowing up arbitrary function ⃗v (t).
Such examples were discussed by Giga, Inui and Matsui in 1999 when they considered
non-decaying initial data for the Navier–Stokes problem [210], and by Koch, Nadirashvili,
Seregin and Šverák in 2007 [265] when they considered a Liouville theorem for the NavierStokes problem and had to rule out those “parasitic solutions.”
1.4
Boundaries and the Navier–Stokes Clay Millennium Problem
On the website of the Clay Mathematics Institute, the problem is presented in these
words (http://www.claymath.org/Millennium/Navier-Stokes− Equations/):
Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe
that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier–Stokes
equations. Although these equations were written down in the 19th Century, our
understanding of them remains minimal. The challenge is to make substantial
progress toward a mathematical theory which will unlock the secrets hidden in
the Navier–Stokes equations.
Thus, the aim is to eventually understand turbulence. But some severe doubts have been
raised about the model case proposed for the Millennium Prize. For instance, Tartar writes
in the preface of his book [464]:
Reading the text of the conjecture to be solved for winning that particular prize
leaves the impression that the subject was not chosen by people interested in
12
The Navier–Stokes Problem in the 21st Century (2nd edition)
continuum mechanics, as the selected question has almost no physical content.
Invariance by translation or scaling is mentioned, but why is invariance by rotations not pointed out and why is Galilean invariance omitted, as it is the
essential fact which makes the equation introduced by Navier much better than
that introduced by Stokes? If one used the word ‘turbulence’ to make the donator believe that he would be giving one million dollars away for an important
realistic problem in continuum mechanics, why has attention been restricted to
unrealistic domains without boundary (the whole space R3 , or a torus for periodic solutions), as if one did not know that vorticity is created at the boundary
of the domain? The problems seem to have been chosen in the hope that they
will be solved by specialists of harmonic analysis.
However, the question of regularity of the solutions to the Navier–Stokes equations even
when neglecting the influence of the boundary is generally considered as a major issue in
hydrodynamics. For instance, Moffatt says about singularities in fluid dynamics [366]:
Singularities may be associated with the geometry of the fluid boundary or with
some singular feature of the motion of the boundaries; they may arise spontaneously at a free surface as a result of viscous stresses and despite the smoothing
effect of surface tension; or they may conceivably occur at interior points of a
fluid due to unbounded vortex stretching at high (or infinite) Reynolds number. In the last case, we are up against the unsolved and extremely challenging
‘finite-time-singularity’ problem for the Euler and/or Navier–Stokes equations.
The question of existence of finite-time singularities is still open. Solution of
this problem would have far-reaching consequences for our understanding of the
smallest-scale features of turbulent flow.
When dealing with models of fully developed turbulence, one often uses an asymptotic model, which has spatially homogeneous statistical properties; then, one neglects the
boundary effects and works in the setting of the whole space (and in most of the cases in a
space-periodical setting, as it is easier to define and compute the statistical quantities that
are involved in the model).
In this book, we shall stick to the boundary-free3 Navier–Stokes problem, i.e., when the
domain is the whole space R3 .
3 I.e.,
on a domain without boundary (and not on a domain with free boundary).
Chapter 2
The Physical Meaning of the Navier–Stokes
Equations
In this chapter, we try to give a short presentation of each term in the Navier–Stokes
equations, to explain how and why they are introduced in fluid mechanics. A classical
treaty on hydrodynamics from the physicists’ point of view is the book by Landau and
Lifschitz [302], and a rapid introduction can be found in Feynman’s lecture notes [175]. The
mathematicians’ point of view can be found in the classical treaty of Batchelor [25], or for
a modern point of view in the book by Childress [121].
2.1
Frames of References
Fluid theory is based on a continuum hypothesis [25] which states that the macroscopic
behavior of a fluid is the same as if the fluid was perfectly continuous: density, pressure,
temperature, and velocity are taken to be well-defined at infinitely small points and are
assumed to vary continuously from one point to another.
Since the seminal memoir of Euler [167], one describes the laws of fluid mechanics as
applied to fluid parcels, very small volumes δV of fluids that contain many molecules but
whose size is “infinitesimal” with respect to the macroscopic scale. Then the physical properties of the parcels are defined as averages of the associated continuously
R varying quantities:
1
for instance, the temperature θδV of the parcel is given by θδV = |δV
| δV θ(t, x) dx, where
θ is the temperature defined at point x and at time t.
There are then two representations of the fluid motion and of the associated physical
quantities. In the Eulerian reference frame, the reference frame is fixed while the fluid
moves. Thus, the quantities are measured at a position x attached to the fixed frame (one
often speaks of the “laboratory frame”). The velocity ⃗u(t, x) is the velocity at time t of
the fluid parcel that occupies the position x at that very instant t. In the Lagrangian
reference frame, the reference frame is the initial state of the fluid. The quantities are
attached to the parcels as they move.
More precisely, if Xx0 (t) is the position of the parcel at time t whose position at time
0 was x0 , and if Q is some quantity attached to the parcels, we have two descriptions of
the distribution of the values taken by Q at time t: the value Q(t, x) taken at time t for
the parcel which is located at this time at position x, and Qx0 (t) the value taken at time
t for the parcel which was located at time 0 at position x0 . In particular, the velocity field
d
Xx0 (t) = ⃗u(t, Xx0 (t)). This
⃗u(t, x) describes the velocities of the parcels as they move: dt
gives us the link between the variations of Qx0 (t) and those of Q(t, x): from the chain rule
for differentiation, we get
3
X
d
d
Qx0 (t) = ∂t Q(x, t)|x=Xx0 (t) +
∂i Q(x, t)|x=Xx0 (t)
Xx0 ,i (t)
dt
dt
i=1
DOI: 10.1201/9781003042594-2
13
14
The Navier–Stokes Problem in the 21st Century (2nd edition)
d
The quantity dt
Qx0 (t) is called the material derivative of Q and is designed as
have thus obtained the following formula:
D
Dt Q.
We
The material derivative
3
X
D
Q = ∂t Q(x, t) +
ui (t, x)∂i Q(x, t)
Dt
i=1
2.2
(2.1)
The Convection Theorem
If we consider a volume V0 at time 0 filled of fluid parcels, and define Vt the volume
filled by the parcels as they moved, we have
Vt = {y ∈ R3 / y = Xx (t) for some x ∈ V0 }.
The volume element dy of Vt is given by J(t, x) dx, where J is the Jacobian of the transform
x 7→ Xx (t). We have
∂
yj
.
J = det
∂xi
1≤i,j≤3
∂
Let J (t, x) = det ∂x
y
; we have
j
i
1≤i,j≤3
3
∂t
X ∂
∂
∂
∂
∂
yj =
∂t yj =
uj (t, y) =
uj (t, y)
yk
∂xi
∂xi
∂xi
∂yk
∂xi
k=1
and thus
∂
∂
∂
∂
∂
∂
y1 ,
y2 ,
y3 ) + det( y1 , ∂t y2 ,
y3 )
∂x
∂x
∂x
∂x
∂x
∂x
∂
∂
∂
+ det( y1 ,
y2 , ∂t y3 )
∂x
∂x
∂x
3
X
∂
∂
∂
∂
=
u1 (t, y) det( yk ,
y2 ,
y3 )
∂yk
∂x
∂x
∂x
∂t J = det(∂t
k=1
+
3
X
∂
∂
∂
∂
u2 (t, y) det( y1 ,
yk ,
y3 )
∂yk
∂x
∂x
∂x
k=1
+
3
X
∂
∂
∂
∂
u3 (t, y) det( y1 ,
y2 ,
yk )
∂yk
∂x
∂x
∂x
k=1
= div ⃗u(t, y) J
so that, since J(0, x) = 1,
J(t, x) = e
Rt
0
div ⃗
u(s,Xx (s)) ds
(2.2)
The Physical Meaning of the Navier–Stokes Equations
15
Thus, we have seen that the divergence of ⃗u is the quantity that governs the deflation
or the inflation of the volume of Vt .
R
Now, if f (t, x) is a time-dependent field over R3 , we may define F (t) = Vt f (t, y) dy.
We have
Z
F (t) =
f (t, Xx (t))J(t, x) dx
V0
D
We use the fact that ∂t [f (t, Xx (t))] = Dt
f (t, y) and ∂t J(t, x) = div ⃗u(t, y)J(t, x) and
J(t, x) dx = dy to get the convection theorem:
The convection theorem
d
dt
Z
Z
f (t, y) dy =
Vt
Vt
D
f (t, y) + f (t, y) div ⃗u(t, y) dy
Dt
(2.3)
D
⃗ + f div ⃗u = ∂t f + div(f⃗u), and using Ostrogradski’s
Writing Dt
f + f div ⃗u = ∂t f + ⃗u.∇f
formula, we find, writing dσ for the surface element of the boundary ∂Vt and ⃗ν for the
normal at ∂t V pointing outward:
Z
Z
Z
d
f (t, y) dy =
∂t f dy +
f⃗u.⃗ν dσ
(2.4)
dt Vt
Vt
∂Vt
This is a special case of Reynolds’ transport theorem.
2.3
Conservation of Mass
We apply the convection theorem to the mass m of the parcels included
in the volume
R
Vt . If ρ(t, y) is the density at time t and at position y, we have m = Vt ρ(t, y) dy. When
d
the parcels move, their mass is conserved, so we find that dt
m = 0. For this identity to be
valid for any initial volume V0 , this gives the equation of conservation of mass:
Conservation of mass
D
ρ + ρ div ⃗u = 0
Dt
(2.5)
When the fluid is incompressible, the density of a given parcel cannot change, so that
= 0, hence we find (in absence of vacuum or null-density areas)
D
Dt ρ
Incompressibility
div ⃗u = 0
(2.6)
16
The Navier–Stokes Problem in the 21st Century (2nd edition)
This is consistent with Equation (2.2): if div ⃗u = 0, then the volume occupied by a parcel
never varies.
⃗ If the fluid is homogeneous, the
For an incompressible fluid, we find that ∂t ρ = −⃗u.∇ρ.
d
ρ(t) = 0; the density is constant in
density does not depend on the position, thus we find dt
time and in space:
Incompressibility and homogeneity
ρ = Constant
2.4
(2.7)
Newton’s Second Law
We apply Newton’s second
R law to a moving parcel of fluid. The momentum of the parcel
at time t is given by M = Vt ρ(t, y)⃗u(t, y) dy. If f⃗(t, y) is the force density at time t and
R
position y, the force applied to the parcel is F⃗ = Vt f⃗(t, y) dy. Newton’s second law of
mechanics then gives that
d
M = F⃗ .
dt
The convection theorem gives then
Z
D
(ρ⃗u) + ρ⃗u div ⃗u − f⃗ dy = 0
Vt Dt
Equation (2.5) gives
D
Dt ρ
+ ρ div ⃗u = 0, hence we have (taking infinitesimal volume V0 )
Newton’s second law
ρ
D
⃗u = f⃗
Dt
(2.8)
This can be written as well as
⃗ u = f⃗
ρ ∂t ⃗u + (⃗u.∇)⃗
(2.9)
⃗ = P3 ui ∂i . Of course, there remains to describe the force density f⃗. This is the
where ⃗u.∇
i=1
resultant of several forces: exterior forces (such as gravity) and internal forces. In the next
sections, we consider two important types of internal forces: the force induced by pressure
and the force induced by friction.
Remark: This balance of momentum is classical in fluid mechanics since the seminal memoir of Euler [167]. However, it has been recently disputed by H. Brenner [65] who argues
that one must distinguish between the (Eulerian) mass transportation velocity ⃗um and
the (Lagrangian) particle velocity ⃗uv . Thus, we would have instead of (2.1) the equation
P3
D
i=1 um,i (t, x)∂i Q(x, t), the continuity Equation (2.5) would become
Dt Q = ∂t Q(x, t) +
The Physical Meaning of the Navier–Stokes Equations
17
D
+ ρ div ⃗um = 0 and the balance of momentum (2.8) would become ρ Dt
⃗uv = f⃗. One
then needs a constitutive law to describe the difference ⃗uv − ⃗um . Brenner proposed the law
D
Dt ρ
⃗
⃗uv − ⃗um = K ∇ρ.
Thus, the equations should be modified in case of compressible fluids with high density
gradients, while for uncompressible homogeneous fluids the classical equations of fluid mechanics would still be valid.
A study of the Brenner model has been performed by Feireisl and Vasseur [172] who
showed that the weak solutions for this model are more regular than the weak solutions for
the classical Navier–Stokes equations for highly compressible fluids.
The story does not stop with Brenner’s model, which remains disputed. Various models
of extended Navier–Stokes or Euler equations have been recently discussed, as for instance
by Svärd in 2018 [457] or by Reddy, Dadzie, Ocone, Borg and Reese in 2019 [407].
2.5
Pressure
When a fluid is in contact with a body, it exerts on the surface of the body a force that
is normal to the surface and called the pressure. The pressure is a scalar quantity, which
does not depend on the direction of the normal. Positive pressure gives a compression force
that points inward of the body, so that is opposed to the normal.
Internal pressure (or static pressure) is defined in an analogous way. The fluid parcel
occupies Ra volume δV ; the force exerted on the parcel induced by the pressure is then
F⃗P = − ∂δV p ⃗ν dσ. This can be rewritten with Ostrogradski’s formula into the following
equation:
Z
⃗ dx.
F⃗P = −
∇p
V
This gives us the density for the pressure force:
Force density for the pressure
⃗
f⃗P = −∇p
2.6
(2.10)
Strain
Fluids are not rigid bodies. Thus, their motion implies deformations. Those deformations
may be illustrated through the strain tensor. If the velocities and their derivatives are small
enough, we may estimate for two initial points x0 and y0 how the distance of the parcels
will evolve. Indeed, if x(t) = Xx0 (t) and y(t) = Xy0 (t), we have
∥x − y∥2 = ∥x0 − y0 ∥2 + 2
Z
t
(x(s) − y(s)).(⃗u(s, x(s)) − ⃗u(s, y(s))) ds
0
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The Navier–Stokes Problem in the 21st Century (2nd edition)
and, neglecting terms of higher order, we get
Z t
∥x − y∥2 ≈ ∥x0 − y0 ∥2 + 2
(x(s) − y(s)).Du(s, x(s))(x(s) − y(s)) ds
0
where the matrix Du is the matrix
Du = (∂j ui (s, x))1≤i,j≤3 .
(2.11)
Cauchy’s strain tensor ϵ is defined as the symmetric part of Du:
ϵ=
1
Du + (Du)T .
2
(2.12)
The antisymmetric part has a null contribution to the integral, and we find:
Z t
∥x − y∥2 ≈ ∥x0 − y0 ∥2 + 2
(x(s) − y(s)).ϵ(s, x(s))(x(s) − y(s)) ds
0
Cauchy’s strain tensor
The strain tensor at time t and position x is the matrix ϵ given by
ϵi,j =
1
(∂i uj + ∂j ui )
2
for 1 ≤ i, j ≤ 3
(2.13)
If we look at the infinitesimal displacement of y, we have
D
1
y = ⃗u(t, y) = ⃗u(t, x) + ϵ(y − x) + (Du − (Du)T )(y − x) + O((y − x)2 ).
Dt
2
⃗u(t, x) does not depend on y: it corresponds to an (infinitesimal) translation; 12 (Du−(Du)T )
does not contribute to the distortion of distances, it corresponds to an (infinitesimal) rotation. ϵ corresponds to the (infinitesimal) deformation.
2.7
Stress
When a fluid is viscous, it reacts like an elastic body that resists deformations. Applying
the theory of elasticity to the fluid motion, one can see that the deformations induce forces.
If δV is a small parcel, the deformation of the parcel induces a force exerted on the border
of δV ; this force F⃗visc is given by a tensor T (the viscous stress tensor)1 and we have
Z
F⃗visc =
T ⃗ν dσ
∂δV
or, equivalently,
Z
Fvisc,i =
3
X
Ti,j νj dσ.
∂δV j=1
1 The
stress tensor is the sum T − pI3 , where p is the hydrostatic pressure.
The Physical Meaning of the Navier–Stokes Equations
19
Ostrogradski’s formula gives us the force density f⃗visc associated to the stress:
fvisc,i =
3
X
∂j Ti,j = div Ti,.
(2.14)
j=1
When the fluid velocity and its derivatives are small enough, Stokes has shown that the
relation between the stress tensor and the strain tensor is linear. In the case of an isotropic
fluid (so that the linear relation is the same at all points) we find that f⃗visc is a sum of
second derivatives of ⃗u. But, due to the isotropy of the fluid, a change of referential through
a rotation should not alter the relation between the force and the velocity. This gives that
f⃗visc is determined only by two viscosity coefficients2 :
Force density associated to the stress
In an isotropic fluid with small velocities, we have
⃗
f⃗visc = µ∆⃗u + λ∇(div
⃗u)
(2.15)
Equation (2.15) corresponds to a very simple relationship between the tensor ϵ and the
tensor T:
T = 2µϵ + η tr(ϵ) I3
(2.16)
with tr(ϵ) = ϵ1,1 + ϵ2,2 + ϵ3,3 and λ = µ + η. µ is called the dynamical viscosity of the
fluid, and η the volume viscosity of the fluid. Fluids for which the relation (2.16) holds are
called Newtonian fluids. All gases and most liquids which have simple molecular formula
and low molecular weight such as water, benzene, ethyl alcohol, etc. are Newtonian fluids.
In contrast, polymer solutions are non-Newtonian.
Stokes [451] has expressed the notion of internal pressure in a very general principle that
allowed, a hundred years later, Reiner [408] and Rivlin [414] to describe a more general class
of fluids. For a Stokesian fluid, the stress tensor T is still related to the strain tensor ϵ in
a homogeneous and isotropic way, but the relationship is no longer linear. Following Serrin
[431, 432] and Aris [6], a Stokesian fluid satisfies the following four assumptions:
ˆ the stress tensor T is a continuous function of the strain tensor ϵ and the local thermodynamical state, but independent of other kinematical properties
ˆ T does not depend explicitly on x (fluid homogeneity)
ˆ the fluid is isotropic
ˆ when there is no deformation (ϵ = 0), the fluid is hydrostatic (T = 0).
Then, using the symmetries induced by the principle of material objectivity or of frame
indifference (see Noll and Truesdell [377]) which states that “the constitutive laws governing
the internal conditions of a physical system and the interactions between its parts should
not depend on whatever external frame of reference,” Serrin showed that the viscous stress
tensor can be expressed as
T = α I3 + β ϵ + γϵ2
(2.17)
2 This
is expressed by Feynman [175] in the following terms:
the most general form of second derivatives that can occur in a vector equation is a sum of a
term in the Laplacian (∇.∇v = ∇2 v), and a term in the gradient of the divergence (∇(∇.v)).
20
The Navier–Stokes Problem in the 21st Century (2nd edition)
where α(0, 0, 0) = 0 and α = α(Θ, Φ, Ψ), β = β(Θ, Φ, Ψ) and γ = γ(Θ, Φ, Ψ) are functions
of the three invariants of the symmetric matrix ϵ : if the eigenvalues of ϵ are λ1 , λ2 and λ3 ,
then Θ = λ1 + λ2 + λ3 = tr(ϵ), Φ = λ1 λ2 + λ2 λ3 + λ3 λ1 and Ψ = λ1 λ2 λ3 = det(ϵ).
2.8
The Equations of Hydrodynamics
Let us consider a Newtonian isotropic fluid. We have seen that we have
D
ρ + ρ div ⃗u = 0
Dt
and
ρ
D
⃗u = f⃗.
Dt
The force density f⃗ is a superposition of external forces f⃗ext and internal forces f⃗int . In the
external forces, one may have the gravity, or the Coriolis force. In the internal forces, one
has seen the force due to the pressure:
⃗
f⃗P = −∇p
and the force due to the viscosity:
⃗
f⃗visc = µ∆⃗u + λ∇(div
⃗u)
In the absence of other internal forces, we obtain the equations of hydrodynamics:
The equations of hydrodynamics
For a Newtonian isotropic fluid, we have
and
ρ
D
ρ + ρ div ⃗u = 0
Dt
(2.18)
D
⃗ + µ∆⃗u + λ∇(div
⃗
⃗u = −∇p
⃗u) + f⃗ext
Dt
(2.19)
Those equations are in number of four scalar equations with five unknown scalar quantities (u1 , u2 , u3 , ρ and p). The fifth equation depends on the nature of the fluid: it is a
thermodynamical equation of state that links the pressure, the density and the temperature (one usually assumes that temperature is constant).
Remark:
1. In the case of an incompressible fluid, the equation of state is very simple:
ρ = Constant
2. When there is no viscosity, one speaks of ideal fluids: λ = µ = 0.
The Physical Meaning of the Navier–Stokes Equations
21
⃗
3. Writing µ∆⃗u + λ∇(div
⃗u) as the divergence of the symmetrical tensor
T = µ(∂i uj + ∂j ui )1≤i,j≤3 + η (div ⃗u) I3
with η = λ − µ, we find that the trace of T is given by (2µ + 3η) div ⃗u; it leads to add
to the gradient of the (thermodynamical) pressure another gradient of pressure; the
total mechanical pressure is then p − (2µ + 3η) div ⃗u. The coefficient 2µ + 3η is called
the bulk viscosity. An important case is the Stokes hypothesis where the tensor
T has no trace: 2µ + 3η = 0. This corresponds to λ = 0.
Sometimes, one considers other internal forces, such as those linked to electric or thermal
conductivity of the fluid. One then has to add new internal forces to the equations that
are dependent on the velocity and influence the velocity. One then quits the domain of
hydrodynamics and enters the domain of magnetohydrodynamics (a discipline founded by
the 1970 Nobel Prize winner Alfvén) or of the Boussinesq equations that link the velocity
and the temperature.
2.9
The Navier–Stokes Equations
In this section we consider the case of a Newtonian, isotropic, homogeneous and incompressible fluid. The equations of hydrodynamics (2.18) and (2.19) then are transformed into
the Navier–Stokes equations. Since ρ is constant, it is customary to divide the equations
by ρ, and to replace the force density f⃗ext with a reduced density f⃗r = ρ1 f⃗ext , the pressure
p with a reduced pressure pr = ρ1 p (which is called the kinematic pressure), and the
dynamical viscosity µ by the kinematic viscosity3 ν = ρ1 µ. We then have:
The Navier–Stokes equations
⃗ u = −∇p
⃗ r + ν∆⃗u + f⃗r
∂t ⃗u + (⃗u.∇)⃗
(2.20)
div ⃗u = 0
(2.21)
ν is positive for a viscous fluid. In case of an ideal fluid, (ν = 0), we obtain the Euler
equations:
The Euler equations
⃗ u = −∇p
⃗ r + f⃗r
∂t ⃗u + (⃗u.∇)⃗
(2.22)
div ⃗u = 0
(2.23)
3 In the 19th century, the difference between kinetics and kinematics was a keystone in mechanics. This
difference seems to be less understood in the 21st century: on the website www.answers.com, one can read
kinematic is the study of state of motion of a body i.e. includes both rest and moving bodies..
but kinetic is study of moving bodies only....
(https://www.answers.com/Q/Difference− between− kinetic− and− kinematic)
22
2.10
The Navier–Stokes Problem in the 21st Century (2nd edition)
Vorticity
The Navier–Stokes equations may be rewritten to underline the role played by vorticity.
We start from the identity
2
⃗ |⃗u| = (⃗u.∇)⃗
⃗ u
(curl ⃗u) ∧ ⃗u + ∇
2
We thus can write the Navier–Stokes equations as
Another formulation of the Navier–Stokes equations
⃗ r + ν∆⃗u + f⃗r
∂t ⃗u + ω
⃗ ∧ ⃗u = −∇Q
(2.24)
div ⃗u = 0
(2.25)
where ω
⃗ = curl ⃗u is the vorticity of the flow and Qr the (reduced) total pressure
The total pressure Q = ρQr is thus the sum of the hydrostatic pressure p and the
dynamic pressure q = ρ 21 |⃗u|2 .
Taking the curl of the Navier–Stokes equations gives the following equations for ω
⃗:
⃗ ω = ν∆⃗
⃗ u + curl f⃗r .
∂t ω
⃗ + (⃗u.∇)⃗
ω + (⃗
ω .∇)⃗
(2.26)
We find again the phenomenon of diffusion (induced by ∆⃗
ω ), the advection by the vector
⃗ ω
⃗ u, which corresponds
field ⃗u (described by the term (⃗u.∇)
⃗ ) and we have a third term (⃗
ω .∇)⃗
to stretching forces. This term is very important in 3D fluid mechanics. When the fluid is
⃗ u = 0.
planar ⃗u(t, x1 , x2 , x3 ) = (u1 (x1 , x2 ), u2 (x1 , x2 ), 0), the stretching force vanishes: (⃗
ω .∇)⃗
2.11
Boundary Terms
To make the Navier–Stokes system complete, one must specify the conditions at the
boundary of the domain of the fluid. In this book, all along, we will consider a problem
with no boundary (the fluid fills the whole space). However, in this section, we shall give a
few words on the boundary value problem.
When the fluid occupies only a domain Ω, the problem of the boundary conditions
is raised. The domain may vary with time. A particular problem is the free-boundary
problem: the boundary of Ω evolves through a partial differential equation which describes
the evolution of the curvature of the boundary through the action of the deformation tensor
of the fluid (see the paper by Solonnikov [445]).
For a rigid domain, one has to prescribe the behavior at the boundary and at infinity
(when the domain is unbounded). The most used condition is the no-slip condition which
says that, at a point of the border, the normal part of the velocity should vanish (⃗u.⃗ν = 0)
and the tangential part of the velocity should equal the velocity of the solid point of the
boundary (if the boundary is moving). If the boundary points do not move, the no-slip
condition is the homogeneous Dirichlet condition: ⃗u|∂Ω = 0.
The Physical Meaning of the Navier–Stokes Equations
23
For Euler equations on a fixed domain, the no-slip condition is replaced by an impermeability condition (that expresses that no fluid crosses over the boundary) ⃗u.⃗ν = 0 on
∂Ω.
The no-slip condition was introduced by Stokes [451] in 1849 and has been in accordance
with many experimental data. However, there are some cases where some slip is to be
considered, as for instance in microfluidics (see the review paper [304]) that deals with very
small quantities of fluids (between an attoliter [10−18 l.] and a nanoliter [10−9 l.]), where
the macroscopic properties of fluids are no longer valid. For such fluids, the slip condition
introduced by Navier in 1822 [373] has been experimentally validated. The Navier slip
condition stipulates that the normal part of the fluid velocity at the boundary vanishes,
but that the tangential part is governed by the stress tensor: if Q∥ is the projection Q∥ (⃗g ) =
⃗g − ⟨⃗g |⃗ν ⟩⃗ν on the tangent plane to the boundary, Q∥ ⃗u is proportional to Q∥ (T.⃗ν ).
For the Navier slip condition, one assumes more precisely that we have, for a constant
σ ≥ 0, the equality
Q∥ (T⃗ν + αu) = 0.
α is called the friction coefficient. A popular choice is α = 0, the pure slip condition. The
pure slip condition may be rewritten in the following way. If τ is a tangent vector in the
tangent plane of ∂Ω, we have the identity
⃗ ⊗ ⃗u · ⃗ν ⊗ ⃗ν )⃗ν .
Q∥ (T⃗ν ) = 2µϵ⃗ν − 2µ(ϵ⃗ν · ⃗ν )⃗ν = 2µϵ⃗ν − 2µ(∇
Thus, Q∥ (T⃗ν ) = 0 if and only for every tangent vector ⃗τ of ∂Ω, we have
⃗τ · ϵ⃗ν = 0.
(2.27)
The study of the Navier–Stokes equations with this pure slip boundary condition has been
initiated by Solonnikov and Ščadilov in 1973 [446], while studying a model for flow with
free boundary.
Recently, another type of boundary condition has been considered. Equation (2.27) may
be rewritten, due to the identity
3
X
1
⃗ i ) = 1 ⃗τ · (⃗
νi ∇u
ω ∧ ⃗ν ) + ⃗τ · (
ω ∧ ⃗ν ) + ∂τ (⃗u · ⃗ν ) − ⃗u · ∂τ ⃗ν
⃗τ · ϵ⃗ν = ⃗τ · (⃗
2
2
i=1
and due to the fact that on the boundary ⃗u · ⃗ν = 0 so that the tangential derivative
∂τ (⃗u · ⃗ν ) = 0, as
1
⃗τ · (⃗
ω ∧ ⃗ν ) = ⃗u · ∂τ ⃗ν .
2
In the regions where the boundary of Ω is flat (so that the normal ⃗ν is constant), we thus
have (for all tangential directions)
1
⃗τ · (⃗
ω ∧ ⃗ν ) = 0
2
or equivalently
ω
⃗ ∧ ⃗ν = 0.
(2.28)
The boundary conditions ⃗u · ⃗ν = 0 and ω
⃗ ∧ ⃗ν = 0 on general (non-flat) domains were
considered by Xiao and Xin [508] and Beirão da Veiga and Crispo [32] for the study of
the inviscid limit of the equations. Those equations were coined as Hodge–Navier–Stokes
equations by Mitrea and Monniaux [363], since those boundary conditions are natural for
the Hodge-Laplacian operator.
24
The Navier–Stokes Problem in the 21st Century (2nd edition)
2.12
Blow-up
Let us consider the Clay Millennium Problem for the Navier–Stokes equations in absence
of external forces. As we shall see, a classical result on the Navier–Stokes equations shows
that the Cauchy initial value problem will have a smooth solution as long as the velocity
⃗u remains bounded. Thus, in order to have a breakdown in regularity, the L∞ norm must
blow up. But this blow-up has no physical meaning; for various reasons, one has to drop
the equations long before the blow-up can occur. For instance,
ˆ the incompressibilty of the fluid is an approximation that is valid only if the velocity
of the fluid is much smaller than the speed of sound
ˆ the Newtonian character of the fluid was derived under the hypothesis of small velocities and small derivative of the velocities
ˆ when velocities are too important, classical mechanics should be corrected into relativistic mechanics
Thus, the blow-up issue is essentially a mathematical problem, not a physical one. However,
it is hoped that the understanding of the mechanism that leads to blow-up or blocks it would
shed a good light on the mechanism that leads physical fluids to turbulent states.
2.13
Turbulence
Smooth flows are called laminar, whereas disordered flows are called turbulent. For turbulent flows, it is quite hopeless to try and find a description of all the fluid parcels, as
the number of degrees of freedom is too important. Since the works of Reynolds (1894) and
Taylor (1921), one tries only to describe the evolution of the flow on a large scale, and to
discuss the behavior of the flow at small scales as a dissipative correction of the equations
for the large scales.
This separation between the large-scale components and the small-scale ones relies on
several physical observations. The large-scale components are sensitive to the geometry of
the boundary and to the nature of external forces that are impressed on the fluid, whereas
the small scale components can be analyzed in a more universal way.
To separate the large-scale component from the small-scale component, one uses an
averaging process that gives a mean value ū of the velocity ⃗u. The Navier–Stokes equations
then give new equations for ū:
⃗ = ν∆ū − ∇p̄
⃗ + f̄ + div R
∂t ū + ū.∇ū
(2.29)
(together with div ū = 0 and u|t=0 = u0 ) where the Reynolds stress R is given by
R = ū ⊗ ū − ⃗u ⊗ ⃗u.
(2.30)
As the mean value ⃗u ⊗ ⃗u does not depend on the mean value ū, those equations are not
closed. The problem is then to give a satisfying modelization of the Reynolds stress.
The theory of Kolmogorov (1941) gives a modelization of ⃗u − ū as a random field
obeying some universal laws due to the (local) homogeneity and isotropy of the fluctuations.
Whereas this theory has been confirmed experimentally, it remains far from being completely
understood and is the core of a very active research field (see the classical book of Monin
and Yaglom [367]).
Chapter 3
History of the Equation
In this chapter, we sketch some points of the history of the Navier–Stokes equations. The
reader will find a comprehensive study of the period 1750–1900 in Darrigol’s book Worlds
of flow [145], which studies the origin of the equations as well from the mathematical
theoretical point of view as from the point of view of physical experiments and observation.
Other stimulating references on the infancy of mathematical hydrodynamics are the papers
of Truesdell [479, 480, 481].
3.1
Mechanics in the Scientific Revolution Era
Hydrodynamics appeared in 1738. The word hydrodynamica was coined by D.
Bernoulli in his treatise Hydrodynamica, sive De viribus et motibus fluidorum commentarii
[37], where he wanted to propose a unified theory of hydrostatics and hydraulics.
Hydraulics is a very old science. Irrigation has been known since the 6th millennium
BCE in Ancient Persia. Managing water supply for human settlements and irrigation has
been an important technique in human development. The technology of “qanats” has been
developed by Iranians in the early 1st millennium BCE and then spread toward Asia, Africa
and Europe. In the kingdom of Saba’ (now, in Yemen), dams were constructed as soon as
2000 BCE in order to irrigate the crops; the great Dam of Ma’rib (built about the 8th
century BCE) is counted as one of the most wonderful feats of engineering in the ancient
world (its remains were severely damaged by a Saudi airstrike in 2015). Working machines
using hydraulic power, such as the force pump, have been developed by Hellenistic scientists
(as, for instance, Hero of Alexandria) and by Roman engineers for raising water. Modern
hydraulics was initiated in Italy, in the 16th century as an experimental science, then in the
17th century in a more theoretical approach with the influential treatises of Castelli (1628)
and Fontana (1696).
Hydrostatics has ancient roots as well. The phenomenon of buoyancy has been explained
by Archimedes in the 3rd century BCE. Pressure has been explained in the 17th century: a
fluid is a substance that continually deforms under an applied shear stress; thus, in hydrostatics (the science that studies fluids at rest), there cannot exist a shear stress; however,
fluids can exert pressure normal to any contacting surface. Due to gravitation, liquids exert
pressure on the sides of a container as well as on anything within the liquid itself. This
pressure is transmitted in all directions and increases with depth, as established by Pascal.
Atmospheric pressure had been revealed by Torricelli who invented barometers.
The 17th century is the century of the so-called Scientific Revolution. Mechanics was
deeply refounded in that period, culminating with the work of Newton (1687). Basic concepts of physics emerged throughout the century. Kepler gave in 1609 the laws ruling planetary motion. The study of free fall by Galileo (1638) clearly put in light the notion of
accelerated motion. Aristotle’s notion of uniform velocity and of proportionality to describe
DOI: 10.1201/9781003042594-3
25
26
The Navier–Stokes Problem in the 21st Century (2nd edition)
motions had already been criticized by many medieval authors, including the philosopher
Buridan and the Oxford Calculators (as Bradwardine), but the mathematical law of motion
in free fall was stated and experimentally checked by Galileo. Huygens replaced Buridan’s
impetus with momentum (1673). Newton’s second law expresses the variation of momentum through the action of forces. While free fall was caused by gravity, other forces were
explored in this century: hydrostatic pressure (Pascal’s law in 1648, extending previous work
of Stevin (1586)), tension in elasticity (Hooke’s law in 1660), resistance to motion, etc. Another important concept emerged in 1676: the vis viva introduced by Leibniz in the study
of elastic shocks, which has been fiercely debated all along the 18th century and became
the kinetic energy in the 19th century.
Physics had been reshaped through two main tools: experimentation, with the invention
of new observation devices, and mathematization. The celebrated sentence of Galileo states:
La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto
innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non
s’impara a intender la lingua, e conoscer i caratteri, ne’quali è scritto. Egli è
scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure
geometriche, senza i quali mezzi è impossibile a intenderne umanamente parola;
senza questi è un aggirarsi vanamente per un oscuro laberinto1 .
Mathematics in the 17th century was drastically reshaped as well, in a parallel dynamics.
The new algebra introduced by Viète (1591) allowed symbolic computations. Algebraization
of geometry was then proposed by Descartes (1637), through the use of numerical coordinates in a reference frame. Fermat’s rule for the determination of maxima and minima
and Barrow’s duality principle between problems on tangents and problems of area [7] lead
to the foundation of modern calculus by Newton and Leibniz (1684), with the notion of
derivative and primitive functions.
3.2
Bernoulli’s Hydrodymica
Mathematicians tried to apply those new tools to explain the empirical rules of physics.
Understanding the laws of statics could be reduced to geometrical reasoning, as in the work
of Stevin, who discovered the hydrostatic paradox [449] in 1586: the downward pressure of
any given liquid is independent of the shape of the vessel, and depends only on its height
and base. This was illustrated in Pascal’s barrel experiment in 1646 [392]: Pascal inserted
a 10-m long vertical tube into a barrel filled with water; when water was poured into the
vertical tube, the increase in pressure caused the barrel to burst.
Understanding the laws of dynamics needed the invention of calculus. The model for
mathematicians was then the derivation by Newton of Kepler’s laws on planetary motion.
In fluid motion, one of the first laws investigated was Torricelli’s law on the efflux (1644)
[474]: the speed of efflux of a fluid through a sharp-edged hole at the bottom of a tank
filled to a depth h is the same as the speed that a body would acquire in falling freely from
the same height h. As soon as 1695, Varignon, developing analytic dynamics by adapting
Leibniz’s calculus to the inertial mechanics of Newton’s Principia, proposed a derivation
1 Philosophy is written in this grand book – I mean the universe – which stands continually open to
our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret
the characters in which it is written. It is written in the language of mathematics, and its characters are
triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a
single word of it; without these, one is wandering around in a dark labyrinth.
History of the Equation
27
of Torricelli’s law based on the momentum principle. Varignon’s derivation was based on
the hypothesis that the force causing the outflow was given by the weight of the column
of water over the opening. This assumption was proven to be false, but is very common
in the attempts of mathematical derivations of Torricelli’s law in the early 18th century
(as, for instance, Hermann (1716) and J. Bernoulli (1716)). In the section X “Principes de
l’hydrodynamique” of Part II of his famous treatise Traité de méchanique analitique [299],
Lagrange comments on Varignon’s proof and on Newton’s attempt (in the second edition
of the Principia, (1713)) of Torricelli’s law. (For a discussion of those various attempts, see
Mikhailov [360] or Blay [45]).
D. Bernoulli’s approach in Hydrodynamica [37] is totally different. He does not rely on the
momentum principle, but on the Leibnizian theory of vis viva. Leibniz’s theory was contested
both by the Newtonians in England and by the Cartesians in France, but was gaining
stronger support: ’s Gravesande in 1722 made an experiment in which brass balls were
dropped with varying velocity onto a soft clay surface; the results of the experiment clearly
proved that their penetration depth was proportional to the square of their impact speed.
The French physicist and mathematician Émilie du Châtelet recognized the implications of
the experiment and published an explanation in 1740 in her influential treatise on physics
[104]. Bernoulli’s treatise was a determining example of the interest of the principle of
conservation of vis viva. In 1757, D’Alembert could write in the Encyclopédie [140]:
On peut voir par différens mémoires répandus dans les volumes des académies
des Sciences de Paris, de Berlin, de Petersbourg, combien le principe de la conservation des forces vives facilite la solution d’un grand nombre de problemes
de Dynamique; nous croyons même qu’il a été un tems où on auroit été fort
embarrassé de résoudre plusieurs de ces problemes sans employer ce principe2 .
Bernoulli’s treatise has been considered as the first successful attempt of mathematical
derivation of Toricelli’s law. It contained other results on hydraulics, such as the prefiguration of Bernoulli’s law that explains how the pressure exerted by a moving fluid is lesser
than the pression of the fluid at rest.
3.3
D’Alembert
The solution proposed by D. Bernoulli for the derivation of Torricelli’s law was felt
insecure by many mathematicians as it relied on a controversial principle. There was at
that time a strong discussion of what was the meaning of forces that put bodies in motion.
Cartesians insisted on the momentum, a quantity that was clearly defined for a moving mass
point (the mass times the velocity). Newtonians insisted on the variation of momentum,
hence gave an important role to acceleration, according to Newton’s second law:
II. The change of motion is proportional to the motive force impressed, and it
takes place along the right line in which that force is impressed.
But the nature of the motive force remained obscure, and coined as metaphysical by Cartesians who could not accept the principle of distant action and especially the theory of universal attraction to explain gravity. Leibnizians, following ideas from Huygens and Leibniz,
2 One can see through various memoirs that can be found in the volumes of the science academies in Paris,
Berlin or Petersburg, how the principle of conservation of living forces eases the solution of many problems
in Dynamics; we even believe that there has been a time when one would have been most embarassed to
solve many of those problems without using this principle.
28
The Navier–Stokes Problem in the 21st Century (2nd edition)
used the notions of vis viva and vis morta, and expressed the laws of motion as a conversion
of vis morta into vis viva. This conception was rejected by Cartesians as metaphysical as
well, since the vis viva seemed an inherent property attached to the moving bodies, and not
a measurable kinetic quantity.
Johann Bernoulli, Daniel’s father, published Hydraulica, a treatise in 1742, with the
aim of rewriting his son’s results rather with help of Newtonian mechanics rather than of
Leibnizian vis viva. Johann Bernoulli had to identify precisely the acceleration of the fluid,
and he was thus led to describe the convective part of the acceleration and the internal
pressure, i.e., the pressure that the moving parts of the fluid exerted on the other parts.
Later, those two innovations would be crucial elements for Euler’s derivation of the equations
of hydrodynamics.
D’Alembert tried to avoid any use of the concept of force, as it seemed to be linked to
metaphysical issues. In 1743, he founded his Traité de dynamique [136] on a principle that
avoided the use of internal forces to describe the motion of a constrained system of bodies.
Tout ce que nous voyons bien distinctement dans le Mouvement d’un Corps,
c’est qu’il parcourt un certain espace, & qu’il employe un certain tems à le
parcourir. C’est donc de cette seule idée qu’on doit tirer tous les Principes de la
Méchanique, quand on veut les démontrer d’une manière nette & précise; ainsi
on ne sera point surpris qu’en conséquence de cette réflexion, j’aie, pour ainsi
dire, détourné la vûe de dessus les causes motrices, pour n’envisager uniquement
que le Mouvement qu’elles produisent; que j’aie entièrement proscrit les forces
inhérentes au Corps en Mouvement, êtres obscurs & Métaphysiques, qui ne sont
capables que de répandre les ténèbres sur une Science claire par elle-même.3
He claimed that one did not need to use Newton’s second law:
Pourquoi donc aurions-nous recours à ce principe dont tout le monde fait usage aujourd’hui, que la force accélératrice ou retardatrice est proportionnelle à
l’élément de la vitesse? principe appuyé sur cet unique axiome vague & obscur,
que l’effet est proportionnel à sa cause.4
He based his theory of dynamics on three principles:
Le Principe de l’équilibre joint à ceux de la force d’inertie & du Mouvement
composé, nous conduit donc à la solution de tous les Problèmes où l’on considère
le Mouvement d’un Corps.5
According to those three principles, he decomposed the motion of a constrained body into
a natural one, described through the law of inertia, and the motion due to the presence
of constraints; for this latter one, his principle of equilibrium asserts that the forces corresponding to the accelerations due to the presence of constraints form a system in static
equilibrium.
3 All that we can distinctly see in the Motion of a Body is the fact that it covers a certain space and that
it takes a certain time to cover that space. One must draw all the Principles of Mechanics from that sole
idea, when one wants to give a neat and precise demonstration of them. Thus, it won’t be a surprise that,
as a consequence of this reflection, I have turned my view away from the motive forces and considered but
the Motion they produce; that I entirely banished the forces inherent to the Body in Motion, as obscure
and Metaphysical beings that can only shed darkness on a Science that is clear by itself.
4 Why should we appeal to that principle used by everybody nowadays, that the accelerating or retarding
force is proportional to the element of velocity, a principle resting only on that vague and obscure axiom
that the effect is proportional to the cause?
5 The principle of equilibrium joined with the principles of the law of inertia and of the composition of
motions leads us to the solution of all the problems where the Motion of a Body is considered.
History of the Equation
29
With those simple principles, D’Alembert was able to prove the conservation of living
forces. In 1744, right after the Traité de dynamique, he published the Traité des fluides
[137] where he applied his dynamical theory to the proof of Daniel Bernoulli and Johann
Bernoulli’s results. In the Traité des fluides as well as in D. Bernoulli’s Hydrodynamica or J.
Bernoulli’s Hydraulica, the fluid considered has only one degree of freedom: in their models,
the fluid is decomposed into horizontal slices and the velocity is uniform on each slice.
In 1747, in his treatise Réflexions sur la cause générale des Vents [138], he developed
the notion of a velocity field, with velocities that depended on the position. The differential
equations were then turned into partial differential equations. D’Alembert is known as a
pioneer of the use of partial derivatives in mathematical physics, with the famous example
of the wave equation which he gave in 1749 for describing vibrating strings. While partial
differential equations were already known in the setting of the prehistory of variational
calculus, D’Alembert was the first to use them in a mechanical context. Later, D’Alembert
worked on the resistance opposed to the motion of an immersed body, as in his 1752 treatise
Essai d’une nouvelle théorie de la résistance des fluides [139].
In 1768, he noticed that his theory of (inviscid) incompressible fluids led to a paradox, the
celebrated D’Alembert paradox [141]. He considered an axisymmetric body with a head-tail
symmetry, immersed in an inviscid incompressible fluid and moving with constant velocity
relative to the fluid, and proved that the drag force exerted on the body is then zero. This
result was in direct contradiction to the observation of substantial drag on bodies moving
relative to fluids:
Je ne vois donc pas, je l’avoue, comment on peut expliquer par la théorie, d’une
manière satisfaisante, la résistance des fluides. Il me paroı̂t au contraire que
cette théorie, traitée & approfondie avec toute la rigueur possible, donne, au
moins en plusieurs cas, la résistance absolument nulle; paradoxe singulier que
je laisse à éclaircir aux Géomètres.6
This paradox, and the fact that the equations derived for the description of fluid motions
had in general no easily computed solutions, caused a deep gap between mathematicians
dealing with fluid mechanics and engineers dealing with hydraulics. This situation lasted
for decades, before eventually the mathematical theory evolved to a frame more adapted to
the real-world situations, taking into account the viscosity effects.
3.4
Euler
Then Euler came. . .
Newtonian mechanics implied a new vision of geometry, as it has been underlined by
Bochner [47]:
Several significant physical entities of the Principia, namely, velocities, moments,
and forces are, by mathematical structure, vectors, that is, elements of vector
fields, and vectorial composition and decomposition of these entities constitute
an innermost scheme of the entire theory. This means that the mathematical
6 Thus, I do not see, I admit, how one can satisfactorily explain by theory the resistance of fluids. On
the contrary, it seems to me that the theory, developed in all possible rigor, gives, at least in several cases,
a strictly vanishing resistance, a singular paradox which I leave to future Geometers to elucidate.
30
The Navier–Stokes Problem in the 21st Century (2nd edition)
space of the Principia, in addition to being the Greek Euclidean substratum,
also carries a so-called affine structure, in the sense that with each point of the
space there is associated a three-dimensional vector space over real coefficients,
and that parallelism and equality between vectors which emanate from different
points are also envisaged.
However, the celebrated Newton formula f⃗ = m⃗a (where ⃗a is the acceleration) was not
expressed in such a vectorial form in the Principia and was not well understood in the fifty
years following the release of the Principia. MacLaurin in 1742 [346] and Euler in 1747 [165]
were the first ones to express Newton’s second law in its full 3D expression.
In 1750, Euler [166] applied Newton’s second law to the mechanics of continuous media.
He expressed the opinion that no other mechanical principles were needed. Euler’s mechanics
is an important turning point: Newton’s mechanics was essentially a kinematic theory for
a mass point; Euler extended this theory to the case of a continuous medium.
In 1755, Euler [167] presented a memoir (published in 1757) entitled Principes généraux
du mouvement des fluides, where he applied his theory to the theory of fluid motions. While
his predecessors worked on incompressible flows with one degree of freedom (D. Bernoulli
and J. Bernoulli) or two degrees of freedom (D’Alembert), Euler could derive the equations
for a general fluid, compressible or not, in the presence of arbitrary external forces.
In his seminal memoir, Euler described the laws of fluid mechanics as applied to fluid
parcels, very small volumes of fluids that are fictitiously isolated. With this notion of parcels,
he could introduce the internal pressure (or static pressure), as the density of the force
exerted on the parcel by the other parcels of fluid. With those two ideas, he could derive
Euler’s equation for an ideal fluid submitted to external forces (with force density f⃗ext ): the
equation expressing the conservation of mass
D
ρ + ρ div ⃗u = 0
Dt
(3.1)
and the equation corresponding to Newton’s second law
ρ
D
⃗ + f⃗ext
⃗u = −∇p
Dt
(3.2)
Lagrange underlined the importance of Euler’s equations [299]:
C’est à Euler qu’on doit les premières formules générales pour le mouvement
des fluides, fondées sur les lois de leur équilibre, et présentées avec la notation simple et lumineuse des différences partielles. Par cette découverte, toute la
Mécanique des fluides fut réduite à un seul point d’analyse, et si les équations
qui la renferment étaient intégrables, on pourrait, dans tous les cas, déterminer
complètement les circonstances du mouvement et de l’action d’un fluide mû par
des forces quelconques; malheureusement, elles sont si rebelles, qu’on n’a pu,
jusqu’à présent, en venir à bout que dans des cas très-limités.7
Truesdell sketches the legacy of Euler in those words [481]:
Judged from a positivist philosophy, Euler’s hydrodynamic researches are misconceived and unsuccessful: Their basic assumptions cannot be established experimentally, nor did Euler obtain from them numbers which can be read on a
7 Euler gave the first general formulas for the motion of fluids, based on the laws of their equilibrium, and
presented with the simple and bright notation of partial differences. By this discovery, the entire mechanics
of fluids was reduced to a single point of analysis, and if the equations which include it were integrable,
one could determine completely the circumstances of motion and of action of a fluid moved by any forces.
Unfortunately, they are so rebellious that up to the present time only a few very limited cases have been
worked out.
History of the Equation
31
dial. Yet, after Euler’s death, special solutions of his equations have given us the
theories of the tides, the winds, the ship, and the airplane, and every year new
practical as well as physical discoveries are found by their aid.
Euler’s success in this most difficult matter lay in his analysis of concepts.
After years of trial, sometimes adopting some semi-empirical compromise with
experimental data, Euler saw that experiments had to be set aside for a time.
They concerned phenomena too complicated for treatment then; some remain not
fully understood today. By creating a simple field model for fluids, defined by
a set of partial differential equations, Euler opened to us a new range of vision
in physical science. It is the range we all work in today. In this great insight,
looking within the interior moving fluid, where neither eye nor experiment may
reach, he called upon the “imagination, fancy, and invention” which Swift could
find neither in music nor in mathematics.
3.5
Laplacian Physics
The mathematical physics developed in the 18th century by Euler, D’Alembert and Lagrange rested on partial differential equations describing the regular behavior of continuous
quantities. However powerful this theory turned out to be, it sufffered from many drawbacks
for engineers as well as for physicists.
The equations obtained in this setting remained unsolved but in some very special
cases. Moreover, they described idealized situations that were very different from the real
life events. They could not explain the deformation of solid bodies, nor the creation of eddies
in turbulent flows. Engineers went on applying empirical formulae that were not derived
from those theories (and sometimes were in contradiction with those theories).
In opposition to Lagrange’s analytical mechanics, Laplace tried to develop a molecular
model of nature that could explain the laws of physics through the role of inter-molecular
forces, in analogy to the Newtonian theory of celestial mechanics and the Laplacian theory
of capillarity. This Laplacian physics was fiercely sustained by Poisson, Laplace’s disciple
[402]:
Il serait à désirer que les géomètres reprissent sous ce point de vue physique
et conforme à la nature, les principales questions de la mécanique. Il a fallu
les traiter d’une manière tout à fait abstraite, pour découvrir les lois générales
de l’équilibre et du mouvement, et en ce genre de généralité et d’abstraction,
Lagrange est allé aussi loin qu’on puisse le concevoir, lorsqu’il a remplacé les
liens physiques des corps par des équations entre les coordonnées des différents
points, c’est là ce qui constitue la mécanique analytique; mais à côté de cette admirable conception, on pourrait maintenant élever la mécanique physique dont
le principe unique serait de ramener tout aux actions moléculaires qui transmettent d’un point à un autre l’action des forces données et sont l’intermédiaire de
leur équilibre.8
8 Translated in [145]: It would be desirable that geometers reconsider the main equations of mechanics
under this physical point of view which better agrees with nature. In order to discover the general laws of
equilibrium and motion, one had to treat these questions in a quite abstract manner; in this kind of generality
and abstraction, Lagrange went as far as can be conceived when he replaced the physical connections of
bodies with equations between the coordinates of their various points: this is what analytical mechanics
is about; but next to this admirable conception, one could now erect a physical mechanics, whose unique
32
The Navier–Stokes Problem in the 21st Century (2nd edition)
Molecular models or atomistic ones were as old as the antique science. History of atomism
and molecular theories is well documented in the books by Whyte [500] or Kubbinga [283].
Atomism was proposed as a model by Leucippus and Democritus in the 5th century BCE.
This model was revived in the 17th century by Basson, Beeckman, Gassendi (who coined
the word molecula) and Boyle.
In 1745 Bošković [54, 55] published in De Viribus Vivis an explanation of elasticity and
inelasticity of collisions through an atomistic theory of matter, that tried to find a middle
way between Isaac Newton’s gravitational theory and Gottfried Leibniz’s metaphysical theory of monad-points. In this theory, however, atoms are no longer the ontological primitive
of nature: forces become the primary property of the material world. This was underlined
by Nietzsche [376]:
Während nämlich Kopernikus uns überredet hat zu glauben, wider alle Sinne,
dass die Erde nicht fest steht, lehrte Boscovich dem Glauben an das Letzte,
was von der Erde ‘feststand’, abschwören, dem Glauben an den ‘Stoff ’, an die
‘Materie’, an das Erdenrest - und Klümpchen-Atom; es war der grösste Triumph
über die Sinne, der bisher auf Erden errungen worden ist.9
In the model of Bošković, molecules have no extension, they are just points that are center
of forces. Those forces attract or repell the other molecules: when the distance r is large,
the force is attractive (with a decrease in 1/r2 to fit Newton’s theory of gravitation), while
when the distance is small, the force is repelling (and becomes infinite for vanishing distance) in order to avoid direct contact between distinct molecules. Molecules thus remain
at a positive distance from the other ones, and they are separated by vacuum. This force of
interaction was expected to provide the explanation for all the properties of matter: gravitation, collision, cohesion, flexibility, sound propagation, crystalline states, phase transition,
and so on.
The molecular model proposed in 1808 by Laplace [303] explained as well many physical
phenomena such as optical refraction, elasticity, hardness and viscosity as the result of
short-range forces between molecules. Laplace, together with his friend Berthollet, played
a prominent role in the scientific field at the beginning of the 19th century [182] and many
French physicists developed Laplace’s model as a key to understand the physical phenomena
on what they felt as a firm and non-hypothetical basis. However, the claim that the molecular
model developed by Laplace could explain all the physical phenomena was rapidly discarded,
as alternative methods were developed by Fourier (1822: theory of heat), Fresnel (1818: wave
optics) and Germain (1821: elasticity theory).
Poisson fought the Lagrangian method of virtual works and promoted Laplace’s discrete
molecular distributions and inter-molecular forces instead of the forces of constraint of the
continuous media used in analytical mechanics. In his treatise L’évolution de la mécanique
(1905) [161], Duhem quotes de Saint-Venant and Boussinesq as Poisson’s followers in the
rejection of forces of constraint and in the privileged use of molecular forces. In the same
treatise, Duhem shows how physical experiments on elasticity (such as Wertheim’s experiments on metals), however, eventually disproved Poisson’s hypotheses and confirmed the
results that Cauchy, Green and Lamé obtained by means of analytical mechanics. Duhem’s
conclusion is as severe as Poisson’s influence was still strong:
principle would be to reduce everything to molecular actions that transmit from one point to another the
given action of forces and mediate their equilibrium.
9 Translated in [499]: While Copernicus has persuaded us, against all senses, that the Earth does not stand
still, Boscovich taught us to renounce belief in the last thing of earth to “stand fast,” belief in “substance,”
in “matter,” in the last remnant of Earth, the corpuscular atom: it was the greatest triumph over the senses
achieved on Earth to this time.
History of the Equation
33
Il est donc impossible de garder les principes sur lesquels Poisson voulait faire
reposer la Mécanique physique, à moins d’avoir recours à des subtilités et à des
faux-fuyants.10
Fifty years later, the conclusion of Truesdell [478] was less severe on the Poissonian
approach, even if Truesdell prefered to employ a continuum analysis:
There are two methods of constructing a theory of elasticity or fluid dynamics.
The first, used originally by Boscovich, Navier, Cauchy, and Poisson and after
long discredit now again in favor among physicists, deduces macroscopic equations from special assumptions relative to the behavior of the supposed ultimate
discrete entities comprising the medium. In the present article I employ only the
continuum approach of Clairaut, D’Alembert, Euler, Lagrange, Fresnel, Cauchy,
Green, St. Venant, and Stokes, in which molecular speculations are avoided, and
gross phenomena are described in gross variables and gross hypotheses alone.
3.6
Navier, Cauchy, Poisson, Saint-Venant and Stokes
The discovery of the Navier–Stokes equations is linked to new formulations for elasticity
theory. Elasticity was an important issue at the beginning of the 19th century. While engineers were facing the absence of a convincing theory for the problem of beam flexions, there
had been a fierce debate around the prize proposed by the French Académie des Sciences
on the problem of explaining Chladni’s experiment of vibrating plates in 1808. The prize
was eventually won by Germain in 1818: her work was based on and enriched by Lagrange’s
contributions and violently criticized by Poisson who derived in 1814 a molecular model
based on the Laplacian system.
In this context, Navier contributed to the emergence of a new understanding of elasticity. In 1820, he proposed a Lagrangian approach of the problem of vibrating plates. He
analyzed the continuous deformations of the plates as composed of isotropic stretching - as
in Lagrange’s computations - and anisotropic flexion.
In 1821, he gave two proofs of his results in elasticity, one was based this time on a
Laplacian molecular model and the other one was based on the Lagrangian method which
relied on the balance of virtual moments [372]. The equations he derived were valid for more
general elastic bodies.
The idea developed by Navier was that the restoring forces appearing in elasticity could
be modelized as a response to the change of distances between molecules. For small deformations, this intermolecular force would be proportional (and opposite) to the variation of
the distance (with a proportionality coefficient depending on the distance).
In 1822, Navier [373] extended his theory to hydrodynamics. Once again, he introduced
restoring forces generated by the opposition to the change of distances between molecules.
Of course, those changes are due to the difference of velocities between molecules, and the
computations led Navier to the introduction to a new internal force in Euler’s equations of
⃗ (a force
hydrodynamics: the internal forces included not only the pressure gradient −∇p
which was present in static fluids as well as in moving fluids), but a new force µ(∆⃗u +
⃗
2∇(div
⃗u)) which was generated by the motion of the fluid (and more precisely by the
non-uniformity of the motion of the fluid).
10 It is thus impossible to keep the principles on which Poisson wanted to base physical Mechanics, unless
resorting to subtleties and to evasions.
34
The Navier–Stokes Problem in the 21st Century (2nd edition)
Truesdell [479] comments on Navier’s approach:
In 1821 Navier, a French engineer, constructed imaginary models both for
solid bodies and for fluids by regarding them as nearly static assemblages of
‘molecules’, mass-points obeying certain intermolecular force laws. Forces of cohesion were regarded as arising from summation of the multitudinous intermolecular actions. Such models were not new, having occurred in philosophical
or qualitative speculations for millenia past. Navier’s magnificent achievement
was to put these notions into sufficiently concrete form that he could derive
equations of motion from them.
As soon as 1822, Cauchy [95] gave a new interpretation of Navier’s results. From the
theory of Navier, he could see that the internal force exerted on the surface of a fluid
parcel was no longer perpendicular to the surface but contained a tangential part. Thus,
he developed a theory of elastic bodies, introducing the notion of internal stress that would
generate forces exerted on the surface of (imaginarily isolated) small elements of the body.
In a modern language, the force exerted on the surface element with normal ⃗ν would be
given by a vector f⃗surf that depend linearly on ⃗ν (but no longer directed in the direction
of ⃗ν ). He obtained a relationship f⃗surf = σ⃗ν , where σ is now defined as a 3 × 3 matrix. For
physicists, σ is a second-rank tensor.
Bochner [47] underlined the importance of Cauchy’s stress tensor:
Archimedes also accomplished basic work, perhaps his most famous one, in the
mechanics of floating bodies. Here again he did not introduce the physical concept which is central to the subject matter, namely, the concept of hydrostatic
pressure. But in this case Archimedes may be “excused”. Modern mechanics had
great difficulties in conceptualizing the notion of pressure, although Stevin immediately mapped out the task of doing so and everybody after him was pursuing
it. Even Newton was not yet quite certain of it. In a sense the clear-cut mathematization of the concept of pressure was arrived at only in the course of the
nineteenth century beginning perhaps with work by A. Cauchy on equations of
motion for a continuous medium in general. In the nineteenth century the mathematical “image” of pressure became a tensor, albeit a very special one, and
the actual formalization of the concept of a tensor and a full realization of its
mathematical status took a long time to emerge.
Cauchy showed that the tensor σ should be symmetrical. Then, he studied the quadratic
form associated to this symmetric matrix, and compared it to the quadratic form induced
by the strain tensor (or tensor of deformations: the quadratic form corresponds to the firstorder development of the variation of distance between two close points). For an isotropic
body, he argued that the stress tensor should have the same principal axes as the strain
tensor; then, generalizing Hooke’s law on elastic deformations, he assumed that the tensors
were proportional, with a proportionality coefficient independent of the deformation.
Applying his theory to hydrodynamics, he then obtained Navier’s equations, except that
⃗
⃗
the new force was given µ(∆⃗u + ∇(div
⃗u)) (instead of µ(∆⃗u + 2∇(div
⃗u))).
Poisson fought against Navier (who used Lagrangian methods of virtual works) and of
Cauchy (who studied continuous media) in the name of a strict and rigorous Laplacian
molecular approach. He proposed, as well as Cauchy, a strictly molecular theory; both
⃗
[96, 403] rederived Navier equations (with a more general force (µ + λ)∆⃗u + 2µ∇(div
⃗u),
i.e., the general form for a compressible Newtonian fluid). The arguments between Poisson,
Cauchy and Navier are described in Darrigol’s paper [144].
Saint-Venant tried to concile the experimental laws of engineers with a rigorous mathematical derivation of physical laws for elasticity. He applied his theory to fluid mechanics in
History of the Equation
35
an unpublished memoir to the Académie des Sciences in 1834 [418, 419]. The main idea in
Saint-Venant’s approach was the introduction of a varying viscosity and a non-linear dependency of the stress tensor on the strain tensor. A cause for the variations of the viscosity,
he indicated in 1850, was to be found in the presence of eddies in the flow [420], an idea
that turned out to be very influential in turbulence theory.
Stokes’ first academic works were devoted to hydrodynamics. In his first paper, in 1842
[450], he studied steady flows and introduced the seminal notion of stability, underlining
the fact that the mathematical possibility of a given motion did not imply its existence if
this motion were unstable. In 1845, Stokes [451] derived his own modelization for elasticity
and hydrodynamics. Studying the variation d⃗u of the velocity, he decomposed
d⃗u =
3
X
∂i ⃗u dxi = (∂i uj )d⃗x
i=1
into
1
1
((∂i uj ) + (∂i uj )T )d⃗x + ((∂i uj ) − (∂i uj )T )d⃗x.
2
2
He identified the antisymmetric part to an infinitesimal rotation; nowadays, the matrix
1
T
⃗ = curl ⃗u, we have 12 ((∂i uj ) −
i uj ) ) is identifiedwith the vorticity: if ω
2 ((∂i uj ) − (∂
0
ω3 −ω2
0
ω1 . The symmetrical part corresponded to a symmetric ten(∂i uj )T ) = 12 −ω3
ω2 −ω1
0
sor, whose principal axes described the infinitesimal deformation axes. Thus, he found back
Cauchy’s tensor ϵ = 12 ((∂i uj ) + (∂i uj )T ) which gives the infinitesimal distortion (d⃗x)T ϵ d⃗x
of the distances.
Stokes then required that the shear pressure be given by a tensor whose axes were
superposed with the axes of the infinitesimal deformation, and whose coefficients were determined as functions of the tensor ϵ. Further, for small velocity gradients, he privileged
a linear relation between the stress tensor and the strain tensor (based on a principle of
superposition11 ). For symmetry (or frame-indifference) reasons, he obtained that the stress
tensor should be a combination of the strain tensor ϵ and of tr(ϵ) I3 . This gives, taking the
⃗
divergence, an internal viscous force expressed as the sum µ∆⃗u + λ∇(div
⃗u). Thus, he found
again the Navier–Stokes equations. Stokes’ derivation is commented by Truesdell [479]:
d⃗u =
Stokes /. . . / derived the same equations as had Poisson, but in doing so he put
the theory on a sound and clear phenomenological basis. As far as the received
theory of fluids with linear viscous response is concerned, this paper was final.
Stokes thoroughly investigated the case of creeping flows, where the velocities are so
⃗ u can be neglected in a first-order approximation. The
small that the advective term ⃗u.∇⃗
equations then become linear, and thus Stokes could give analytical formulas to express the
solutions. Those equations are now labeled as the Stokes equations. He discussed at length
the boundary conditions to impose on the velocity and privileged the no-slip condition,
while Navier and Poisson used a tangential slip condition. Navier was influenced by former
works of Girard on capillar vessels [213]. Stokes disagreed with the conclusions of Navier
and, in 1850, in his memoir on the pendulum [452], where he computed the resistance to the
motion of a sphere through a fluid (Stokes’ law), he explained the physical reasons why the
no-slip condition should rather be privileged. In his book Recherches sur l’hydrodynamique
(1904) [160], Duhem explains the various hesitations of hydrodynamicians between the two
11 This very principle that D’Alembert condemned as “a principle resting only on that vague and obscure
axiom that the effect is proportional to the cause.”
36
The Navier–Stokes Problem in the 21st Century (2nd edition)
kinds of boundary conditions throughout the 19th century, and how the experiments of
Poiseuille (1846) [401], Warburg (1870) [497] and Couette (1890) [132] eventually gave the
advantage to Stokes’ no-slip condition.
3.7
Reynolds
Following the works of Navier and Stokes, a theory had been established that enjoyed
a good experimental validation in the case of laminar flows. However, the Navier–Stokes
equations seemed inappropriate to describe turbulent flows, a major concern for practical
applications.
Even nowadays, turbulence is difficult to define. Roughly speaking, laminar flows move
peacefully and it is easy to follow their streamlines, while turbulent flows are constantly eddying, new vortices being generated from old vortices in a process that drastically increases
energy dissipation, drag forces and heat transfers.
Whirling flows had interested scientists and artists for centuries. In the Renaissance
era, Leonardo sketched many drawings of turbulent flows. Frisch [184] quotes a fragment of
the Codice Atlantico where Leonardo uses the term “turbolenza” to describe the whirling
flows:12
Doue la turbolenza dellacqua rigenera,
doue la turbolenza dellacqua simantiene plugho,
doue la turbolenza dellacqua siposa
Frisch underlines that those lines of Leonardo point exactly to the characteristic features
of turbulence that are at the basis of the modern scientific theories of Richardson and
Kolmogorov.
Later, hydraulicians like Venturi (1797) [488] commented on the retardation effect of
the creation of whirls in the streaming of rivers. Saint-Venant was deeply interested in this
retardation effect of eddies. In 1850 [420], he vindicated that the presence of eddies in the
flow generated an extra internal friction that modified the viscosity of the flow. He viewed
the eddies as local variations around an average value, and thus distinguished two scales:
the average value corresponded to a laminar flow obeying the Navier–Stokes equations,
while smaller structures were oscillating and provoking extra viscosity. This theory of eddy
viscosity was further extended by Boussinesq (1870) [53].
At the same time, the dynamics of vortices had been explored by Helmholtz. In 1858
[233], he studied fluid motion in the presence of dissipative forces. As the forces could not
derive from a potential, the vorticity could not be equal to 0. Helmholtz identified vorticity
with an infinitesimal rotation and wanted to exhibit the dynamics vorticity generated.
12 Codice
Atlantico, Biblioteca Ambrosiana di Milano, f. 74v. In modern Italian, this reads as
dove la turbolenza dell’acqua si genera,
dove la turbolenza dell’acqua si mantiene per lungo,
dove la turbolenza dell’acqua si posa
and in English as
Where the turbulence of water is generated
Where the turbulence of water maintains for long
Where the turbulence of water comes to rest.
History of the Equation
37
For an ideal fluid in a potential field of forces, he defined vortex lines as lines that were
everywhere tangent to the vorticity vector and vortex filaments13 as the union of all vortex
lines crossing a given surface element of the fluid, and he showed that those vortex filaments
were stable structures of the fluid.
He introduced the Helmholtz decomposition of a velocity field into its irrotational part
and its divergence-free part, and showed that the formula that reconstructs a divergence-free
vector field from its curl was analogous to the Biot-Savart law in electromagnetism.
For fluid mechanicians, there were two regimes of flows. The laminar flows were very
regular and obeyed the Navier–Stokes equations. Turbulent flows were very irregular, with
vortices of all scales making impossible to describe the flow except for average values. The
term “turbulent” was coined by Thomson (Lord Kelvin) in 1887 [472] in a paper entitled
On the propagation of laminar motion through a turbulently moving inviscid liquid. Navier
distinguished “linear” flows and “non-linear flows,” Reynolds “direct” flows and “sinuous”
flows [410]. Later, Oseen [385] would call turbulent the blowing up solutions of the Navier–
Stokes equations, and he was followed in this by Leray [328] and Ladyzhenskaya [293].
Experimental investigation of turbulence was initiated by Reynolds in 1883 [410]. The
sudden transition from laminar flows to turbulent flows in pipes has been first decribed by
Hagen in 1839 [227] and 1854 [228]. However, Reynolds was the first to try and understand
the dichotomy between the two regimes of flows:
The internal motion of water assumes one or other of two broadly distinguishable
forms – either the elements of the fluid follow one another along lines of motion
which lead in the most direct manner to their destination, or they eddy about in
sinuous paths the most indirect possible.
Reynolds made a decisive experiment on the visualization of the transition from laminarity
to turbulence. Injecting dye in a moving fluid inside a glass tank, he could visualize the
streak lines and show that when the velocity of the fluid at the entrance of the pipe was
increased the flow began to develop eddies, and, varying the velocities and the pipes, he
could show that the passage from laminar flows to turbulent flows was determined by the
size of a dimensionless number, which is now called the Reynolds number Re and is given
by Re = UνL , where ν is the kinematical viscosity, U is the characteristic velocity of the fluid
and L the characteristic length of the device (such as the radius of the pipe, for instance).
Note that Reynolds’ original apparatus is still used for experiments on turbulence transition in pipe flow (see Eckhardt in 2008 [162] or Mullin in 2011 [371]).
Another important contribution of Reynolds was his analytical study of turbulence,
published in 1895 [411], with the introduction of the decomposition of the flow into mean
and fluctuating parts. Averaging the velocity to get the mean part gives an equation (the
Reynolds equations) on this mean velocity that is not closed, due to the non-linearity of the
⃗ u leads to a correction of the Navier–Stokes
Navier–Stokes equations: the advective term ⃗u.∇⃗
equations due to the interaction with the fluctuating part, that modifies the stress tensor
with the Reynolds stress.
The theory of turbulence would then be developed by the modelization of this Reynolds
stress and by the study of the transfer of kinetic energy from the mean flow to turbulent
parts.
13 Note
that vortex filaments for mathematicians are highly more singular. In fluid mechanics, vortex
1
filaments are a tube of vortex lines with cross sectional radius δ = O(Re− 2 ), where Re is the Reynolds
number. In the vanishing viscosity limit, the tube is reduced to a line.
38
3.8
The Navier–Stokes Problem in the 21st Century (2nd edition)
Oseen, Leray, Hopf and Ladyzhenskaya
Lorentz was awarded the 1902 Nobel prize for his works on the electron and paved the
way to Einstein’s relativity theory by discovering in 1904 the key role of the Lorentz group.
But Lorentz gave as well some important contributions to hydrodynamics. In 1896 [342],
he studied Stokes’ steady creeping flows. Looking at the flow associated to the motion of a
sphere of radius R and velocity c, and letting R go to 0, he obtained the Green function Jν
for the steady Stokes equations
⃗
−ν∆⃗u = F⃗ − ∇p,
div ⃗u = 0
Jν is a second-rank tensor and we have ⃗u(x) = Jν (x − y)F⃗ (y) dy. This Green function is
now called a stokeslet, as proposed in 1953 by Hancok [229] (though Kuiken states that it
should rather be called lorentzlet [285] and it is sometimes called the Oseen tensor).
In 1911, Oseen [384, 385] extended the work of Lorentz to the case of evolutionary Stokes
equations and then to the Navier–Stokes equations. He obtained an explicit tensor Oν (the
Oseen tensor) such that the Stokes equations
R
⃗
∂t ⃗u = ν∆⃗u + F⃗ − ∇p,
have the solution, for positive t,
Z tZ
⃗u(t, x) =
0
div ⃗u = 0,
⃗u(0, x) = 0
Oν (t − s, x − y)F⃗ (s, y) dy ds.
R3
Then, he turned the Navier–Stokes equations
⃗ u + f⃗ − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
(3.3)
into an integro-differential equation
Z
⃗u(t, x) =
Wν (t, x − y)⃗u0 (y) dy
R3
Z tZ
+
0
R3
Oν (t − s, x − y) f⃗(s, y) −
3
X
(3.4)
!
ui (s, y)∂i ⃗u(s, y)
dy ds
i=1
where Wν is the heat kernel associated to the heat equation ∂t G = ν∆G. He was then
able to get a solution for a small positive time interval [0, T ] when the initial data ⃗u0 and
the force f⃗(t, x) were regular and localized. More precisely, he considered the solution ⃗uϵ
associated to the initial value ϵ⃗u0 and the force ϵf⃗, and obtained a power series expansion
of ⃗uϵ with respect to the powers of ϵ (by identification of the coefficients of the expansion,
which were computed inductively as solutions of linear Stokes equations) with a convergence
radius greater than 1; taking ϵ = 1 gives the solution of the Navier–Stokes equations.
In 1934, Leray [328] studied the problem of turbulent solutions that Oseen left open:
when the estimates found by Oseen blow up, what can be saidR of the solutions? Leray
found that one had an estimate that did not blow up: the energy R3 |⃗u(t, x)|2 dx. However,
the control of the L2 norm of ⃗u is not enough to ensure that the solution does not blow
up. Leray introduced a new concept of solutions, that he called turbulent solutions and
History of the Equation
39
are now called weak solutions, for which the derivatives in the differential equations were
no longer classical derivatives, but generalized derivatives (now called derivatives in the
sense of distributions14 ). Two years before Sobolev [443], he thus introduced the space
of functions that are Lebesgue measurable, square-integrable and that have a generalized
square-integrable gradient: this space would later be called the Sobolev space H 1 .
He was then able to prove that Oseen’s classical solutions may be extended to global
turbulent solutions, the loss of control on the size of ⃗u and its derivatives being compensated
by a modification of the meaning of derivatives in the equations. The existence of those
new solutions were proved by compactness arguments, due to the strong development of
topological theory in the beginning of the 20th century. However, the prize to pay for using
those methods (replacing a unique limit by a (possibly non-unique) limit point) was severe:
uniqueness of solutions to the Cauchy initial value problem was no longer granted. The
issue of uniqueness of weak solutions or of globalness of strong solutions has remained open
since Leray’s seminal thesis.
Leray went even further in the use of topological methods, by introducing with Schauder
[331] index methods of algebraic topology to get solutions for functional equations, a method
he could apply to the stationary Navier–Stokes problem in a domain. This theorem was a
major turning point in the resolution of equations, as underlined by Leray [330]:15
Pour nous, résoudre une équation, c’est majorer les inconnues et préciser leur
allure le plus possible; ce n’est pas en construire, par des développements compliqués, une solution dont l’emploi pratique sera presque toujours impossible.
To construct his weak solutions, Leray used the formulas derived by Lorentz and Oseen
for hydrodynamic potentials. Such formula were explicitly known only for very simple domains, and not available for more complex domains. Hopf [238] in 1951 and Ladyzhenskaya
in 1957 [262] used another approach, by approximating the equations on finite-dimensional
subspaces of L2 , which is now widely used in the numerical analysis of the equations [471].
This is the Faedo–Galerkin method, initially introduced by Galerkin for solving elliptic
equations, then extended by Faedo for evolution problems; Hopf’s work was one of the first
applications of this method to non-linear equations.
Ladyzhenskaya developed a full mathematic theory for the use of weak solutions for
partial differential equations, beginning in 1953 with her book on hyperbolic equations
[291]. She described her theory in a review paper on the Clay millenium prize [296]. She
comments on those weak solutions in those terms:
This ideology (program) was partially contained in the formulations of the 19th
and the 20th Hilbert problems. Namely, these Hilbert problems contain the important idea of seeking solutions of variational problems in spaces dictated by the
functional rather than in spaces of smooth functions. After those ‘bad’ solutions
14 Lützen
underlines the influence of Leray on the birth of distribution theory [344] :
Leray used the test function generalization in a third way, namely to generalize the divergence
operator. His consistent use of this generalization method occupies a central position in the
prehistory of the theory of distribution as he taught it to his student L. Schwartz at the Ecole
Normale.
15 See the translation in the review of Mawhin [352]: “For us, to solve an equation consists in bounding
its unknowns and precise their shape as much as possible; it is not to construct, through complicated
developments, a solution whose practical use will be almost always impossible.”
In contrast, see the comments by Devlin [152] about the Clay millenium problem for the Navier–Stokes
equations: “There is just one problem. No one has been able to find a formula that solves the Navier–Stokes
equations. In fact, no one has been able to show in principle whether a solution even exists! (More precisely,
we do not know whether there is a mathematical solution - a formula that satisfies the equations. Nature
“solves” the equations every time a real fluid flows, of course.)”
40
The Navier–Stokes Problem in the 21st Century (2nd edition)
are snared, one can then consider them in more detail, that is, study their actual
smoothness in dependence on the smoothness of the data.
The use of weak solutions in mathematical physics was then unusual, as she recalls:
I still remember years (the 1940s and 50s) when the majority of maı̂tres (and first
and foremost I.G. Petrovskii) regarded a problem as unsolved if on the chosen
path of investigation the researcher did not guarantee the existence of a classical
solution.
3.9
Turbulence Models
All along the 20th century, engineers, physicists and mathematicians paid a great attention to the complexity of flows, focusing on instabilities and on turbulence. There are many
references on the history of turbulence, including the historical introduction to the book of
Monin and Yaglom [367], the book by Frisch [184], the review by Lumley and Yaglom [343]
or the biographical book A voyage through turbulence [146].
Turbulence turned out to be a crucial issue with the development of aviation. Engineers
were mostly interested in understanding the turbulence effects on large scales, as it had
important consequences on drag forces. The description of the large scales involves the
interaction with the boundaries, while in the small scales turbulence has a universal behavior
which does not depend on the geometry.
The main tools for the large-scale description of turbulence were introduced in the
twenties and the thirties by Prandtl and von Kármán with logarithmic velocity profile laws
or logarithmic skin friction laws. The log-law [252] that expresses the profile of the mean
velocity of a turbulent flow bounded by parallell walls in terms of the logarithm of the
normal distance to the walls, was published in 1930 by von Kármán.
Another important contribution to the understanding of turbulence was in 1904 the
boundary layer theory of Prandtl [405], which modelizes the behavior of the fluid when the
viscosity vanishes.
The study of small scales was initiated by Taylor and Richardson. Richardson introduced
in 1922 [413] his model of energy cascade in turbulent flows. To describe the energy transfer
from large scales to small scales, he wrote a parody of a poem of Swift:
Big whirls have little whirls
that feed on their velocity.
Little whirls have lesser whirls
and so on to viscosity
. . . in the molecular sense.16
Let us remark, however, that the model of energy cascade has been criticized recently:
according to Tsinober [484], the non-local aspects of incompressible fluid mechanics (i.e.,
16 Swift’s
poem reads as:
So, naturalists observe, a flea
Hath smaller fleas that on him prey :
And these have smaller fleas to bite ’em
And so proceed ad infinitum.
Thus every poet in his kind,
Is hit by him that comes behind.
History of the Equation
41
the non-local dependence of the pressure on the velocity [Laplace equation] and of velocity
on the vorticiity [Biot-Savard law])
contradict the idea of cascade in physical space, which is local by definition
so that Richardson’s verse should be replaced by Betchov’s [42]:
Big whirls lack smaller whirls
to feed on their velocity.
They crush and form the finest curls
permitted by viscosity.
Homogeneous and isotropic turbulence was introduced in 1935 by Taylor [465] as a
statistical modelization of turbulence. Whereas such a turbulent model cannot be applied to
an actual fluid (because of the presence of boundaries that prevent isotropy), this model can
be applied to the asymptotical behavior of the fluid at small scales and the simplifications
induced by the isotropy hypothesis allows an easier handling of the equations.
In 1941, Kolmogorov [268, 269] developed the analysis of fully developed turbulence
based on a stochastic modelization, through the precise description of random fields he
introduced in [267]. The basis of Kolmogorov’s theory was a modelization of the universal
equilibrium regime of small-scale components. Discrepancies between experimental results
and his theory led Kolmogorov to modify his theory in 1962 in order to take into account
intermittency in the distribution of dissipative structures in turbulent flows [270].
Nowadays, numerical simulation of turbulence provides a large number of quantitative
data to support or disprove the theoric ideas that are formulated on the qualitative behavior
of turbulent flows. The main technique is the direct numerical simulation (DNS), introduced
in 1972 by Orszag and Patterson [382]. At the same time appeared the large eddy simulation
(LES) where only the large scales are numerically resolved, the fine scales being parametrized
(Deardorff (1970) [150], using a numerical model introduced in 1962 by Smagorinsky and
Manabe [441]).
Chapter 4
Classical Solutions
In this chapter, we study classical solutions of the Cauchy initial value problem for the
Navier–Stokes equations (with reduced (unknown) pressure p, reduced force density f⃗ and
kinematic viscosity ν > 0):
Navier–Stokes equations
Given a divergence-free vector field ⃗u0 on R3 and a force f⃗ on (0, +∞) × R3 , find a
positive T and regular functions ⃗u and p on [0, T ] × R3 solutions to
⃗ u + f⃗ − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
(4.1)
We are going to solve Equations (4.1). We shall use only classical tools of differential
calculus, as they were used in the end of the 19th century or the beginning of the 20th
century. More precisely, we will stick to the spirit of Oseen’s paper, which was published in
1911 [384, 385]. A similar treatment can be found in a 1966 paper of Knightly [263].
4.1
The Heat Kernel
The heat kernel Wt is the function
Wt (x) =
1
2
x
1
− x4t
√
W
(
)
=
e
t3/2
(4πt)3/2
t
(4.2)
This kernel is used to solve the heat equation:
Heat equation
Theorem 4.1.
Let u0 be a bounded continuous function on R3 . Then the function
Z
√
u(t, x) = u0 (x − ty)W (y) dy
DOI: 10.1201/9781003042594-4
(4.3)
42
Classical Solutions
43
is continuous on [0, +∞)×R3 and C ∞ on (0, +∞)×R3 and solution of the heat equation
∂t u = ∆u
u(0, x) = u0 (x)
on (0, +∞) × R3
(4.4)
on R3
Proof. From the formula (4.3), we see that u is continuous on [0, +∞) × R3 and that
u(0, .) = u0 . From the equality, for t > 0,
Z
u(t, x) = Wt ∗ u0 = Wt (x − y)u(y) dy,
we see that u is C ∞ on (0, +∞) × R3 , since all the derivatives of Wt have exponential decay
in the space variable.
We have (∂t − ∆)u = u ∗ (∂t − ∆)Wt = 0 since ∂i W (x) = − x2i W (x), hence
∂t
3
x
1 3 X x2i
x
√
W
(
)
=
−
(
−
)W ( √ )
3/2
5/2
2 i=1 4t
t
t
t
t
1
and
∆
3
3
X
x
1
x
1 3 X x2i
x
√
√
W
(
)
=
−
∂
x
W
(
)
=
−
(
−
)W ( √ ).
i
i
3/2
5/2
5/2
2 i=1 4t
t
2t
t
t
t
t
i=1
1
The theorem is proved.
We may consider as well classical solutions of the non-homogeneous heat equations,
where we add a forcing term f :
Theorem 4.2.
Let f be a continuous function on [0, +∞) × R3 , which is C 1 in the space variable on
(0, +∞)×R3 and is uniformly bounded with uniformly bounded spatial derivatives. Then
the function
Z tZ
√
F (t, x) =
f (s, x − t − s y)W (y) dy ds
(4.5)
0
3
is continuous on [0, +∞) × R , C 1 on (0, +∞) × R3 and is C 2 in the space variable on
(0, +∞) × R3 . Moreover, F is solution of the heat equation
∂t F = ∆F + f
F (0, x) = 0
on (0, +∞) × R3
(4.6)
on R3
Proof. Just write that
Z tZ
∂i f (s, x −
∂i ∂j F (t, x) =
0
√
t − s y)∂j W (y) dy √
ds
.
t−s
44
The Navier–Stokes Problem in the 21st Century (2nd edition)
Remark: If ⃗u0 is C 2 with bounded derivatives, the solutions u and F given in Theorems
4.1 and 4.2 are clearly unique in the class of continuous functions on [0, +∞) × R3 with the
property that, for all T > 0,
sup
0<t<T,x∈R3 ,|α|≤2
|∂xα u(t, x)| + |∂xα F (t, x)| < +∞.
By linearity, we may assume that u0 = 0 or f = 0 and prove that u or F is 0. This is
obvious: if ∂t u = ∆u and u(0, .) = 0, we write
Z
Z
Z
d
2 −|x|
−|x|
|u(t, x)| e
dx = 2 u ∂t u e
dx = 2 u ∆u e−|x| dx
dt
Z
Z
3
X
xj
⃗ 2 e−|x| dx + 2 ue−|x|
∂j u dx
= −2 |∇u|
|x|
j=1
Z
1
|u(t, x)|2 e−|x| dx
≤
2
R
t R
so that |u(t, x)|2 e−|x| dx ≤ e 2 |u(0, x)|2 e−|x| dx = 0.1
If we consider now a heat equation with a viscosity ν > 0, we have to modify very
slightly the formula: the solution u of the equation
∂t u = ν∆u + f
on (0, +∞) × R3
u(0, x) = u0 (x)
on R3
(4.7)
is given by
Z
u = Wνt ∗ u0 +
t
Wν(t−s) ∗ f (s, .) ds.
(4.8)
0
4.2
The Poisson Equation
We now solve the Poisson equation with help of the Green function
G(x) =
1
4π|x|
p
where |x| = x21 + x22 + x23 .
We begin with an easy lemma:
Lemma 4.1.
Let 0 < γ < 3 and let
Z
Iγ (x) =
1
1
dy.
(1 + |x − y|)4 |y|γ
We have
Iγ (x) ≤ Cγ (1 + |x|)−γ .
1 Actually, according to Tychonov [485], we even have uniqueness in the class of functions that grow no
2
faster than O(eC|x| ), but uniqueness fails in the class of smooth functions with faster increase.
Classical Solutions
45
Proof. We first write:
Z
Iγ (x) ≤
|y|≤1
1
dy +
|y|γ
Z
|y|≥1
1
dy
(1 + |x − y|)4
so that Iγ ∈ L∞ . We then write
Z
Z
2γ
1
1
1
Iγ (x) ≤
dy +
3 |y|γ dy
|x| |x|γ (1 + |x − y|)4
|x|
|y|≥ 2
|x−y|≥ 2 |x − y|
which gives |x|γ Iγ ∈ L∞ .
Corollary 4.1. If 0 < γ < 3, then, for t > 0, we have
Z
1
1
Wt (x − y) γ dy ≤ Cγ √
|y|
( t + |x|)γ
and
Z
Wt (x − y)
1
1
dy ≤ Cγ
γ
(1 + |y|)
(1 + |x|)γ
where the constant Cγ does not depend on t nor x.
Similarly, we have
Z
1
1
1
1
√
dy ≤ C √
( t + |x − y|)4 (1 + |y|)γ
t (1 + |x|)γ
and
Z
1
1
1
1
√
dy ≤ C √ √
4
γ
( t + |x − y|) ( t + |y|)
t ( t + |x|)γ
√
√
t
Proof. Let us notice that Wt (x − y) ≤ C (√t+|x−y|)
4 . We have
Z
Z
Z
1
1
1
1
dz
√
√
√
dy
≤
dy
=
.
γ
4
4
(1 + |z|)4
( t + |x − y|) (1 + |y|)
( t + |x − y|)
t
√
On the other hand, letting y = tz, we get
Z
Z
1
1
1
1
1
√
√
dy
=
dz
x
γ
4
4
1+γ
√
(1 + | t − z|) |z|γ
( t + |x − y|) |y|
( t)
and we conclude by Lemma 4.1.
The Poisson equation
Theorem 4.3.
Let u0 be a C 1 function on R3 such that
sup sup (1 + |x|)4 |∂ α u0 (x)| < +∞.
|α|≤1 x∈R3
Then the function
Z
U (x) =
u0 (x − y) G(y) dy
(4.9)
46
The Navier–Stokes Problem in the 21st Century (2nd edition)
is C 2 on R3 and solution of the Poisson equation
∆U = −u0
(4.10)
Moreover, we have
sup (1 + |x|)|U (x)| < +∞
x∈R3
and
sup
sup (1 + |x|)2 |∂ α U (x)| < +∞.
1≤|α|≤2 x∈R3
x
j
Proof. Let Gj (x) = − 4π|x|
3 . We have ∂j U = u0 ∗ Gj and ∂i ∂j U = ∂i u0 ∗ Gj . Thus, we
control U and its derivatives by
Z
1
1
|U (x)| ≤ C
4 |y| dy
(1 + |x − y| )
and, for 1 ≤ |α| ≤ 2,
Z
1
dy.
(1 + |x − y| ) |y|2
Lemma 4.1 then gives the control of the sizes of U and of its derivatives.
In particular, we have
Z
3
X
∆U (x) =
lim
∂j u0 (x − y)∂j G(y) dy
|∂ α U (x)| ≤ C
ϵ→0,R→+∞
1
4
ϵ<|y|<R j=1
so that, by Ostrogradski’s divergence theorem, we have
Z
3
X
∆U (x) =
lim
ϵ2
u0 (x − ϵσ)
σj ∂j G(ϵσ) dσ
ϵ→0,R→+∞
S2
− R2
j=1
Z
u0 (x − Rσ)
S2
Z
+
3
X
σj ∂j G(Rσ) dσ
j=1
u0 (x − y)∆G(y) dy .
ϵ<|y|<R
P3
1
σj ∂j G(Rσ) = − 4πR
2 , we get that
Z
1
∆U (x) =
lim
−
u0 (x − ϵσ) − u0 (x − Rσ) dσ
ϵ→0,R→+∞
4π S 2
Since ∆G = 0 on R3 − {0}, and
j=1
so that ∆U (x) = −u0 (x).
4.3
The Helmholtz Decomposition
A direct consequence of Theorem 4.3 is the following decomposition theorem of vector
fields:
Classical Solutions
47
The Helmholtz decomposition
Theorem 4.4.
Let F⃗0 be a C 1 vector field on R3 such that
sup sup (1 + |x|)4 |∂ α F⃗0 (x)| < +∞
|α|≤1 x∈R3
and assume that the divergence of F⃗0 is C 1 with
sup sup (1 + |x|)4 |∂ α div F⃗0 (x)| < +∞.
|α|=1 x∈R3
⃗ such that:
Then there exists unique C 1 vector fields F⃗ and H
⃗
• F⃗0 = F⃗ + H
• F⃗ is solenoidal (i.e., divergence free): div F⃗ = 0
⃗ is irrotational: curl H
⃗ =0
• H
• sup0≤|α|≤1 supx∈R3 (1 + |x|)2 |∂ α F⃗ (x)| < +∞.
⃗
• sup0≤|α|≤1 supx∈R3 (1 + |x|)2 |∂ α H(x)|
< +∞.
Moreover, there exists a unique function p which is C 2 on R3 with supx∈R3 (1 +
|x|)|p(x)| < +∞ and
⃗ = ∇p.
⃗
H
Proof. We begin by proving the uniqueness. Let us assume that we have two solutions
⃗ 1 ) and (F⃗2 , H
⃗ 2 ). Since H
⃗2 − H
⃗ 1 is irrotational and C 1 , one can find a C 2 function q
(F⃗1 , H
⃗2 − H
⃗ 1 = ∇q.
⃗ We then have
such that H
⃗2 − H
⃗ 1 ) = div(F⃗1 − F⃗2 ) = 0.
∆q = div(H
Thus q is harmonic; its derivatives are harmonic functions that are O(|x|−2 ) at infinity, hence
⃗2 = H
⃗ 1 , and F⃗2 = F⃗1 .
its derivatives are equal to 0 by the maximum principle. Thus, H
⃗ as H
⃗ = ∇p,
⃗
For proving the existence, it is enough to use Theorem 4.3 and to define H
where p solves the Poisson equation ∆p = div F⃗0 .
Definition 4.1 (Leray projection operator).
⃗ into the sum of
For a regular vector field F⃗0 and its Helmholtz decomposition F⃗0 = F⃗ + H
⃗
⃗
a solenoidal vector field F and an irrotational vector field H, we shall write
F⃗ = PF⃗0 .
The operator P : F⃗0 7→ F⃗ is called the Leray projection operator.
We may define F⃗ = PF⃗0 as the unique solution of the Poisson equation
⃗ ∧ (∇
⃗ ∧ F⃗0 )
−∆F⃗ = ∇
which is o(1) at infinity.
48
4.4
The Navier–Stokes Problem in the 21st Century (2nd edition)
The Stokes Equation
We have gathered enough results to be able to solve the Stokes equations, i.e., the
Navier–Stokes equations when the convective bilinear term is neglected.
The Stokes problem
Given a divergence-free vector field ⃗u0 on R3 and a force f⃗ on [0, +∞) × R3 , find
regular functions ⃗u and p on (0, +∞) × R3 solutions to
⃗
∂t ⃗u = ν∆⃗u + f⃗ − ∇p
div ⃗u = 0
(4.11)
⃗u|t=0 = ⃗u0
The solution of this problem is easy: we use the Helmholtz decomposition of f⃗ into
⃗ where F⃗ is divergence free and H
⃗ = ∇q
⃗ is irrotational. Then the Helmholtz
f⃗ = F⃗ + H,
decomposition of ∂t ⃗u will give
⃗ − ∇p.
⃗
∂t ⃗u = ν∆⃗u + F⃗ and 0 = ∇q
Thus, we know that p is determined through the Poisson equation ∆p = div f⃗, while ⃗u is a
solution of the heat equation with initial value ⃗u0 and forcing term F⃗ = Pf⃗. We thus use
Theorem 4.1, Theorem 4.2 and Theorem 4.4 and get the following result:
The Stokes equation
Theorem 4.5.
Let ⃗u0 be a C 2 divergence-free vector field on R3 and let f⃗ be a time-dependent vector
field such that:
• sup|α|≤2 supx∈R3 (1 + |x|)2 |∂ α ⃗u0 (x)| < +∞
• for |α| ≤ 1, ∂xα f⃗ is continuous on [0, +∞) × R3
• sup|α|≤1 supt≥0,x∈R3 (1 + |x|)4 |∂xα f⃗(t, x)| < +∞
• we have furthermore a control on the derivatives of div f⃗:
sup
sup (1 + |x|)4 |∂xα div f⃗(t, x)| < +∞.
|α|=1 t>0,x∈R3
Then, there exists a unique solution (⃗u, p) of the Stokes problem
⃗
∂t ⃗u = ν∆⃗u + f⃗ − ∇p
div ⃗u = 0
⃗u|t=0 = ⃗u0
Classical Solutions
49
such that:
• for |α| ≤ 2, ∂xα p is continuous on [0, +∞) × R3
• supt≥0,x∈R3 (1 + |x|) |p(t, x)| < +∞
• sup1≤|α|≤2 supt≥0,x∈R3 (1 + |x|)2 |∂xα p(t, x)| < +∞
• for |α| ≤ 2, ∂xα ⃗u is continuous on [0, +∞) × R3 and, for every 0 < T < +∞,
sup
sup
|α|≤2 0<t<T, x∈R3
(1 + |x|)2 |∂xα ⃗u(t, x)| < +∞
• ∂t ⃗u is continuous on [0, +∞) × R3 .
Proof. We write
∆p = div f⃗
and use Theorem 4.3 to have a control on p and its derivatives:
X
sup (1 + |x|) |p(t, x)|+
(1 + |x|)2 |∂xα p(t, x)|
t≥0,x∈R3
1≤|α|≤2
≤C
X
|α|≤1
sup (1 + |x|)4 |∂xα div f⃗(t, x)|.
t>0,x∈R3
⃗ and we write
Tnen, we let F⃗ = f⃗ − ∇p
Z t
⃗u = Wνt ∗ ⃗u0 +
Wν(t−s) ∗ F⃗ (s, .) ds = ⃗u1 + ⃗u2 .
0
For |α ≤ 2, we have
|∂xα (Wνt ∗ ⃗u0 )| ≤ sup (1 + |y|)2 |∂xα ⃗u0 (y)|
Z
Wνt (x − y)
y∈R3
1
dy
(1 + |y|)2
and Corollary 4.1 gives us the control of the size of ∂xα ⃗u1 . Similarly, we have, for |α| ≤ 1,
Z
1
α
2 α⃗
⃗
|∂x (Wν(t−s) ∗ F (s, .))| ≤ sup (1 + |y|) |∂x F (s, y)| Wν(t−s) (x − y)
dy
3
(1
+
|y|)2
y∈R
and thus
|∂xα ⃗u1 (t, x)| ≤ C
t
(1 + |x|)2
sup
0<s<t,y∈R3
(1 + |y|)2 |∂xα F⃗ (s, y)|.
For |α| = 2, we write ∂xα = ∂i ∂j and we remark that
|∂i (Wν(t−s) (x))| ≤ C p
1
ν(t − s)
Wν (t−s) (x)
2
and get
C
|∂i ∂j (Wν(t−s) ∗ F⃗ (s, .))| ≤ p
sup (1 + |y|)2 |∂j F⃗ (s, y)|
ν(t − s) y∈R3
and finally
Z
Wν t−s (x − y)
2
√
|∂i ∂j ⃗u1 (t, x)| ≤ C √
t
ν(1 + |x|)2
sup
0<s<t,y∈R3
(1 + |y|)2 |∂j F⃗ (s, y)|.
dy
(1 + |y|)2
50
The Navier–Stokes Problem in the 21st Century (2nd edition)
4.5
The Oseen Tensor
In Theorem 4.5, the solution (⃗u, p) is given by
p(t, x) = −
3
X
1
∗ div f⃗(t, x) = −
fj (t, x) ∗ ∂j G
4π|x|
j=1
and
t
Z
⃗
Wν(t−s) ∗ (f⃗(s, x) − ∇p(s,
x)) ds
⃗u = Wνt ∗ ⃗u0 +
0
Thus, the k-th component of ⃗u is given by
Z t
3
X
uk = Wνt ∗ u0,k +
Wν(t−s) ∗ (fk + ∂k
∂j G ∗ fj ) ds
0
j=1
We have
Wν(t−s) ∗ (fk + ∂k
3
X
∂j G ∗ fj ) =
j=1
3
X
fj ∗ (δj,k Wν(t−s) + G ∗ ∂j ∂k Wν(t−s) ).
j=1
Definition 4.2 (Oseen tensor).
The Oseen tensor is the tensor (Oj,k (νt, x))1≤j,k≤3 given by
Oj,k (νt, x) = δj,k Wνt + G ∗ ∂j ∂k Wνt
Let Oj,k (x) = Oj,k (1, x); it is easy to see that
Oj,k (νt, x) =
x
1
Oj,k ( √ ).
(νt)3/2
νt
The functions Oj,k are easily determined through Oseen’s formula2 :
The Oseen tensor
Theorem 4.6.
We have
Oj,k (x) = δj,k W (x) + 2∂j ∂k
Z
1
(4π)3/2 |x|
|x|
!
2
e
− s4
ds .
(4.12)
0
When x is close to 0, it is more convenient to write
Z 1
Oj,k (x) = δj,k W (x) + 2∂j ∂k
W (θx) dθ
(4.13)
0
and when x is close to infinity, it is more convenient to write
1
Oj,k (x) =∂j ∂k
+ δj,k W (x)
4π|x|
!
Z ∞
s2
1
− 2∂j ∂k
e− 4 ds .
(4π)3/2 |x| |x|
(4.14)
2 Lerner has recently given an explicit expression of those kernels that involve the incomplete gamma
function and the confluent hypergeometric functions of the first kind [332].
Classical Solutions
51
Proof. We may write
Oj,k (x) = δj,k W (x) + ∂j ∂k (G ∗ W ).
The function G ∗ W = Φ is radial: Φ(x) = F (|x|), and satisfies −∆Φ = W (x) =
|x|
1
e− 4
(4π)3/2
2
= H(|x|). Thus, we must have F ′′ (r) + 2r F ′ (r) = −H(r), hence (rF )′′ =
Rr
−rH(r) = 2H ′ (r), and finally F (r) = Ar + B + 2r 0 H(s) ds. Since Φ is bounded near 0 and
vanishes at infinity, we find that A = B = 0.
Remark: This proof has been taken in Oseen’s book [385]. Another simple proof is to write
that
Z +∞
Z t
Wu du
∆Wu du =
G ∗ W = lim G ∗ W1 − G ∗ Wt = lim −G ∗
t→+∞
t→+∞
and thus
G ∗ W (x) =
Writing u =
2
|x|
s2
and thus
du
u3/2
1
(4π)3/2
+∞
Z
1
1
e−
|x|2
4u
1
du
.
u3/2
2
3
s |x|
= −2 |x|
3 s3 ds, one gets
G ∗ W (x) =
2
(4π)3/2
Z
|x|
0
e−
s2
4
ds
.
|x|
Theorem 4.6 allows precise estimates on the derivatives of Oj,k :
Corollary 4.2.
Oj,k is C ∞ and satisfies:
• for all α ∈ N3 , |∂xα Oj,k (x)| ≤ Cα (1 + |x|)−3−|α|
• for all α ∈ N3 and |x| > 1, |∂xα (Oj,k (x) − ∂j ∂k G(x)) | ≤ Cα e−
where G is the Green function G(x) =
4.6
x2
8
1
4π|x| .
Classical Solutions for the Navier–Stokes Problem
Interpreting the Navier–Stokes equations with given force f⃗ as a Stokes equation with
⃗ u allows one to turn the differential equations (4.1) into an integrogiven force f⃗ − (⃗u.∇)⃗
differential equation:
Integro-differential Navier–Stokes equations
Given a divergence-free vector field ⃗u0 on R3 and a force f⃗ on (0, +∞) × R3 , find a
positive T and regular functions ⃗u and p on [0, T ] × R3 solutions to
uk = Wνt ∗ u0,k +
Z tX
3
0 j=1
⃗ j ds
Oj,k (ν(t − s), .) ∗ fj − (⃗u.∇)u
(4.15)
52
The Navier–Stokes Problem in the 21st Century (2nd edition)
for k = 1, . . . , 3, and
3 X
p(t, x) = −
⃗ j ∗ ∂j G
fj − (⃗u.∇)u
(4.16)
j=1
Let us write O(ν(t − s)) :: f⃗ for the vector ⃗g = O(ν(t − s)) :: f⃗ with components
P3
gk = j=1 Oj,k (ν(t − s)) ∗ fj , we have to solve the quadratic equation
t
Z
⃗ u) ds
O(ν(t − s)) :: (⃗u.∇)⃗
⃗u = ⃗v0 −
(4.17)
0
with
Z
⃗v0 = Wνt ∗ ⃗u0 +
t
O(ν(t − s)) :: f⃗ ds.
(4.18)
0
Oseen’s idea is to solve the same equation with an extra parameter ϵ:
Z t
⃗ uϵ ds
⃗uϵ = ⃗v0 − ϵ
O(ν(t − s)) :: (⃗uϵ .∇)⃗
(4.19)
0
and to develop the solution ⃗uϵ as a power series in ϵ:
⃗uϵ =
∞
X
ϵn⃗vn .
n=0
We get a cascade of equalities (which amounts to solve a cascade of Stokes equations)
⃗vn+1 = −
n Z
X
k=0
t
⃗ vn−k ) ds.
O(ν(t − s)) :: (⃗vk .∇)⃗
0
We have
⃗ 0
∂t⃗v0 = ν∆⃗v0 + f⃗ − ∇q
(4.20)
div ⃗v0 = 0
⃗v0 (0, .) = ⃗u0
and
∂t⃗vn+1 = ν∆⃗vn+1 −
n
X
⃗ vn−k − ∇q
⃗ n+1
(⃗vk .∇)⃗
k=0
(4.21)
div ⃗vn+1 = 0
⃗vn+1 (0, .) = 0
Thus, in order to find a classical solution for the Navier–Stokes equations (4.1), it will be
enough to find a positive time T such that
∞
X
n=0
sup
0≤t≤T,x∈R3 ,|α|≤2
|∂xα⃗vn (t, x)| < +∞ and
∞
X
n=0
sup
0≤t≤T,x∈R3 ,|α|≤1
|∂xα qn (t, x)| < +∞
Classical Solutions
53
– note that, as well, we will get
∞
X
|∂t⃗vn (t, x)| < +∞.
sup
n=0 0≤t≤T,x∈R
3
This can be easily done under the assumptions of the Millennium problem:
Navier–Stokes equations
Theorem 4.7.
Let ⃗u0 be a C 2 divergence-free vector field on R3 and let f⃗ be a time-dependent vector
field such that:
• sup|α|≤2 supx∈R3 (1 + |x|)2 |∂xα ⃗u0 (x)| < +∞
• for |α| ≤ 2, ∂xα f⃗ is continuous on [0, +∞) × R3
• sup|α|≤2 supt≥0,x∈R3 (1 + |x|)4 |∂xα f⃗(t, x)| < +∞.
Then, there exists a positive time T and a unique solution (⃗u, p) of the Navier–Stokes
problem
⃗ u − ∇p
⃗ on (0, T ) × R3
∂t ⃗u = ν∆⃗u + f⃗ − (⃗u.∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
such that:
• for |α| ≤ 2, ∂xα p and ∂xα ⃗u are continuous on [0, T ] × R3
• sup0≤t≤T,x∈R3 (1 + |x|)|p(t, x)| < +∞.
• sup1≤|α|≤2 sup0≤t≤T,x∈R3 (1 + |x|)2 |∂xα p(t, x)| < +∞.
• sup|α|≤2 sup0≤t≤T,x∈R3 (1 + |x|)2 |∂xα ⃗u(t, x)| < +∞.
• ∂t ⃗u is continuous on [0, T ] × R3 .
Proof. We are going to estimate the size of the vector fields ⃗vn given by the equations (4.20)
and (4.21). First of all, we rewrite the “forces” in those equations in a divergence form: for
n ≥ 0, we have
∂t⃗vn = ν∆⃗vn +
3
X
⃗ n,
∂j ⃗gj,n − ∇q
div ⃗vn = 0,
⃗vn (0, .) = δn,0 ⃗u0
(4.22)
j=1
with
⃗gj,0 = −f⃗ ∗ ∂j G
and
⃗gj,n+1 = −
n
X
k=0
vj,k ⃗vn−k .
(4.23)
(4.24)
54
The Navier–Stokes Problem in the 21st Century (2nd edition)
This gives
⃗vn = δn,0 Wνt ∗ ⃗u0 +
3 Z
X
t
∂j O(ν(t − s)) :: ⃗gj,n ds.
(4.25)
0
j=1
We thus get, for |α| ≤ 2 and 0 ≤ t ≤ T ,
|∂xα⃗vn (t, x)|
1
|
(1 + |x|)2
! ZZ
t
2 α
|(1 + |y|) ∂x ⃗gj,n (s, y)|
|∂j O(ν(t − s), x − y)|
≤δn,0 sup (1 + |y|)2 |∂xα ⃗u0 (y)||Wνt ∗
y∈R3
+
3
X
j=1
sup
0≤s≤T,y∈R3
0
ds dy .
(1 + |y|)2
From Theorem 4.6 (and Corollary 4.2), we have the estimate
1
|∂j O(ν(t − s), x − y)| ≤ C p
( ν(t − s) + |x − y|)4
and thus, from Corollary 4.1,
(1 + |x|)2 |∂xα⃗vn (t, x)|

r
≤ C0 δn,0 sup (1 + |y|)2 |∂xα ⃗u0 (y)| +
y∈R3

3
X
T
sup
|(1 + |y|)2 ∂xα⃗gj,n (s, y)|
ν j=1 0≤s≤T,y∈R3
where the constant C0 does not depend on T nor ν.
We are now going to estimate inductively the size of ⃗vn and of its derivatives. Let us
define
Zn (T ) = sup
sup
(1 + |x|)2 |∂xα⃗vn (t, x)|.
|α|≤2 0<t<T,x∈R3
ˆ From Lemma 4.1, we know that
sup (1 + |x|)2 |∂xα⃗gj,0 (t, x)| ≤ C1
t>0,x∈R3
sup (1 + |x|)4 |∂xα f⃗(t, x)|.
t>0,x∈R3
ˆ Thus, we know that we have
r
Z0 (T ) = sup
sup
2
(1 + |x|)
|α|≤2 0<t<T,x∈R3
|∂xα⃗v0 (t, x)|
≤ C0 (A0 + C1
T
B0 )
ν
with
A0 = sup sup (1 + |x|)2 |∂xα ⃗u0 (x)| and B0 = sup
|α|≤2 x∈R3
sup (1 + |x|)4 |∂xα f⃗(t, x)|.
|α|≤2 0<t,x∈R3
ˆ From equality (4.24), we find (through Leinitz’s rule on derivatives) that
sup
sup
|α|≤2 0<t<T,x∈R3
(1 + |x|)4 |∂xα⃗gj,n+1 (t, x)| ≤ 4
n
X
Zk (T )Zn−k (T ).
k=0
Thus, we get that
r
Zn+1 ≤ 12C0
n
T X
Zk (T )Zn−k (T ).
ν
k=0
Classical Solutions
55
ˆ We shall prove that there exists a constant C2 (which does not depend on T nor ν)
such that we have, for every T > 0 and every n ∈ N,
q
q
n
r
C2 Tν C0 (A0 + C1 Tν B0 )
T
Zn (T ) ≤
C0 (A0 + C1
B0 )
4
(1 + n)
ν
This inequality is true when n = 0. Assume that it is true kor n = 0, . . . , N ; we are
going to prove that it is true for n = N + 1. We have
r
r
r
n
2
T X
T
T
ZN +1 (T ) ≤ 12C0
Zk (T )Zn−k (T ) ≤ 12C0
DN +1 C0 (A0 + C1
B0 )
ν
ν
ν
k=0
with
DN +1 =
N
C2
X
q
T
ν
C0 (A0 + C1
q
T
ν
q
q
k
N −k
B0 )
C2 Tν C0 (A0 + C1 Tν B0 )
(1 + k)4
(1 + (N − k))4
r
r
N
N X
T
T
1
1
= C2
C0 (A0 + C1
B0 )
4
ν
ν
(1 + k) (1 + (N − k))4
k=0
r
r
+∞
X
N
T
T
32
1
≤ C2
C0 (A0 + C1
B0 )
.
ν
ν
(N + 2)4
k4
k=0
k=1
Thus, writing C3 = 32
P+∞
1
k=1 k4 ,
q
ZN +1 (T ) ≤
12C0 C3 C2N
we get
T
ν
C0 (A0 + C1
(N +
q
2)4
T
ν
N +1
B0 )
r
C0 (A0 + C1
T
B0 )
ν
The proof of the induction is over, if we choose C2 = 12C0 C3 .
ˆ Size of ∂xα qn : in order to estimate qn and its derivatives, we just write
3
n X
3 X
3
X
X
∆q0 = div f⃗ and ∆qn+1 = div(
∂j ⃗gj,n+1 ) = −
∂i vj,k ∂j vi,n−k
j=1
k=0 j=1 i=1
and we use Theorem 4.3.
If we fix T small enough to have
r
r
T
T
C2
C0 (A0 + C1
B0 ) ≤ 1,
(4.26)
ν
ν
P+∞
P+∞
we get the normal convergence of n=0 ∂xα⃗vn and of n=0 ∂xα qn , which proves the existence
of a solution on (0, T ).
Uniqueness follows the same lines: if ⃗u and ⃗v are two solutions of the Cauchy problem
for the Navier–Stokes equations on (0, T ) × R3 (with associated pressures p and q) which
fulfill the conclusions of Theorem 4.7, then we find that w
⃗ = ⃗u − w
⃗ is solution of
∂t w
⃗ = ν∆w
⃗−
3
X
j=1
⃗ − q), div w
∂j (uj w
⃗ + wj ⃗v ) − ∇(p
⃗ = 0, w(0,
⃗ .) = 0.
56
The Navier–Stokes Problem in the 21st Century (2nd edition)
In particular, for every S ∈ (0, T ),
(1 + |x|)2 |w(t,
⃗ x)|
sup
0≤t≤S,x∈R3
r
≤ 3C0
S
( sup ∥⃗u(s, .)∥∞ + ∥⃗v (s, .)∥∞ )
sup
(1 + |x|)2 |w(t,
⃗ x)|.
ν 0<s<T
0≤t≤S,x∈R3
q
If S is small enough (so that 3C0 Sν (sup0<s<T ∥⃗u(s, .)∥∞ + ∥⃗v (s, .)∥∞ ) < 1), we find that
⃗u = ⃗v on [0, S]. Then by reiteration from S to 2S and so on, we find that ⃗u = ⃗v on [0, T ].
We shall define regular data and classical solutions as the data that fulfill the assumptions
of Theorem 4.7 and solutions that fulfill its conclusions:
Regular data
Definition 4.3.
Regular data for the initial-value problem for the Navier–Stokes equations on (0, +∞)×
R3 are a C 2 divergence-free vector field ⃗u0 on R3 and a time-dependent vector field
f⃗(t, x) on [0, +∞) × R3 such that:
• sup|α|≤2 supx∈R3 (1 + |x|)2 |∂ α ⃗u0 (x)| < +∞
• for |α| ≤ 2, ∂xα f⃗ is continuous on [0, +∞) × R3
• for every 0 < T , sup
sup
(1 + |x|)4 |∂ α f⃗(t, x)| < +∞.
|α|≤2 0≤t≤T,x∈R3
Classical solutions
Definition 4.4.
For 0 < T1 , a classical solution of the Navier–Stokes problem
⃗ u − ∇p
⃗ on (0, T1 ) × R3
∂t ⃗u = ν∆⃗u + f⃗ − (⃗u.∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
associated to regular data (⃗u0 , f⃗) is a solution (⃗u, p) such that, for every 0 < T < T1 ,
• for |α| ≤ 2, ∂xα p and ∂xα ⃗u are continuous on [0, T ] × R3
• sup0≤t≤T,x∈R3 (1 + |x|)|p(t, x)| < +∞.
• sup1≤|α|≤2 sup0≤t≤T,x∈R3 (1 + |x|)2 |∂xα p(t, x)| < +∞.
• sup|α|≤2 sup0≤t≤T,x∈R3 (1 + |x|)2 |∂xα ⃗u(t, x)| < +∞.
Classical Solutions
4.7
57
Maximal Classical Solutions and Estimates in L∞ Norms
Let (⃗u0 , f⃗) be regular data for the Cauchy problem for Navier–Stokes equations. Let
T0 > 0. We define f⃗T0 (t, x) = f⃗(min(t, T0 ), x), so that f⃗T0 fulfills the assumptions of Theorem
4.7 (uniform in time control on [0, +∞)). The Cauchy problem on (0, T0 )×R3 for data (⃗u0 , f⃗)
or (⃗u0 , f⃗T0 ) coincide. In particular, from inequality (4.26), we know that we have existence
of a solution on (0, T ) with T ≤ T0 as soon as T fulfills the conditions

T ≤T0




1

T ≤ ν
2
2
4C2 C0 (sup|α|≤2 supx∈R3 (1 + |x|)2 |∂xα ⃗u0 (x)|)2
.



ν
1


T ≤
2C2 C0 C1 sup|α|≤2 sup0<t<T ,x∈R3 (1 + |x|)4 |∂xα f⃗(t, x)|
0
Thus, we have local existence of a solution. Moreover, we know that we have uniqueness.
We may then conclude that we have a unique maximal solution:
Maximal classical solution
Proposition 4.1.
Let (⃗u0 , f⃗) be regular data. Let TMAX be the maximal time where one can find a classical
solution ⃗u of the Cauchy problem for the Navier–Stokes equations on (0, TMAX ) × R3 .
If TMAX < +∞, then
lim
sup sup (1 + |x|)2 |∂xα ⃗u(t, x)| = +∞.
t→TMAX |α|≤2 x∈R3
Proof. Assume that TMAX < +∞. Let 0 < T1 < TMAX < T0 < +∞. We have a solution for
the Cauchy problem at time T1 which is defined on [T1 , T1 + T ] × R3 , for T satisfying
T ≤ T0 − T1
T ≤ T2 =
ν
2C2 C0 C1 sup|α|≤2 sup0<t<T
1
0 ,x∈R
and
T ≤
3
(1 + |x|)4 |∂xα f⃗(t, x)|
ν
1
.
4C22 C02 (sup|α|≤2 supx∈R3 (1 + |x|)2 |∂xα ⃗u(T1 , x)|)2
We must have
T + T1 ≤ TMAX
hence, if T1 > TMAX − T2 , we find that
ν
1
≤ TMAX − T1 .
4C22 C02 (sup|α|≤2 supx∈R3 (1 + |x|)2 |∂xα ⃗u(T1 , x)|)2
58
The Navier–Stokes Problem in the 21st Century (2nd edition)
Actually, it is enough to control the L∞ norm of ⃗u:
Theorem 4.8.
Let (⃗u0 , f⃗) be regular data. Let TMAX be the maximal time where one can find a classical
solution ⃗u of the Cauchy problem for the Navier–Stokes equations on (0, TMAX ) × R3 .
If TMAX < +∞, then
|⃗u(t, x)| = +∞.
sup
0≤t<TMAX , x∈R3
Proof. For 0 ≤ t < TMAX , let us define, for 0 ≤ k ≤ 2,
Ak (t) = sup sup (1 + |x|)2 |∂xα ⃗u(t, x)|
|α|=k x∈R3
B0 (t) = sup |⃗u(t, x)|
x∈R3
and
Fk (t) = sup sup (1 + |x|)4 |∂xα f⃗(t, x)|.
|α|=k x∈R3
We write the Navier–Stokes equations as
3 Z
X
⃗u = Wνt ∗ ⃗u0 +
j=1
t
∂j O(ν(t − s)) :: ⃗gj ds.
(4.27)
0
with
⃗gj = −f⃗ ∗ ∂j G − uj ⃗u.
(4.28)
We proved (from Corollaries 4.1 and 4.2) that
(1 + |x|)2 |∂xα ⃗u(t, x)|

≤ C0  sup (1 + |y|)2 |∂xα ⃗u0 (y)| +
y∈R3
Z
0
t
1
p
ν(t − s)

3
X
j=1
sup
0≤s≤t,y∈R3
|(1 + |y|)2 ∂xα⃗gj (s, y)| ds
which gives
r
A0 (t) ≤ C0 A0 (0) + C1
r
A1 (t) ≤ C0 A1 (0) + C1
t
sup F0 (s) + C1
ν 0<s<t
Z
t
sup F1 (s) + C1
ν 0<s<t
Z
t
1
p
B0 (s)A0 (s) ds
ν(t − s)
t
1
p
B0 (s)A1 (s) ds
ν(t − s)
0
0
and
r
A2 (t) ≤ C0 A2 (0) + C1
t
sup F2 (s) + C1
ν 0<s<t
Z
0
t
1
p
(B0 (s)A2 (s) + A1 (s)2 ) ds.
ν(t − s)
We rermark that a Gronwall-like inequality
Z t
ds
α(t) ≤ A + B
α(s) √
t−s
0
Classical Solutions
59
can be reiterated into a Gronwall inequality, as
√
Z t
ZZ
Z t
ds
dτ ds
AB t
2
√
A√
+B
α(τ ) √
= A+
+πB 2
α(τ ) dτ
α(t) ≤ A+B
2
t−s
t−s s−τ
0
0≤τ ≤s≤t
0
and thus into
√
AB t
α(t) ≤ e
(A +
).
2
Thus, we find that, if B0 remains bounded, A0 and A1 remain bounded, and finally A2
remains bounded. We then conclude with Proposition 4.1 that TMAX < +∞ implies that
sup0<t<TMAX ∥⃗u(t, .)∥∞ = 0.
πB 2 t
4.8
Small Data
In case of small regular data, we prove easily that we have global solutions. In particular,
we have the following result:
Global solutions
Theorem 4.9. There exists a positive constant ϵ0 such that, if
• supx∈R3 |x||⃗u0 (x)| < ϵ0 ν,
√
• supt≥0,x∈R3 ( νt + |x|)3 |f⃗(t, x)| < ϵ0 ν 2 ,
R
• supt≥0,R>0 | |x|<R f⃗(t, x) dx| < ϵ0 ν 2 ,
then the classical solution (⃗u, p) associated to the regular data ⃗u0 and f⃗ is defined for
all times and satisfies
√
sup ( νt + |x|) |⃗u(t, x)| < +∞.
0≤t,x∈R3
⃗
Proof. We begin with a remark on the assumptions
P3on f . As we shall see in the proof, we
⃗
⃗
only need to know that f may be written as f = j=1 ∂j ⃗gj , where
√
sup ( νt + |x|)2 |⃗gj (t, x)| < C0 ϵ0 ν 2 .
(4.29)
t≥0,x∈R3
If (4.29) is satisfied, then, by Stokes’ formula,
Z
|
f⃗(t, x) dx| ≤ 4π
|x|<R
3
X
∥|x|2⃗gj ∥∞ ≤ 4πC0 ϵ0 ν 2 .
j=1
Conversely, under the assumptions of Theorem 4.9, we write
f⃗ = −
3
X
j=1
∂j (f⃗ ∗ ∂j G).
60
The Navier–Stokes Problem in the 21st Century (2nd edition)
We have
|f⃗ ∗ ∂j G| ≤ ϵ0 ν 2
Z
1
1
ϵ0 ν 2
1
1
√
dy ≤
∥
∥L∞ ∩L2 ∥ 2 ∥L1 +L2 .
2
3
3
νt (1 + |y|)
|y|
( νt + |x − y|) |y|
On the other hand, we have
f⃗ ∗ ∂j G =
Z
|x−y|>
|x|
2
f⃗(t, x − y)∂j G(y) dy
Z
+
|x−y|<
|x|
2
Z
+
|x−y|<
|x|
2
f⃗(t, x − y)(∂j G(y) − ∂j G(x)) dy
f⃗(t, x − y)∂j G(x) dy
and we get
|f⃗ ∗ ∂j G| ≤ϵ0 ν 2
Z
+ ϵ0 ν
|x|
|x−y|> 2
2
=C1 ϵ0 ν 2
1
1
dy
3
|x − y| 4π|y|2
Z
|x|
|x−y|< 2
1
|x − y|
1
dy + ϵ0 ν 2
|x − y|3 π|y|3
4π|x|2
1
.
|x|2
Finally, let us remark that f⃗ will satisfy the assumptions of Theorem 4.9 whenever
√
3
νt
2
|f⃗(t, x)| <
ϵ0 ν √
.
2π 2
( νt + |x|)4
The proof of Theorem 4.9 relies on the following inequality: there exists a constant
C0 > 0 such that, for all t > 0 and x ∈ R3 , we have
Z tZ
dy ds
C0
p
I(t, x) =
≤ √
.
(4.30)
√
4
2
ν( νt + |x|)
( ν(t − s) + |x − y|) ( νs + |y|)
0
This inequality is proved by Fubini’s theorem and Hölder’s inequality:
Z t
1
1
I(t, x) ≤
∥ p
∥L6/5 (dy) ∥ √
∥L6 (dy) ds
4
( νs + |y|)2
( ν(t − s) + |y|)
0
Z T
1
1
1
1
=C
ds = C ′ √
3/2 (t − s)3/4 s3/4
ν
ν tν
0
and
Z
1
1
∥ √
∥L1 (ds) ∥ √
∥L∞ (ds) dy
( νs + |x − y|)4
( νs + |y|)2
Z
1
1
1
=C
dy = C ′
.
ν|x − y|2 |y|2
ν|x|
I(t, x) ≤
R3
Now, let us consider the regular solution (⃗u, p) on [0, TMAX ). We write
f⃗ = −
3
X
j=1
∂j (f⃗ ∗ ∂j G)
Classical Solutions
and we define
61
√
α(t) = sup ( νt + |x|)|⃗u(t, x)|
x∈R3
and
3
X
√
β(t) = sup ( νt + |x|)2
|⃗gj (t, x)|
x∈R3
j=1
where gj = −f⃗ ∗ ∂j G. We start from the equality
⃗u = Wνt ∗ ⃗u0 +
3 Z
X
t
∂j O(ν(t − s)) :: (⃗gj − uj ⃗u) ds.
(4.31)
0
j=1
By Corollary 4.1, we know that
√
( t + |x|)|Wνt ∗ ⃗u0 (x)| ≤ C1 ∥|x|⃗u0 ∥∞ = C1 α(0).
By Corollary 4.2, we know that
t
Z
|
Z tZ
∂j O(ν(t − s)) :: (⃗gj − uj ⃗u) ds| ≤ C2
0
(|⃗u(s, y)|2 +
0
3
X
dy ds
.
|⃗gj (s, y)|) p
( ν(t − s) + |x − y|)4
j=1
Inequality (4.30) gives us that:
α(t) ≤ C1 α(0) + C0 C2
1
sup (β(s) + α(s)2 ).
ν 0<s<t
Using the assumptions on ⃗u0 and f⃗ (and thus inequality (4.29) on ⃗gj ), we find
α(s)2
.
ν
0<s<t
α(t) ≤ C3 ϵ0 ν + C3 sup
Assuming that α(s) < 3C3 ϵ0 ν on [0, t), we find that
α(t) ≤ C3 ϵ0 ν(1 + 9C3 ϵ0 ).
As α is a continuous function of t, we find that, if ϵ0 <
2C3 ϵ0 ν on the whole interval [0, TMAX ).
1
9C3 ,
α(t) will remain bounded by
Finally, we find that
α(t)
∥⃗u(t, .)∥∞ ≤ √ ≤ 2C3 ϵ0
νt
and Theorem 4.8 gives us that TMAX = +∞.
4.9
r
ν
t
Spatial Asymptotics
In this section, we will show that, even if ⃗u0 has good decay properties, we (generically) cannot hope for a good decay for the solution ⃗u. Dobrokhotov and Shafarevich [154]
proved that, unless some algebraic conditions are satisfied by the initial value ⃗u0 , there is
an instantaneous spreading of the velocity that cannot decay faster than O(|x|−4 ). This
instantaneous spreading has been studied by Brandolese in his Ph.D. [59] and in several
papers [14, 61, 62, 63]:
62
The Navier–Stokes Problem in the 21st Century (2nd edition)
Spatial decay estimates
Theorem 4.10.
Let (⃗u, p) be the classical solution of the Navier–Stokes problem on a strip [0, T ] × R3 ,
associated to the regular data (⃗u0 , f⃗). Assume moreover that we have:

limx→∞ |x|4 |⃗u0 (x)| = 0

(4.32)

sup0≤t≤T,x∈R3 (1 + |x|)5 |f⃗(t, x)| < +∞
then, for fixed t ∈ (0, T ], a necessary condition to ensure that
lim |x|4 |⃗u(t, x)| = 0
x→∞
is that ⃗u satisfies the Dobrokhotov and Shafarevich conditions
R tR

for 1 ≤ i ≤ 3, 0 fi dx ds = 0





R tR
for 1 ≤ i < j ≤ 3, 0 2ui uj + xi fj + xj fi dx ds = 0




R tR

for 1 ≤ i < j ≤ 3, 0 u2i − u2j + xi fi − xj fj dx ds = 0
(4.33)
Following Brandolese’s results, we are going to prove Theorem 4.10 by giving a precise
asymptotic formula for the solution ⃗u. This formula will prove that the Dobrokhotov and
Shafarevich conditions are sufficient as well.
Spatial asymptotics
Theorem 4.11.
Let ⃗u0 be a C 2 vector field on R3 and let f⃗ satisfy the assumptions of Theorem 4.10,
including the decay estimates (4.32). Then the classical solution (⃗u, p) of the Navier–
Stokes equations with initial data ⃗u0 and with forcing term f⃗ has, for any fixed t ∈ (0, T ],
the following asymptotic development when x goes to ∞:
⃗u =
3
X
⃗ i G(x) −
ci (t)∇∂
i,=1
3 X
3
X
⃗ i ∂j G(x) + o(|x|−4 )
di,j (t)∇∂
i=1 j=1
where G is the Green function
G(x) =
1
4π|x|
and where the coefficients ci and di,j are given by


for 1 ≤ i ≤ 3,

for 1 ≤ i ≤ 3, 1 ≤ j ≤ 3,
ci (t) =
di,j (t) =
R tR
0
R tR
0
fi dx ds
ui uj + xi fj dx ds
(4.34)
Classical Solutions
63
Proof. We first write ⃗u as a solution of a Stokes system
Z t
⃗u = Wνt ∗ ⃗u0 +
O(ν(t − s)) :: ⃗g (s) ds
0
with forcing term
⃗ u.
⃗g = f⃗ − (⃗u.∇)⃗
We already know that
|⃗u0 ∗ Wt (x)| ≤ C(1 + |x|)−2 ,
|⃗u(t, x)| ≤ C(1 + |x|)−2 ,
for 1 ≤ i ≤ 3, |∂i ⃗u(t, x)| ≤ C(1 + |x|)−2 ,
and hence that
|⃗g (t, x)| ≤ C(1 + |x|)−4 .
We begin with checking that Wνt ∗ ⃗u0 is small at infinity. We have
Z
Z
|Wνt ∗ ⃗u0 (x)| ≤
|⃗u0 (y)|Wνt (x − y) dy + ∥⃗u0 ∥∞
Wνt (x − y) dy
|y|> 12 |x|
|y|< 12 |x|
4
≤ 16
sup|y|≥ 12 |x| |y| |⃗u0 (y)|
|x|4
= o(|x|
−4
Z
+ C∥⃗u0 ∥∞
|x−y|> 12 |x|
νt5/2
dy
|x − y|8
)
where the remainder o(|x|−4 ) is small with respect to |x|−4 (at fixed ν, uniformly on t in
the compact interval [0, T ]).
We then use the estimate, for any ϵ with 0 < ϵ < 1,
|O(ν(t − s), x − y)| ≤ Cϵ
1
1
.
ν ϵ (t − s)ϵ |x − y|3−2ϵ
This proves that if
|⃗g (t, x)| ≤ Cν |x|−γ
on [0, T ] × R3 with 0 < γ < 3, then
|⃗u(t, x) − Wνt ∗ ⃗u0 (x)| ≤ Cν′ T 1−ϵ |x|2ϵ−γ
From ⃗g = O(|x|−3+ϵ ), we get ⃗u = O(|x|−3+3ϵ ). As ∂j ⃗u = O(|x|−2 ), we find that, for any
γ ∈ (0, 1), ⃗g = O(|x|−4−γ ).
Thus we are led to estimate
Z t
⃗ =
U
O(ν(t − s)) :: ⃗g (s) ds
0
when
sup
(1 + |x|)4+γ |⃗g (t, x)| < +∞.
0≤t≤T,x∈R3
We fix some β ∈ (0, 1) close enough to 1 and cut the integral
Z tZ
Oj,k (ν(t − s), x − y)gj (s, y) dy ds
0
64
The Navier–Stokes Problem in the 21st Century (2nd edition)
into three domains of integration:
∆1 = {y / |y| > |x|β and |x − y| < |x|β },
∆2 = {y / |y| > |x|β and |x − y| > |x|β },
and
∆3 = {y / |y| < |x|β }.
For 1 ≤ p ≤ 3, let
Z tZ
Oj,k (ν(t − s), x − y)gj (s, y) dy ds.
Ip =
0
∆p
We have, provided that ϵ < γ/2,
Z tZ
|I1 | ≤ C
0
1
|x|(4+γ)β
|x−y|<|x|β
t1−ϵ
dy ds
=
O(
)
(t − s)ϵ |x − y|3−2ϵ
|x|β(4+γ−2ϵ)
and similarly
Z tZ
|I2 | ≤ C
0
|y|>|x|β
1
dy ds
t1−ϵ
=
O(
).
|y|4+γ (t − s)ϵ |x|β(3−2ϵ)
|x|β(4+γ−2ϵ)
If we choose β and ϵ such that β(4 + γ − 2ϵ) > 4, we obtain
|I1 | + |I2 | = o(|x|−4 ).
For I3 , we have
I3 = A3 + B3 + C3 + D3
where
Z tZ
(Oj,k (ν(t − s), x − y) − ∂j ∂k G(x − y))gj (s, y) dy ds,
A3 =
0
|y|<|x|β
Z tZ
(∂j ∂k G(x − y) − ∂j ∂k G(x) +
B3 =
0
|y|<|x|β
C3 =
(−∂j ∂k G(x) +
3
X
|y|>|x|β
Z tZ
(∂j ∂k G(x) −
D3 =
∂j ∂k ∂l G(x)yl )gj (s, y) dy ds,
l=1
and
0
∂j ∂k ∂l G(x)yl )gj (s, y) dy ds,
l=1
Z tZ
0
3
X
R3
3
X
∂j ∂k ∂l G(x)yl )gj (s, y) dy ds.
l=1
We check that the three first terms are negligible. We have
Z tZ
ν 5/2 (t − s)5/2
|A3 | ≤ C
∥gj ∥∞ dy ds = o(|x|−4 )
|x|8
0
|y|≤|x|β
Z tZ
|B3 | ≤ C
0
|y|1+ϵ
1
dy ds = o(|x|−4 )
|x|4+ϵ (1 + |y|)4+γ
Z tZ
|C3 | ≤ C
(
0
|y|≥|x|β
1
|y|
1
+ 4 ) 4+γ dy = o(|x|−4 ).
|x|3
|x| |y|
Classical Solutions
65
Thus, only D3 cannot be neglected. Hence, we have obtained
3
X
⃗u =
⃗ i G(x) −
ci (t)∇∂
i,=1
3 X
3
X
⃗ i ∂j G(x) + o(|x|−4 )
di,j (t)∇∂
(4.35)
i=1 j=1
where the coefficients ci and di,j are given by


for 1 ≤ i ≤ 3,

for 1 ≤ i ≤ 3, 1 ≤ j ≤ 3,
ci (t) =
R tR
di,j (t) =
R tR
0
0
gi dx ds
xj gi dx ds
P3
−5−γ
−4−γ
with gi =Rfi − k=1 ∂k (uk ui ). Since
,
R |uk ui | ≤ C(1+|x|)
R and |∂k (uk ui )| ≤ C(1+|x|)
we have ∂k (uk ui ) dx = 0 and xj ∂k (uk uj ) dx = −δj,k uj ui ds. Thus, Theorem 4.11 is
proved.
⃗ + ∇B
⃗ + o(|x|−4 ) with A = P3 ci (t)∂i G(x) (hence
Thus, we have proved that ⃗u = ∇A
i,=1
⃗ is homogeneous of degree −3) and B = − P3 P3 di,j (t)∂i ∂j G(x) (hence, ∇B
⃗ is
∇A
i=1
j=1
homogeneous of degree −4). If ⃗u = o(|x|−4 ), we have that A(t, x) and B(t, x) must be
constant on x ̸= 0. Thus, by homogeneity of A and B, we have A = B = 0 and thus
3
X
ci (t)xi = A(t, x)|x|3 = 0
i,=1
and
3 X
3
X
di,j (t)(δi,j |x|2 − 3xi xj ) = B(t, x)|x|5 = 0.
i=1 j=1
Thus, we get ci (t) = 0 for 1 ≤ i ≤ 3, di,j (t) + dj,i (t) = 0 for 1 ≤ i < j ≤ 3 and
d1,1 (t) = d2,2 (t) = d3,3 (t). Theorem 4.10 is proved.
This proves that we have (generically) instantaneous spreading:
Corollary 4.3.
Under the assumptions of Theorem 4.10:
R
• if for some i, we have fi (0, x) ̸= 0, then there exists a positive time t0 such that for
all t ∈ (0, t0 ), lim supx→∞ |x|3 |⃗u(t, x)| > 0;
• if for some i and j with i ̸= j, we have
Z
2u0,i (x)u0,j (x) + xi fj (0, x) + xj fi (0, x) dx ̸= 0
or
Z
u0,i (x)2 + xi fi (0, x) dx ̸=
Z
u0,j (x)2 + xj fi (0, x) dx
then there exists a positive time t0 such that for all t ∈ (0, t0 ), lim supx→∞ |x|4 |⃗u(t, x)| >
0.
66
4.10
The Navier–Stokes Problem in the 21st Century (2nd edition)
Spatial Asymptotics for the Vorticity
In contrast with the phenomenon of instantaneous spreading for the velocities, there is
no such spreading for the vorticity:
Vorticity’s decay
Theorem 4.12.
Let (⃗u, p) be the classical solution of the Navier–Stokes problem on a strip [0, T ]×R3 ,
associated to the regular data (⃗u0 , f⃗).
Then, we have the following property for the vorticity
⃗ ∧ ⃗u :
ω
⃗ =∇
if for some N ∈ N, we have
sup |x|N |⃗
ω0 (x)| < +∞,
x∈R3
and
sup
|x|N |f⃗(t, x)| < +∞
0≤t≤T,x∈R3
then
sup
|x|N |⃗
ω (t, x)| < +∞.
0≤t≤T,x∈R3
Proof. The proof is easy. It is enough to write ω
⃗ as a solution of a heat equation
Z t
ω
⃗ = Wνt ∗ ω
⃗0 +
Wν(t−s) ∗ ⃗g (s) ds
0
with forcing term
⃗g = curl f⃗ + div(⃗
ω ⊗ ⃗u − ⃗u ⊗ ω
⃗)
(where div(⃗a ⊗ ⃗b) =
P3
j=1
∂j (aj⃗b)). We already know that
|⃗u(t, x)| ≤ C(1 + |x|)−2 ,
so that every information on ω
⃗ = O(|x|−δ ) will be converted into an estimate ω
⃗ = Wνt ∗
Rt
ω
⃗ 0 + 0 curl(Wν(t−s) ∗ f⃗) ds + O(|x|−δ−2 ).
We thus have precise information on the localization of the vorticity, and not on the
velocity. However, there is a relationship between the velocity ⃗u and the vorticity ω
⃗ : from
ω
⃗ = curl ⃗u, we have curl ω
⃗ = −∆⃗u and, since ⃗u vanishes at infinity, we may determine ⃗u
through the Biot-Savart law
⃗
⃗u = ∇G(∧∗)⃗
ω
(4.36)
where the operation ∧∗ is defined with the Fourier transform F by
⃗a(∧∗)⃗b = F −1 (F(⃗a) ∧ F(⃗b)).
Classical Solutions
67
If ω
⃗ is rapidly decaying (|⃗
ω | ≤ C(1 + |x|−5−γ ), we find that
⃗
⃗ +
⃗u = ∇G(x)
∧ A(t)
3
X
⃗ i G(x) ∧ B
⃗ i (t) +
∇∂
i,=1
3 X
3
X
⃗ i ∂j G(x) ∧ C
⃗ i,j (t) + o(|x|−4 ) (4.37)
∇∂
i=1 j=1
with
⃗ =
A(t)
Z
ω
⃗ (t, y) dy,
⃗ i (t) = −
B
Z
yi ω
⃗ (t, y) dy,
⃗ i,j (t) = 1
C
2
Z
yi yj ω
⃗ (t, y) dy.
From the decay of ω
⃗ in O(|x|−5−γ ) and the fact that div ω
⃗ = 0, we find that
⃗ = 0 and ⃗u = O(|x|−3 ).
hence A
If we want ⃗u(t0 , .) = o(|x|−4 ) at some time t0 , we must have
3
X
⃗ i G(x) ∧ B
⃗ i (t0 ) +
∇∂
i,=1
3 X
3
X
R
ω
⃗ dy = 0,
⃗ i ∂j G(x) ∧ C
⃗ i,j (t0 ) = 0.
∇∂
(4.38)
i=1 j=1
If ⃗u0 = o(|x|−4 ), then (4.38) is satisfied at t = 0, and we find
Z t0
Z t0
3
3 X
3
X
X
d ⃗
d ⃗
⃗
⃗
∇∂i G(x) ∧
Bi (t) dt +
∇∂i ∂j G(x) ∧
Ci,j (t) dt = 0.
dt
dt
0
0
i,=1
i=1 j=1
We now write
∂t ω
⃗ = ν∆⃗
ω + curl f⃗ + div(⃗
ω ⊗ ⃗u − ⃗u ⊗ ω
⃗ ) = ν∆⃗
ω + curl f⃗ − curl(div(⃗u ⊗ ⃗u)).
From the decay of f⃗ in |x|−5 , of ⃗u in |x|−3 and of ω
⃗ in |x|−5−γ we find that
Z
Z
d ⃗
⃗ i ) dy
Bi (t) = − yi curl f⃗(t, y) dy = − f⃗(t, y) ∧ ∇(y
dt
and
d ⃗
Ci,j (t) =
dt
Z
3
X
⃗ yi yj +
⃗ k yi yj dy.
f⃗(t, y) ∧ ∇
uk (t, y)⃗u(t, y) ∧ ∇∂
2
2
k=1
This gives
X
⃗ i G(x) ∧ d B
⃗ i (t) =
∇∂
∂i ∂k G(x)
dt
Z
⃗ i ) dy − ∂i2 G(x)
fk (t, y)∇(y
Z
f⃗(t, y) dy
k
and thus (since ∆G = 0 for x ̸= 0)
3
3 Z
X
d X⃗
⃗
⃗
∇∂i G(x) ∧ Bi (t) =
( fk (t, y) dy) ∂k ∇G(x).
dt i=1
k=1
Similarly, we write
3
X
⃗ i ∂j G(x) ∧ d C
⃗ i,j (t) =
∇∂
∂i ∂j ∂l G(x)
dt
Z
∂l
l=1
+
3
X
∂i ∂j ∂l G(x)
Z X
3
l=1
−
3
X
3
X
l=1
∂k ∂l
k=1
Z
∂i ∂j ∂l G(x)
⃗
fl (t, y)∇
l=1
−
yi yj ⃗
f (t, y) dy
2
∂i ∂j ∂l G(x)
Z X
3
k=1
yi yj uk (t, y)⃗u(t, y)dy
2
yi yj dy
2
⃗
uk (t, y)ul (t, y)∂k ∇
yi yj dy
2
68
The Navier–Stokes Problem in the 21st Century (2nd edition)
and thus (using again ∆G(x) = 0 for x ̸= 0)
3 X
3
X
Z
3 X
3
X
⃗ i ∂j G(x) ∧ d C
⃗ i,j (t) = −
⃗
∇∂
∂i ∂l ∇G(x)
fl (t, y)yi dy
dt
i=1 j=1
i=1
l=1
−2
3 X
3
X
⃗
∂i ∂l ∇G(x)
Z
ui (t, y)ul (t, y)∂k (yi ) dy
i=1 l=1
We thus recover the Dobrokhotov and Shafarevich conditions. We can conclude, in the case
of a null force f⃗ = 0 and of a rapidly decaying vorticity ω
⃗ 0 , that we have:
R
• ω
⃗ (t, x)dx = 0
R
R
• if ω
⃗ 0 (x)xi dx = 0 for all 1 ≤ i ≤ 3, then for all t ∈ (0, T ], ω
⃗ (t, x)xi dx = 0
R
• even if ω
⃗ 0 (x)xi xj dx = 0 for all 1 ≤ i, j ≤ 3,
R we may have that for all t ∈ (0, t0 ], t0
small enough, there exists i and j such that ω
⃗ (t, x) xi xj dx ̸= 0
• An easy example of suchRa ω
⃗ 0 is ⃗u0 = (−∂2 ψ, ∂1 ψ, 0) and ω
⃗ 0 = (−∂1 ∂3 ψ, −∂2 ∂3 ψ, (∂12 +
2
∂2 )ψ) with ψ ∈ D and ψ dx = R0 (and ψ ̸= R0). ⃗u does not satisfy the Dobrokhotov
and Shafarevich conditions since u20,1 dx ̸= u20,3 dx.
4.11
Maximal Classical Solutions and Estimates in L2 Norms
Let (⃗u0 , f⃗) be regular data for the Cauchy problem for Navier–Stokes equations. We
know that we have a unique maximal solution defined on an interval [0, TMAX ). In order to
prove that we have a global solution, i.e. that TMAX = +∞, we have seen that it is enough
to get an a priori control on the L∞ norm of the solution (Theorem 4.8). However, we have
no such control (except on the case of small data [Theorem 4.9]). Actually, the only control
we have is a control on the L2 norm:
Energy balance
Proposition 4.2.
Let (⃗u, p) be the classical solution of the Navier–Stokes problem on the strip [0, TMAX )×
R3 , associated to the regular data (⃗u0 , f⃗). Then we have
⃗ ⊗ ⃗u|2 − div (|⃗u|2 + 2p)⃗u + 2⃗u · f⃗
∂t (|⃗u|2 ) = ν∆(|⃗u|2 ) − 2ν|∇
(4.39)
2
⃗ ⊗ ⃗u|2 = P
(where |∇
gj = f⃗ ∗ ∂j G (so that
1≤i,j≤3 |∂i uj | ). In particular, writing ⃗
P
3
f⃗ = − j=1 ∂j ⃗gj ), we have
2
Z
∥⃗u(t, .)∥ + ν
0
t
3 Z
1X t
2
2
⃗
∥∇ ⊗ ⃗u∥2 ds ≤ ∥⃗u0 ∥2 +
∥⃗gj ∥22 ds
ν j=1 0
and, for every finite T with 0 < T ≤ TMAX ,
2
2 1
3
⃗u ∈ L∞
t Lx ∩ Lt Hx ((0, T ) × R ).
(4.40)
Classical Solutions
69
Proof. As ⃗u, f⃗ and p are C 2 , equation (4.39) is obtained easily by writing ∂t (|⃗u|2 ) = 2⃗u · ∂t ⃗u
and using the fact that div ⃗u = 0. Then, due to the decay of ⃗u, f⃗ and p and of their
derivatives, we may integrate this equality on (0, t) × R3 and obtain
Z t
Z tZ
2
2
2
⃗
∥⃗u(t, .)∥ − ∥⃗u0 ∥2 = −2ν
∥∇ ⊗ ⃗u∥2 + 2
⃗u · f⃗ dx ds.
0
0
We then finish the proof by integration by parts:
Z
⃗u · f⃗ dx =
3 Z
X
3
X
∂j ⃗u · ⃗gj dx ≤
j=1
ν∥∂j ⃗u∥22 +
j=1
1
∥⃗gj ∥22 .
ν
Energy estimates can be useful for getting other accurate estimates. Here, we shall give
an example of control on the L2 norms of derivatives, i.e. on Sobolev norms, of the classical
solutions of the Navier–Stokes equations. We begin with energy estimates for the heat kernel:
Energy estimates for the heat kernel
Proposition 4.3.
2
2
⃗
A) if u0 ∈ L2 , then Wνt ∗ u0 ∈ L∞ ((0, +∞), L2 ) and ∇(W
νt ∗ u0 ) ∈ L ((0, +∞), L ).
Moreover,
Z
t
2
2
⃗
∥∇(W
νs ∗ u0 )∥2 ds = ∥u0 ∥2 .
∥Wνt ∗ u0 ∥22 + 2ν
0
Rt
B) if ⃗g ∈ L ((0, +∞), L ) and U = 0 Wν(t−s) ∗ div ⃗g ds, then U ∈ L∞ ((0, +∞), L2 )
⃗ ∈ L2 ((0, +∞), L2 ). Moreover,
and ∇U
2
2
∥U (t, .)∥22
t
Z
⃗ (s, .)∥22 ds = −2
∥∇U
+ 2ν
Z tZ
0
⃗ (s, .) · ⃗g (s, .) dx ds
∇U
0
and thus
∥U (t, .)∥22 + ν
Z
t
⃗ (s, .)∥22 ds ≤
∥∇U
0
2
1
ν
2
Z
t
∥⃗g (s, .)∥22 ds.
0
C) (maximal regularity) if h ∈ L ((0, +∞), L ) and V =
V ∈ L2 ((0, +∞), L2 ) and
ν2
Z
0
+∞
∥V (s, .)∥22 ds ≤
Z
Rt
0
Wν(t−s) ∗ ∆h ds, then
+∞
∥h(s, .)∥22 ds
0
Proof.
A) By Plancherel and Fubini, we have
Z
Z tZ
2
|c
u0 (ξ)2 (e−2t|ξ| − 1) dξ = −2
|ξ|2 e−2s|ξ| |c
u0 (ξ)2 dξ ds
0
B) If ⃗g ∈ D((0, +∞) × R3 ), then we have seen that U and its derivatives is smooth and
rapidly decaying in space (uniformly in time). Moreover, we have
∂t (|U |2 ) = 2U ∂t U = 2νU ∆U + 2U div ⃗g .
70
The Navier–Stokes Problem in the 21st Century (2nd edition)
Intregrating this equatlity between 0 and t with respect to time and space variables
gives (since U (0, .) = 0)
Z t
Z tZ
⃗ (s, .)∥22 ds = −2
⃗ (s, .) · ⃗g (s, .) dx ds.
∥U (t, .)∥22 + 2ν
∥∇U
∇U
0
0
The inequality
Z
⃗ (s, .) · ⃗g (s, .) dx ≤ ∥∇U
⃗ (s, .)∥2 ∥⃗g (s, .)∥2 ≤ ν ∇U
⃗ (s, .) + 1 ∥⃗g (s, .)∥22
∇U
ν
then gives that
∥U (t, .)∥22 + ν
Z
t
⃗ (s, .)∥2 ds ≤
∥∇U
2
0
1
ν
Z
t
∥⃗g (s, .)∥22 ds.
0
Thus, ⃗g 7→ U is linear and continuous (in the L2t L2x norm) from the space D((0, +∞)×
R3 ) to Cb ([0, +∞), L2 (R3 )) ∩ L2 ((0, +∞), Ḣx1 (R3 )) (where the norm of the homoge⃗ ∥2 ). We may then conclude, due to
neous Sobolev space Ḣ 1 is given by ∥f ∥Ḣ 1 = ∥∇f
3
2
the density of the space D((0, +∞) × R ) in L ((0, +∞) × R3 ) ≈ L2 ((0, +∞), L2 ).
C) We define ⃗gj,ϵ = (g1,j,ϵ , g2,j,ϵ , g3,j,ϵ ) by the Fourier transforms
g[
k,j,ϵ (ξ) = −
and
Z
ξj ξk
ĥ(ξ)
ϵ + |ξ|2
t
Wν(t−s) ∗ div ⃗gj,ϵ ds.
Uj,ϵ =
0
We have
Z
+∞
ν
⃗ j,ϵ (s, .)∥22 ds ≤
∥∇U
0
1
ν
Z
+∞
∥⃗gj,ϵ (s, .)∥22 ds.
0
or, equivalently, by the Plancherel formula,
Z ∞Z
Z Z
|ξj |2 |ξ|2
1 +∞ |ξj |2 |ξ|2
2
ν
|V̂ (s, ξ)| dξ ds ≤
|ĥ(s, ξ)|2 dξ ds.
(ϵ + |ξ|2 )2
ν 0
(ϵ + |ξ|2 )2
0
Then, we sum on j and let ϵ go to 0; by monotonous convergence we find
Z ∞Z
Z Z
1 +∞
ν
|V̂ (s, ξ)|2 dξ ds ≤
|ĥ(s, ξ)|2 dξ ds.
ν 0
0
Another interesting application of energy balances is the following result:
Energy equality
Proposition 4.4.
If u ∈ L2 ((0, T ), H 1 ) and ∂t u ∈ L2 ((0, T ), H −1 ) (where H −1 is defined by: f ∈ H −1 ⇔
1
(1 + |ξ|2 )− 2 fˆ ∈ L2 ) then u belongs to C([0, T ], L2 ) and, for 0 ≤ t0 ≤ t ≤ T ,
∥u(t, .)∥22
=
∥u(t0 , .)∥22
Z
t
⟨u|∂t u⟩H 1 ,H −1 ds.
+2
t0
Classical Solutions
71
Proof. We prove the theorem for u ∈ Cc∞ ([0, T ] × R3 ). We have
∂t |u|2 = 2u∂t u
and integration in time and space variables between t0 and t gives the result. In particular,
if θ1 and θ2 are smooth functions on [0, T ] such that 0 ≤ θ1 ≤ 1, θ1 (t) = 1 for t ≤ T /2 and
= 0 for t ≥ 3T /4 while 0 ≤ θ2 ≤ 1, θ2 (t) = 0 for t ≤ T /4 and = 0 for t ≥ T /2, we find that,
for t ≤ T /2
Z T
∥u(t, .)∥22 ≤ 2
θ1 (s)|⟨u|∂t u⟩H 1 ,H −1 | + |θ1′ (s)∥u∥22 ds
t
while, for t ≥ T /2,
∥u(t, .)∥22 ≤ 2
Z
t
θ2 (s)|⟨u|∂t u⟩H 1 ,H −1 | + |θ2′ (s)∥u∥22 ds.
0
Thus, we find that the linear map u ∈ Cc∞ ([0, T ] × R3 ) 7→ u ∈ C([0, T ], L2 ) is bounded for
the norm of ∥u∥L2 H 1 + ∥∂t u∥L2 H −1 . We then finish the proof by a density argument.
An interesting application is the control on the size of ⃗u in the Sobolev space H 3 when
⃗u0 and f⃗ are one-derivative more regular. Let us recall some elementary results on Sobolev
spaces H k , k ∈ N:
• definition: H k is the space of functions in L2 whose derivatives (in the sense of distributions) of order less or equal to k are square integrable, normed with
sX
∥f ∥H k =
∥∂ α f ∥22 .
|α|≤k
• H k is a Hilbert space. Its norm is equivalent with
∥(Id − ∆)k/2 f ∥2 =
1
∥(1 + |ξ|)k/2 fˆ∥2
(2π)3/2
(where fˆ is the Fourier transform of f ) and with
s
X
∥f ∥22 +
∥∂ α f ∥22
|α|=k
• for k ≥ 2 and f ∈ H k , fˆ ∈ L1 , so that H k is continuously embedded in L∞ :
∥f ∥∞ ≤
1
1
∥fˆ∥1 ≤
∥(1 + |ξ|)−k/2 ∥2 ∥(1 + |ξ|)k/2 fˆ∥2
(2π)3
(2π)3
• for f ∈ H 2 , we have
Z
Z
4
|∂i f | dx = −3 f (∂i f )2 ∂i2 f dx ≤ 3∥f ∥∞ ∥∂i f ∥24 ∥∂i2 f ∥2 ≤ C∥f ∥2H 2 ∥∂i f ∥24
• for k ≥ 2 and f, g ∈ H k , f g ∈ H k and ∥f g∥H k ≤ Ck ∥f ∥H k ∥g∥H k [obvious by the
Leibniz rule]
We first collect the results that may be induced from a straightforward adaptation of
Theorems 4.7 and 4.8:
72
The Navier–Stokes Problem in the 21st Century (2nd edition)
Theorem 4.13 (Navier–Stokes equations and C 3 solutions).
Let ⃗u0 be a C 3 divergence-free vector field on R3 and let f⃗ be a time-dependent vector field
such that:
• sup|α|≤3 supx∈R3 (1 + |x|)2 |∂xα ⃗u0 (x)| < +∞
• for |α| ≤ 3, ∂xα f⃗ is continuous on [0, +∞) × R3
• sup|α|≤3 supt≥0,x∈R3 (1 + |x|)4 |∂xα f⃗(t, x)| < +∞.
Then, there exists a positive time T and a unique solution (⃗u, p) of the Navier–Stokes problem
⃗ u − ∇p
⃗ on (0, T ) × R3
∂t ⃗u =ν∆⃗u + f⃗ − (⃗u.∇)⃗
div ⃗u = 0 ⃗u|t=0 = ⃗u0
such that:
• for |α| ≤ 3, ∂xα p and ∂xα ⃗u are continuous on [0, T ] × R3
• sup0≤t≤T,x∈R3 (1 + |x|)|p(t, x)| < +∞.
• sup1≤|α|≤3 sup0≤t≤T,x∈R3 (1 + |x|)2 |∂xα p(t, x)| < +∞.
• sup|α|≤3 sup0≤t≤T,x∈R3 (1 + |x|)2 |∂xα ⃗u(t, x)| < +∞.
Let TMAX be the maximal time where one can find a C 3 classical solution ⃗u of the
Cauchy problem for the Navier–Stokes equations on (0, TMAX ) × R3 . Then we have the a
priori estimate:
TMAX ≥ min(T1 , T2 )
with

1
ν



T1 = C0 (sup|α|≤3 supx∈R3 (1 + |x|)2 |∂xα ⃗u0 (x)|)2
.
ν
1


T
=

 2 C0 sup
4 α⃗
|α|≤3 sup0<t<T ,x∈R3 (1 + |x|) |∂x f (t, x)|
0
and the following blow up criterion: If TMAX < +∞, then
sup
|⃗u(t, x)| = +∞.
0≤t<TMAX , x∈R3
Thus, the lower bound on TMAX seems to depend on ν, and to go to 0 as ν goes to 0.
However, Swann [458] obtained a bound that depends only on the H 3 norm of ⃗u0 and on
the size of f⃗:
Navier–Stokes equations and H 3 norms
Theorem 4.14.
Under the assumptions of Theorem 4.13, we have the following size estimates for
s
X
∥⃗u(t, .)∥H 3 = ∥⃗u∥2 +
∥∂xα ⃗u∥22 :
|α|=3
Classical Solutions
73
for 0 < t < TMAX , we have
∥⃗u(t, .)∥2H 3 ≤ ∥⃗u0 ∥2H 3 +
t
Z
0
3/2
∥f⃗∥H 3 ds + C0
t
Z
∥⃗u(s, .)∥3H 1 ds
0
for a constant C0 which does not depend on ν.
In particular, if T0 < +∞, and if
T1 = min(T0 ,
8C0 (∥⃗u0 ∥2H 3 +
1
R T0
0
3/2
∥f⃗∥H 3 ds)2
)
then TMAX ≥ T1 .
Proof. We have
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u) + Pf⃗
and
Z
⃗u = Wνt ∗ ⃗u0 +
t
Wν(t−s) ∗ (−P div(⃗u ⊗ ⃗u) + Pf⃗) ds.
0
If T < TMAX , we have that ⃗u belongs to L∞ ((0, T ), H 3 ); thus f⃗ − div(⃗u ⊗ ⃗u) belongs to
L ((0, T ), H 2 ). Recall that the Leray projection operator P is defined as
∞
⃗
PF⃗ = F⃗ + ∇(G
∗ div F⃗ )
and thus
b
ξ · F⃗ (ξ)
c
b
PF⃗ (ξ) = F⃗ (ξ) − ξ
;
|ξ|2
c
b
thus, P is bounded on H 2 (as |PF⃗ (ξ)| ≤ |F⃗ (ξ)|). Thus, writing
∆2 ⃗u =
3
X
Z
∂i (Wνt ∗ ∂i ∆⃗u0 ) +
t
Wν(t−s) ∗ ∆ ∆P(f⃗ − div(⃗u ⊗ ⃗u)) ds,
0
i=1
√
we find that ⃗u ∈ L2 ((0, T ), H 4 ): of course, ∥⃗u∥L2 L2 ≤ T ∥⃗u∥L∞ L2 ; on the other hand, using
the maximal regularity of the heat kernel (Proposition 4.3), we get
1
1
∥∆2 ⃗u∥L2 L2 ≤C √ ∥⃗u0 ∥H 3 + C ∥∆P(f⃗ − div(⃗u ⊗ ⃗u))∥L2 L2
ν
ν
√
√
1
′ T ⃗
′ T
≤C √ ∥⃗u0 ∥H 3 + C
∥f ∥L∞ H 2 + C
∥⃗u∥2L∞ H 3 .
ν
ν
ν
Now, we write, for |α| = 3,
∂t ∂xα ⃗u = ν∂xα ∆⃗u − ∂xα P div(⃗u ⊗ ⃗u) + ∂xα Pf⃗;
74
The Navier–Stokes Problem in the 21st Century (2nd edition)
as ⃗u ∈ L2 H 4 , we find that ∂xα ⃗u ∈ L2 H 1 and ∂t ∂xα ⃗u ∈ L2 H −1 , so that (Proposition 4.4) we
have
Z t
α
2
α
2
∥∂x ⃗u(t, .)∥2 =∥∂x ⃗u0 ∥2 + 2
⟨∂xα ⃗u|ν∂xα ∆⃗u − ∂xα P div(⃗u ⊗ ⃗u) + ∂xα Pf⃗⟩H 1 ,H −1 ds
0
Z tZ
α
2
α
2
⃗
=∥∂x ⃗u0 ∥2 − 2ν∥∇ ⊗ ∂x ⃗u∥2 + 2
∂xα ⃗u · ∂xα f⃗ dx ds
0
Z t
⃗ u)⟩H 1 ,H −1 ds
−2
⟨∂xα ⃗u|∂xα (⃗u · ∇⃗
0
Z tZ
⃗ ⊗ ∂xα ⃗u∥22 + 2
=∥∂xα ⃗u0 ∥22 − 2ν∥∇
∂xα ⃗u · ∂xα f⃗ dx ds
0
Z tZ
X
α!
⃗ u dx ds
−2
∂xα ⃗u · (∂xα−γ ⃗u) · ∂xγ ∇⃗
γ!(α − γ)! 0
γ≤α,γ̸=α
Z t
⃗ xα ⃗u)⟩H 1 ,H −1 ds.
−2
⟨∂xα ⃗u|⃗u · ∇∂
0
For ⃗v ∈ H 3 and w1 , w2 ∈ H 1 , we have
⃗ 2 ⟩H 1 ,H −1 = −⟨w2 |⃗v · ∇w
⃗ 1 ⟩H 1 ,H −1 −
⟨w1 |⃗v · ∇w
Z
w1 w2 div ⃗v dx
so that
⃗ xα ⃗u)⟩H 1 ,H −1 = 0.
⟨∂xα ⃗u|⃗u · ∇∂
On the other hand, we have:
* when γ = 0,
⃗ u∥2 ≤ ∥∂xα ⃗u∥2 ∥∇
⃗ ⊗ ⃗u∥∞ ≤ C∥⃗u∥2 3
∥(∂xα−γ ⃗u) · ∂xγ ∇⃗
H
* when |γ| = 1,
⃗ u∥2 ≤ ∥∂ α−γ ⃗u∥4 ∥∂ γ ∇
⃗ ⊗ ⃗u∥4 ≤ C∥⃗u∥2 3
∥(∂xα−γ ⃗u) · ∂xγ ∇⃗
x
x
H
* when |γ| = 2,
⃗ u∥2 ≤ ∥∂xα−γ ⃗u∥∞ ∥∂xγ ∇
⃗ ⊗ ⃗u∥2 ≤ C∥⃗u∥2 3 .
∥(∂xα−γ ⃗u) · ∂xγ ∇⃗
H
Thus, if ∥⃗u∥H 3 is defined as
s
X
∥⃗u∥2 +
∥∂xα ⃗u∥22 ,
∥⃗u(t, .)∥H 3 =
|α|=3
we find that
∥⃗u(t, .)∥2H 3 ≤ ∥⃗u0 ∥2H 3 + 2
t
Z
∥⃗u∥H 3 ∥f⃗∥H 3 + C
Z
0
t
∥⃗u(s, .)∥3H 3 ds
0
and finally
∥⃗u(t, .)∥2H 3 ≤ ∥⃗u0 ∥2H 3 +
Z
0
t
3/2
∥f⃗∥H 3 ds + C0
We may now easily finish the proof. If T0 < +∞, and if
0 ≤ t < TMAX
Z
0
t
∥⃗u(s, .)∥3H 3 ds.
Classical Solutions
and
t ≤ T1 = min(T0 ,
8C0 (∥⃗u0 ∥2H 3
+
75
1
R T0
0
3/2
∥f⃗∥H 3 ds)2
)
we get that
∥⃗u(t, .)∥2H 3 ≤ 2(∥⃗u0 ∥2H 3 +
Z
0
T0
3/2
∥f⃗∥H 3 ds).
In particular,
sup
∥⃗u(t, .)∥∞ < +∞
0<t<min(TMAX ,T1 )
and TMAX ≥ T1 .
4.12
Intermediate Conclusion
What have we seen in this chapter? Given regular initial value ⃗u0 and forcing term f⃗, as
in the setting of the Clay Millenium problem, we have been able to prove with elementary
tools of calculus the following results:
• existence of a (classical) solution on [0, T ] × R3 for a positive time T
• global existence of the solution when the data ⃗u0 and f⃗ are small
• instantaneous spreading of the velocity (so that the assumptions of the Clay problem
on the initial value cannot be kept)
• localization of the vorticity
In the following chapters, we are going to extend the class of solutions (weak solutions
instead of classical ones) in order to grant global existence, and to use tools from functional
analysis and real harmonic analysis to try and get better insight into the properties of those
extended solutions. As a matter of fact, when the data are large, we do not know whether
the solutions that we are able to construct are unique, nor whether they are regular.
Chapter 5
A Capacitary Approach of the Navier–Stokes
Integral Equations
In Chapter 4, we have studied classical solutions of the Cauchy initial value problem for the
Navier–Stokes equations (with reduced (unknown) pressure p, reduced force density f⃗ and
kinematic viscosity ν > 0): given a divergence-free vector field ⃗u0 on R3 and a force f⃗ on
(0, +∞) × R3 , find a positive T and regular functions ⃗u and p on [0, T ] × R3 solutions to
⃗ u + f⃗ − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
(5.1)
We have reformulated this problem into an integral equation: find ⃗u such that
⃗u = Wνt ∗ ⃗u0 −
Z tX
3
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G + uj ⃗u ds
(5.2)
0 j=1
We can see in the formulation of Equation (5.2) that we do not need any regularity on
⃗u to compute the right-hand side of the equation, but just integrability properties. In this
chapter, we shall discuss the existence of measurable solutions of the integral equation, and
we shall see in Chapter 6 to what extent they are a solution of the differential equations.
5.1
The Integral Navier–Stokes Problem
Throughout the chapter, we are going to study generalized solutions of the Navier–Stokes
equations. More precisely, starting with the initial data ⃗u0 and the force f⃗, we assume that
⃗ (t, x) = Wνt ∗ ⃗u0 −
U
Z tX
3
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G
0 j=1
is a measurable function of t and x such that, for all R > 0, we have, for 1 ≤ k ≤ 3,
Z TZ
|Uk (t, x)| dx dt < ∞
0
B(0,R)
and we study the measurable functions ⃗u(t, x) defined on (0, T )×R3 such that, for all R > 0,
we have, for 1 ≤ j, k, l ≤ 3,
Z TZ
Z tZ
|∂j Ok,l (ν(t − s), x − y)| |uj (s, y)||ul (s, y)| dy ds dx dt < ∞
0
B(0,R)
0
DOI: 10.1201/9781003042594-5
76
A Capacitary Approach of the Navier–Stokes Integral Equations
and such that
⃗ −
⃗u = U
Z tX
3
∂j O(ν(t − s)) :: uj ⃗u ds.
77
(5.3)
0 j=1
5.2
Quadratic Equations in Banach Spaces
Solving the Navier–Stokes equations when written as integro-differential equations is
solving a quadratic equation in the unknown ⃗u (Equation (5.3)). In this section, we show
how to solve general quadratic equations with small data in a Banach space:
Quadratic equations
Theorem 5.1.
Let B be a bounded bilinear operator on a Banach space E:
∥B(u, v)∥E ≤ C0 ∥u∥E ∥v∥E .
Then, when ∥u0 ∥E ≤
1
4C0 ,
the equation
u = u0 + B(u, u)
has a unique solution in E such that ∥u∥E ≤
1
2C0 .
Moreover, ∥u∥E ≤ 2∥u0 ∥E .
Proof. The method used by Oseen, which we developed in Chapter 4, Section 4.6, is very
efficient. We introduce a development of the solution uϵ P
of the equation uϵ = u0 + ϵB(uϵ , uϵ )
∞
as a power series in ϵ. We find (at least formally) uϵ = n=0 ϵn un , where
uk+1 =
k
X
B(un , uk−n ).
n=0
Thus, we have
∥uk+1 ∥E ≤ C0
k
X
∥un ∥E ∥uk−n ∥E .
n=0
The norm of un is thus dominated by αn where α0 = ∥y0 ∥E , and
αk+1 = C0
k
X
αk αk−n .
n=0
The function
αϵ =
∞
X
ϵn αn
n=0
is solution of
αϵ = α0 + C0 ϵ αϵ2 .
(5.4)
78
The Navier–Stokes Problem in the 21st Century (2nd edition)
We find that the series in Equation (5.4) converges for 1 − 4C0 ϵα0 ≥ 0 and that
√
1 − 1 − 4C0 ϵ α0
2α0
√
αϵ =
=
.
2C0 ϵ
1 + 1 − 4C0 ϵ α0
This proves the existence of the solution u of u = u0 + B(u, u) when 4C0 ∥u0 ∥E ≤ 1 and
that ∥u∥E ≤ 2∥u0 ∥E .
An alternative way to describe this series expansion of the solution is the following one.
Consider B as an internal operation on E. We write B(u, v) = u ⊛ v. This operation is not
associative, hence we must use parentheses when defining the “product” of three (or more)
terms: (u ⊛ v) ⊛ w is not the same as u ⊛ (v ⊛ w). Let An be the number of different ways to
introduce parentheses for defining the product of n terms. Obviously, we have A1 = A2 = 1
(no need of parentheses) and A3 = 2. For defining the product of n terms, n ≥ 2, we must
choose the order of priority in the computations of the ⊛ products: the product that will
be computed at last will involve a product of k terms on the left-hand side and n − k terms
on the right-hand side, so that we see easily that
An =
n−1
X
Ak An−k .
k=1
Now, let us call a word an expression that is defined inductively in the following way:
ˆ u0 is a word
ˆ if w1 and w2 are two words, then w1 ⊛ w2 is a word.
Let Wn be the set of words where u0 appears n times in the word, and W = ∪+∞
n=1 Wn .
The cardinal of the set Wn is P
An . Moreover, the norm of a word w ∈ Wn is controlled
by
P
+∞
C0n−1 ∥u0 ∥nE . If u0 is such that n=1 An C0n−1 ∥u0 ∥nE < +∞, then the series u = w∈W w is
normally convergent. A word is either equal to u0 or of the form w1 ⊛ w2 , so that
X
X
X
u = u0 +
w1 ⊛ w2 = u0 + (
w1 ) ⊛ (
w2 ) = u0 + B(u, u).
w1 ∈W
(w1 ,w2 )∈W ×W
w2 ∈W
Hence, we find a solution of u = u0 + B(u, u). In the case of E = C and u ⊛ v = uv, we find
that
+∞
X
X
w=
An un0
w∈W
n=1
while u = u0 + u2 ; thus the generating series of the sequence An is given by
A(z) =
+∞
X
An z n =
1−
√
n=1
1 − 4z
2
√
√
(where the determination of the square root is given on ℜ(z) ≥ 0 by ( z)2 = z and 1 = 1).
As the numbers An are positive
P and A(1/4) < +∞, we find that the series is convergent for
|z| ≤ 1/4. Thus, the series w∈W w will be normally convergent for C0 ∥u0 ∥E ≤ 1/4. The
Oseen solution then consists in writing the solution as
u=
+∞ X
X
(
w).
n=1 w∈Wn
A Capacitary Approach of the Navier–Stokes Integral Equations
79
As a final remark, let us recall that the numbers An are very well known: they are the
Catalan numbers which are widely used in combinatorics.
When 4C0 ∥u0 ∥E < 1, we can yet use another proof, based on the Banach contraction principle. We consider the map v 7→ F (v) = u0 + B(u, u) on the ball B0 = {v ∈
E / 2C0 ∥v∥E ≤ 1}. We have ∥F (v)∥E ≤ ∥u0 ∥E + C0 ∥v∥2E ≤ ∥u0 ∥E + 4C1 0 = δ0 < 2C1 0 .
Moreover, on the ball B1 = {v ∈ E / ∥v∥E ≤ δ0 }, we have
∥F (v) − F (w)∥E = ∥B(v, v − w) + B(v − w, w)∥E ≤ C0 ∥v − w∥E (∥v∥E + ∥w∥E )
≤ 2δ0 C0 ∥v − w∥E
with 2C0 δ0 < 1. Thus, F is contractive on B1 and has a unique fixed point in B0 .
In the case when ∥u0 ∥ = 4C1 0 , we already know that there exists a solution u in the ball
B0 . If 2C0 ∥u∥E < 1, then we see easily that this solution is unique in B0 , since, for v ∈ B0 ,
we have
1
∥u − (u0 + B(v, v))∥E ≤ ( + C0 ∥u∥E )∥u − v∥E .
2
In order to finish the proof of Theorem 5.1, it remains to deal with uniqueness in the case
4C0 ∥u∥E = 1. We follow the proof of Auscher and Tchamitchian [12]. LetPv0 ∈ B0 and
∞
let vn+1 = FP
(vn ). Recall that the fixed-point u has been given as u =
n=0 un with
∞
1
1
∥un ∥E ≤ αn , n=0 αn ≤ 2C0 ; thus, if ∥u∥E = 2C0 , we must have ∥un ∥E = αn .
P∞
We have ∥v0 ∥E ≤ 2C1 0 = n=0 ∥un ∥E . We are going to prove inductively that ∥vk −
Pk−1
P∞
n=0 un ∥E ≤
n=k ∥un ∥E . We write
vk+1 −
k
X
un = F (vk ) − F (
n=0
k−1
X
k−1
k
X
X
un ) + F (
un ) −
un = Ak + Bk .
n=0
n=0
n=0
We have
Ak = B(vk −
k−1
X
un , vk −
k−1
X
un ) + B(vk −
n=0
n=0
k−1
X
un ,
k−1
X
un ) + B(
n=0
n=0
k−1
X
un , vk −
n=0
n=0
Hence, we find
∥Ak ∥E ≤ C0 ∥vk −
k−1
X
un ∥E ∥vk −
n=0
≤ C0
∞
X
un ∥E + 2∥
n=0
∥un ∥E (
n=k
k−1
X
∞
X
∥un ∥E + 2
n=k
k−1
X
un ∥E
n=0
k−1
X
∥un ∥E )
n=0
On the other hand, we have
Bk = u0 + B(
k−1
X
n=0
=
k−1
X k−1
X
q=1 p=k−q
un ,
k−1
X
un ) − u0 −
n=0
B(up , uq )
p−1
k X
X
p=1 q=0
k−1
X
B(uq , up−1−q )
un ).
80
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
∥Bk ∥E ≤ C0
k−1
X k−1
X
∥up ∥E ∥uq ∥E .
q=1 p=k−q
Thus, we get
∥vk+1 −
k
X
un ∥E ≤ C0 (
n=0
∞
X
∥un ∥E )2 − (
n=0
∞
X
∞
X
∥un ∥E )(
n=0
∞
X
∥un ∥E )2
n=0
≤ 2C0 (
=
k−1
X
∥un ∥E )
n=k+1
∥un ∥E
n=k+1
Thus, vn converges to u and u is the unique fixed point of F in B0 .
We may again interpret the proof through Picard’s iterates in terms of the formal expansion in words of the form w1 ⊛ w2 . Indeed, the set W of words generated from u0
through repeated combination of words may be described inductively: starting from the set
V0 = {u0 }, we describe inductively the set Vn by: w ∈ W belongs to Vn+1 if either w = u0
or there exists two words w1 and w2 in Vn such that w = w1 ⊛w2 . ThenPwe easily check that
Vn ⊂ Vn+1 and that W = ∪n∈N
≤ 14 , we know that w∈W ∥w∥E < +∞.
P Vn . If C0 ∥u0 ∥E P
Thus, we have, writing vn = w∈Vn w and u = w∈W w, limn→+∞ ∥u − vn ∥E = 0. As we
have
X
X
X
vn+1 = u0 +
w1 ⊛ w2 = u0 + (
w1 ) ⊛ (
w2 ) = u0 + B(vn , vn ),
w1 ∈Vn
(w1 ,w2 )∈Vn ×Vn
w2 ∈Vn
we find that vn is the n-th iterate of Picard starting from v0 = u0 (for the function v 7→
F (v) = u0 + v ⊛ v).
The method used by Oseen for solving a non-linear equation is based on a power series
development, hence on analyticity. Combining with the method of analytic majorization
introduced by Cauchy, we have obtained uniqueness in the limit case ∥u0 ∥ = 2C1 0 .
Oseen’s method works in a more general setting than quadratic equations. In order to
solve a non-linear equation y = y0 + N (y) in a (complete) vector space, Oseen’s method
solves more generally yϵ = y0 + ϵN (yϵ ). If yϵ is analytical with respect to ϵ, we search for a
P∞
P∞
1 dk
Taylor development yϵ = n=0 ϵn yn , where yk = k!
( n=0 ϵn yn )|ϵ=0 . If N is analytical
dϵk
P
P∞
∞
with respect to y, we will have a Taylor series for N ( n=0 ϵn yn ) given by N( n=0 ϵn yn ) =
P∞ n
P∞ n
Pk
1 dk
1 dk
n
. Thus, we
n=0 ϵ An with Ak = k! dϵk (N (
n=0 ϵ yn ))|ϵ=0 = k! dϵk N (
n=0 ϵ yn )
|ϵ=0
find that
yk+1 =
1 dk
k! dϵk
N(
k
X
n=0
!
ϵn yn )
.
(5.5)
|ϵ=0
Hence, we get a cascade of equations. The problem is then to solve the equations and to
show that the radius of convergence is greater than 11 .
1 The old method used by Oseen has known modern developments in numerical analysis, where it is
known as Adomian’s decomposition method, a method introduced by Adomian in the eighties in George
Adomian, Solving Frontier problems of Physics: The decomposition method, Kluwer Academic Publishers,
1994.
A Capacitary Approach of the Navier–Stokes Integral Equations
81
We have followed Oseen’s method in Section 4.6. In modern texts on the Navier–Stokes
equations, such as the book of Cannone for instance [81], one uses a less stringent approach:
Picard’s iteration method. This method was introduced by Picard in 1890 [398] for solving
PDEs: one starts from z0 = y0 and one defines inductively
zk+1 as zk+1 = y0 + N (zk ) and
P∞
yk+1 as yk+1 = zk+1 − zk ; if N is a contraction, the series n=0 yn converges to the solution
y. This is known as the Banach contraction principle, as Banach stated the principle in
abstract vector spaces in 1922 [16].
5.3
A Capacitary Approach of Quadratic Integral Equations
In this section, we discuss the general integral equation
Z
f (x) = f0 (x) +
K(x, y)f 2 (y) dµ(y)
(5.6)
X
where µ is a non-negative σ-finite measure on a space X (X = ∪nN Yn with µ(Yn ) < +∞),
and K is a positive measurable function on X × X : K(x, y) > 0 almost everywhere. We
shall make a stronger assumption on K: there exists a sequence Xn of measurable subsets
of X such that X = ∪n∈N Xn and
Z Z
dµ(x) dµ(y)
< +∞.
(5.7)
K(x, y)
Xn Xn
We start with the following easy lemma:
Lemma 5.1.
Let f0 be non-negative and measurable and let fn be inductively defined as
Z
fn+1 (x) = f0 (x) +
K(x, y)fn2 (y) dµ(y)
(5.8)
X
Let f = supn∈N fn (x). Then either f = +∞ almost everywhere or f < +∞ almost everywhere. If f < +∞, then f is a solution to Equation (5.6).
Proof. Due to the inequalities f0 ≥ 0 and K ≥ 0, we find by induction that 0 ≤ fn ,
so that fn+1 is well defined (with values in [0, +∞]); we get moreover (by induction, as
well) that fn ≤ fn+1R. We thus may apply the theorem of monotone convergence and get
that
f (x) = f0 (x) + X K(x, y)f 2 (y) dµ(y). If f = +∞ on a set of positive measure, then
R
K(x, y)f 2 (y) dµ(y) = +∞ almost everywhere and f = +∞ almost everywhere.
X
We see that if f0 is such that Equation
R (5.6) has a solution f which is finite almost
everywhere, then we have f0 ≤ f and X K(x, y)f 2 (y) dµ(y) ≤ f (x). This is almost a
characterization of such functions f0 :
Proposition 5.1.
Let CK be the set
R of non-negative measurable functions Ω such that Ω < +∞ (almost
everywhere) and X K(x, y)Ω2 (y) dµ(y) ≤ Ω(x). Then
A) if Ω ∈ CK and if f0 is a non-negative measurable function such that f0 ≤ 14 Ω, Equation
(5.6) has a solution f which is finite almost everywhere. Moreover, we have
R
K(x,
y)f02 (y) dµ(y) ≤ 12 Ω(x).
X
82
The Navier–Stokes Problem in the 21st Century (2nd edition)
B) If Ω ∈ CK and if f0 is a non-negative measurable function such that
Z
1
K(x, y)f02 (y) dµ(y) ≤
Ω(x),
16
X
Equation (5.6) has a solution f which is finite almost everywhere.
Proof. A) Take the sequence of functions (fn )n∈N defined in Lemma 5.1. By induction,
we see that fn ≤ 12 Ω, and thus f = supn fn ≤ 12 Ω.
B) Take the sequence of functions (fn )n∈N defined in Lemma 5.1. By induction, we see
that fn ≤ f0 + 14 Ω, so that
Z
Z
1
1
2
K(x, y)fn (y) dµ(y) ≤ 2 K(x, y)(f0 (y)2 + Ω(y)2 ) dµ(y) ≤ Ω(x).
16
4
Thus, f = supn fn ≤ f0 + 14 Ω.
This remark leads us to define a Banach space of measurable functions in which it is
natural to solve Equation (5.6):
Proposition 5.2.
Let EK be the space of measurable functions f on X such that there exists λ ≥ 0 and Ω ∈ CK
such that |f (x)| ≤ λΩ almost everywhere. Then:
• EK is a linear space.
• The function f ∈ EK 7→ ∥f ∥K = inf{λ / ∃Ω ∈ Ck |f | ≤ λΩ} is a semi-norm on EK .
• ∥f ∥K = 0 ⇔ f = 0 almost everywhere.
• The normed linear space EK (obtained from EK by quotienting with the relationship
f ∼ g ⇔ f = g a.e.) is a Banach space.
• If f0 ∈ EK is non-negative and satisfies ∥f0 ∥K < 14 , then Equation (5.6) has a nonnegative solution f ∈ EK .
Proof. Since t 7→ t2 is a convex function, we find that CK is a balanced convex set and thus
that EK is a linear space and ∥ ∥K is a semi-norm on EK .
Next, we see that, for Ω ∈ CK , p, q ∈ N, we have
R R dµ(x) dµ(y)
Z
Xq
Ω(x) dµ(x) ≤
Xq
K(x,y)
(5.9)
(µ(Yp ∩ Xq ))2
Yp ∩Xq
This is easily checked by writing that
Z Z
Ω(y) dµ(x) dµ(y) ≤
(Yp ∩Xq )2
sZ
Xq
Z
Xq
dµ(x) dµ(y)
K(x, y)
sZ
Z
[
(5.10)
K(x, y)Ω2 (y)
dµ(y)] dµ(x)
Yp ∩Xq
Thus we find that, when ∥f ∥K = 0, we have
that f = 0 almost everywhere.
RR
Yp ∩Xq
|f (x)| dµ(x) = 0 for all p and q, so
A Capacitary Approach of the Navier–Stokes Integral Equations
83
P
P
Similarly, we find that if λn ≥ 0, Ωn ∈ CK and n∈N λn = 1, then, if Ω = n∈N λn Ωn ,
we have (by dominated convergence),
R
Z
Ω(x) dµ(x) ≤
Yp ∩Xq
dµ(x) dµ(y)
K(x,y)
(µ(Yp ∩ Xq ))2
R
Xn
Xn
(5.11)
so that Ω < +∞ almost everywhere. Moreover (by dominated convergence), we have Ω ∈
CK . From that, we easily get that EK is complete.
Finally, existence of a solution of (5.6) when ∥f0 ∥K < 41 is a consequence of Proposition
5.1.
Remark: We have obviously (by Proposition 5.1) that, for a measurable function f ,
Z
f ∈ EK ⇔ |f | ∈ EK ⇔ K(x, y)f (y)2 dµ(y) ∈ EK .
An easy corollary of Proposition 5.2 is the following one:
Proposition 5.3.
If E is a Banach space of measurable functions such that:
• f ∈ E → |f | ∈ E and ∥ |f | ∥E ≤ CE ∥f ∥E
R
• ∥ X K(x, y)f 2 (y) dµ(y)∥E ≤ CE ∥f ∥2E
then E is continuously embedded into EK .
Proof. By Theorem 5.1, we know that the equation
Z
Ω = Ω0 +
K(x, y)Ω(y)2 dµ(y)
X
has a unique solution in E when ∥Ω0 ∥E ≤ 4C1E . Moreover, this solution Ω is non-negative
if Ω0 is non-negative (as it is obtained by the series method), and thus Ω ∈ CK . Thus, for
2
f ∈ E, 4C 2 1∥f ∥E |f | ∈ CK , and ∥f ∥EK ≤ 4CE
∥f ∥E .
E
Now, we recall a result of Kalton and Verbitsky that characterizes the space EK for a
general class of kernels K [250].
Kalton and Verbitsky’s theorem
Theorem 5.2.
Assume that the kernel K satisfies:
• ρ(x, y) =
1
K(x,y)
is a quasi-metric:
1. ρ(x, y) = ρ(y, x) ≥ 0
2. ρ(x, y) = 0 ⇔ x = y
3. ρ(x, y) ≤ κ(ρ(x, z) + ρ(z, y))
• K satisfies the following inequality: there exists a constant C > 0 such that,
for all x ∈ X and all R > 0, we have
84
The Navier–Stokes Problem in the 21st Century (2nd edition)
Z
R
Z
dµ(y)
0
ρ(x,y)<t
dt
≤ CR
t2
Z
+∞
Z
dµ(y)
R
ρ(x,y)<t
dt
t3
(5.12)
Then the following assertions are equivalent for a measurable function f on X:
• (A) f ∈ EK
• (B) There exists a constant C such that, for all g ∈ L2 , we have
Z
Z
2
|f (x)|2
K(x, y)g(y) dµ(y) dµ(x) ≤ C∥g∥22
X
• (C) There exists a constant C such that, for almost every x,
Z
Z
Z
K(x, y)( K(y, z)f 2 (z)dµ(z))2 dµ(y)) ≤ C
K(x, y)f 2 (y) dµ(y)
X
5.4
(5.13)
X
X
(5.14)
X
Generalized Riesz Potentials on Spaces of Homogeneous Type
A direct consequence of Theorem 5.2 concerns generalized Riesz potentials on spaces of
homogeneous type.
Definition 5.1.
(X, δ, µ) is a space of homogeneous type if the quasi-metric δ and the measure µ satisfy:
• for all x, y ∈ X, δ(x, y) ≥ 0
• δ(x, y) = δ(y, x)
• δ(x, y) = 0 ⇔ x = y
• there is a positive constant κ such that:
for all x, y, z ∈ X, δ(x, y) ≤ κ(δ(x, z) + δ(z, y))
(5.15)
• there exists postive A, B and Q which satisfy:
for all x ∈ X, for all r > 0, ArQ ≤
Z
dµ(y) ≤ BrQ
(5.16)
δ(x,y)<r
Q is the homogeneous dimension of (X, δ, µ).
Riesz potentials
Theorem 5.3.
Let (X, δ, µ) be a space of homogeneous type, with homogeneous dimension Q. Let
Kα (x, y) =
1
δ(x, y)Q−α
(5.17)
A Capacitary Approach of the Navier–Stokes Integral Equations
85
(where 0 < α < Q/2) and EKα the associated Banach space (defined in Proposition
5.2). Let Iα be the Riesz operator associated Kα :
Z
Iα f (x) =
Kα (x, y)f (y) dµ(y).
(5.18)
X
We define two further linear spaces associated to Kα :
• the Sobolev space W α defined by
g ∈ W α ⇔ ∃h ∈ L2 g = Iα h
(5.19)
• the multiplier space V α defined by
Z
1/2
α
α
< +∞
f ∈ V ⇔ ∥f ∥V =
sup
|f (x)|2 |Iα h(x)|2 dµ(x)
∥h∥2 ≤1
(5.20)
X
(so that pointwise multiplication by a function in V α maps boundedly W α to L2 ).
Then, we have (with equivalence of norms) for 0 < α < Q/2:
EKα = V α .
Q
Proof. It is enough to see that At Q−α ≤
and that 1 <
Q
Q−α
(5.21)
Q
R
ρ(x,y)<t
dµ(y) ≤ Bt Q−α (with ρ(x, y) =
1
K(x,y)
)
< 2, then use Theorem 5.2.
As V α is defined as the space of pointwise multipliers from W α to L2 , we shall write
V = M(W α 7→ L2 ). This space of multipliers is not easy to handle: it can be characterized
through capacitary inequalities (for the case of Riesz potentials on Rn , this is a theorem of
Maz’ya [357]).
The space of multipliers however can be compared to easier spaces, the Morrey–
Campanato spaces.
α
Definition 5.2.
The (homogeneous) Morrey–Campanato space Ṁ p,q (X) (1 < p ≤ q < +∞) is the space of
the functions that are locally Lp and satisfy
Z
1/p
1
1
∥f ∥Ṁ p,q = sup sup RQ( q − p )
|f (y)|p dµ(y)
< +∞.
(5.22)
x∈X R>0
|δ(x,y)|<R
Remark that Lq ⊂ Ṁ p,q (X), as a direct consequence of Hölder inequality.
We shall need two technical lemmas on Morrey–Campanato spaces. The first lemma
deals with the Hardy–Littlewood maximal function:
Lemma 5.2.
Let Mf be the Hardy–Littlewood maximal function of f :
Z
1
Mf (x) = sup
|f (y)| dµ(y)
R>0 µ(B(x, R)) B(x,R)
(5.23)
where B(x, R) = {y ∈ X / δ(x, y) < R}. Then there exist constants Cp and Cp,q such that:
86
The Navier–Stokes Problem in the 21st Century (2nd edition)
• for every f ∈ L1 and every λ > 0,
µ({x ∈ X / Mf (x) > λ}) ≤ C1
∥f ∥1
λ
• for 1 < p ≤ +∞ and for every f ∈ Lp
∥Mf ∥p ≤ Cp ∥f ∥p
• for every 1 < p ≤ q < +∞ and for every f ∈ Ṁ p,q (X)
∥Mf ∥Ṁ p,q ≤ Cp,q ∥f ∥Ṁ p,q
Proof. The weak type (1,1) of the Hardy–Littlewood maximal function is a classical result
(see Coifman and Weiss [125] for the spaces of homogeneous type). The boundedness of the
maximal function on Lp for 1 < p ≤ +∞ is then a direct consequence of the Marcinkiewicz
interpolation theorem [215].
Thus, we shall be interested inRthe proof for Ṁ p,q (X). Let f ∈ Ṁ p,q (X). For x ∈ X
and R > 0, we need to estimate B(x,R) |Mf (y)|p dµ(y). We write f = f1 + f2 , where
f1 (y) = f (y)1B(x,2κR) (y). We have Mf ≤ Mf1 + Mf2 . We have
Z
p
Mf1 (y)p dµ(y) ≤ (Cp ∥f1 ∥p )p ≤ Cpp ∥f ∥pṀ p,q (2κR)Q(1− q ) .
B(x,R)
On the other hand, for δ(x, y) ≤ R,
Z
1
1
1
Mf2 (y) = sup
|f2 (z)| dµ(z) ≤ sup
∥f ∥Ṁ p,q ρQ(1− q )
Q
ρ>R µ(B(y, ρ)) B(y,ρ)
ρ>R Aρ
so that 1B(x,R) Mf2 ≤
Z
∥f ∥Ṁ p,q
1
and
AR q
Mf2 (y)p dµ(y) ≤ µ(B(x, R))∥1B(x,R) Mf2 ∥p∞ ≤
B(x,R)
p
B
∥f ∥pṀ p,q RQ(1− q ) .
Ap
The second lemma is a pointwise estimate for the Riesz potential, known as the Hedberg
inequality [3, 232].
Lemma 5.3 (Adams–Hedberg inequality).
If f ∈ Ṁ p,q (X) and if 0 < α < Q
q , then
Z
αq
αq
1
Q
|
f (y) dµ(y)| ≤ Cp,q,α (Mf (x))1− Q ∥f ∥Ṁ
.
p,q
Q−α
X δ(x, y)
(5.24)
Proof. Let R > 0. We have
Z
|
ρ(x,y)<R
+∞ Z
X
f (y)
≤
dµ(y)|
δ(x, y)Q−α
j=0
≤
+∞
X
j=0
≤B
R
2j+1
B2
≤ρ(x,y)< Rj
2
−jα
|f (y)|
dµ(y)
δ(x, y)Q−α
1
R
µ(B(x, 2−j R))
α
1
Rα Mf (x)
1 − 2−α
Z
|f (y)| dµ(y)
B(x,2−j R)
A Capacitary Approach of the Navier–Stokes Integral Equations
87
and
+∞ Z
Z
|
ρ(x,y)≥R
X
f (y)
dµ(y)| ≤
Q−α
δ(x, y)
j=0
≤
+∞
X
j=0
2j R≤ρ(x,y)<2j+1 R
|f (y)|
dµ(y)
δ(x, y)Q−α
1
1
1
1
1
B 1− p (2j+1 R)Q(1− p ) (2j+1 R)Q( p − q ) ∥f ∥Ṁ p,q
(2j R)Q−α
1
2Q(1− q )
1
≤ B 1− p
α− Q
q
1−2
We then end the proof by taking R
Q
q
=
∥f ∥Ṁ p,q
Mf (x)
Q
Rα− q ∥f ∥Ṁ p,q
.
As a direct corollary of Lemma 5.3, we get the following result of Adams [2] on Riesz
potentials2 :
Corollary 5.1.
p q
p,q
(X) to Ṁ λ , λ (X), with λ =
For 0 < α < Q
q , the Riesz potential Iα is bounded from Ṁ
1 − αq
Q.
We may now state the comparison result between spaces of multipliers and Morrey–
Campanato spaces, a result which is known as the Fefferman–Phong inequality [170]:
Proposition 5.4.
Let 0 < α < Q/2 and 2 < p ≤
Q
α.
Then we have:
Q
Q
Ṁ p, α (X) ⊂ V α = M(W α 7→ L2 ) ⊂ Ṁ 2, α (X)
Q
Q
p
(5.25)
Q
Proof. For f ∈ Ṁ p, α (X) and g ∈ Ṁ p, α (X), we have f g ∈ Ṁ 2 , 2α (X). We have p/2 > 1
Q
p, Q
α (X). Thus, from
, hence, since λ = 1 − αq
and α < Q/q with q = 2α
Q = 1/2, Iα (f g) ∈ Ṁ
Q
Proposition 5.3, we see that Ṁ p, α (X) ⊂ V α .
Q
The embedding V α ⊂ Ṁ 2, α (X) is easy to check. Indeed, if F = 1B(x,2κR) , we have for
y ∈ B(x, R)
Z
dµ(z)
µ(B(y, R)
Iα F (y) ≥
≥
≥ ARα
Q−α
ρ(z,
y)
RQ−α
ρ(z,y)<R
hence, for f ∈ V α ,
Z
B(x,R)
|f (y)|2 dµ(y) ≤
B(2κ)Q
∥F ∥22
2
∥f
∥
∥f ∥2V α RQ−2α .
α ≤
V
A2 R2α
A2
Remark: The embeddings are strict. For a proof in the case of the Euclidean space, see
for instance [318].
For the Navier–Stokes equations, we shall be interested in two examples of Riesz potenn
tials:
pPnclassical Riesz potentials on the usual Euclidean spacenR (with δ(x, y) = |x − y| =
2
i=1 |xi − yi | ) and parabolic Riesz potentials on R × R (with the parabolic [quasi]distance δ2 ((t, x), (s, y)) = |t − s|1/2 + |x − y|).
2 This
is sometimes called the Olsen inequality; see the paper by Olsen on Schrödinger potentials [380].
88
The Navier–Stokes Problem in the 21st Century (2nd edition)
Riesz potentials on Rn
Proposition 5.5.
pPn
2
In the case of the usual Euclidean space Rn with δ(x, y) = |x − y| =
i=1 |xi − yi | ,
α
α
W is the homogeneous Sobolev space Ḣ , i.e., the Banach space of tempered distributions
such that their Fourier transforms fˆ are locally integrable and satisfy
R 2α
ˆ
|ξ| |f (ξ)|2 dξ < +∞.
Thus, V α (Rn ) = M(Ḣ α 7→ L2 ).
Proof. Just check that the Fourier transform of
constant cα,n .
1
|x|n−α
is equal to cα,n |ξ|1α for a positive
Parabolic Riesz potential on R × Rn
Proposition 5.6.
Let δα be the parabolic (quasi)-distance
δα ((t, x), (s, y)) = |t − s|1/α + |x − y|
(5.26)
on R × Rn , where 0 < α. The associated homogeneous dimension (for the Lebesgue
measure) is Q = n + α.
For 0 < β < α, we consider the kernel
Kα,β (t − s, x − y) =
1
δα ((t, x), (s, y))Q−(α−β)
(5.27)
1
(|t|1/α + |x|)n+β
(5.28)
or equivalently
Kα,β (t, x) =
For 0 < α − β < Q/2, we consider the associated Banach spaces W α,β = Kα,β ∗ L2 and
V α,β = M(W α,β 7→ L2 ).
1− β ,α−β
If, moreover, β < 2, we define the Banach space Ḣt,x α
of tempered distributions
such that their Fourier transforms fˆ are locally integrable and satisfy
Z Z
β
(|ξ|α−β + |τ |1− α )2 |fˆ(τ, ξ)|2 dξ dτ < +∞
(5.29)
1− β ,α−β
Then (for β < 2), V α,β = M(Ḣt,x α
7→ L2 ).
Proof. We shall use the Landau notation Ω(.): F ≈ Ω(G) if there are two positive constants
c1 and c2 such that c1 < F/G < c2 . The proposition will be proved through the following
lemma:
Lemma 5.4.
Let Wβ,n (x) be defined as
Wβ,n (x) =
1
(2π)n
Z
Rn
β
e−|ξ| ei x.ξ dξ
(5.30)
A Capacitary Approach of the Navier–Stokes Integral Equations
89
Let Kα,β (t, x) be defined on R × Rn as
Kα,β (t, x) =
1
|t|
n+β
α
Wβ,n
x
(5.31)
1
|t| α
Then, for 0 < β < 2,:
Kα,β (t, x) ≈ Ω(Kα,β (t, x)).
(5.32)
Let Mα,β (τ, ξ) be the Fourier transform of Kα,β (t, x). Then
!
1
Mα,β (τ, ξ) ≈ Ω
.
β
|ξ|α−β + |τ |1− α
(5.33)
The first step of the proof is the estimation of Wβ,n (x). When β = 2, we get the Gaussian
function
|x|2
1
(5.34)
W2,n (x) =
e− 4 .
n/2
(4π)
When 0 < β < 2, we have a subordination of Wβ,n to W2,n :
+∞
Z
Wβ,n (x) =
0
1
σ n/2
x
W2,n ( √ ) dµβ (σ)
σ
(5.35)
where dµβ is a probability measure on (0, +∞) [421].
We have the following important result of Blumenthal and Getoor [46]: for 0 < β < 2,
there exists a positive constant cβ,n such that
lim
|x|→+∞
Wβ,n (x)|x|n+β = cβ,n .
(5.36)
Thus, we have
Wβ,n (x) ≈ Ω(
Recall that
1
).
(1 + |x|)n+β
(5.37)
1
(|t|1/α + |x|)n+β
Kα,β (x, y) =
which may be rewritten as
Kα,β (x, y) =
1
|t|
1
n+β
α
|x| n+β
)
|t|1/α
(1 +
≈ Ω(Kα,β (t, x)).
We now compute the Fourier transform Mα,β (τ, ξ) of Kα,β as the Fourier transform in
the time variable t of the Fourier transform N (t, ξ) in the space variable x of Kα,β . We have
N (t, ξ) =
so that
Z
1
|t|
β
α
β
1
Mα,β (τ, ξ) = C
β
R
β
e−|t| α |ξ|
|τ − η|1− α
1
Wβ
|ξ|α α ,1
(5.38)
η
|ξ|α
dη
(5.39)
Thus, we have
Z
Mα,β (τ, ξ) ≈ Ω
β
R
!
|ξ|β
1
β
|τ − η|1− α (|ξ|α + |η|)1+ α
dη .
(5.40)
90
The Navier–Stokes Problem in the 21st Century (2nd edition)
We may rewrite that estimate as
1
Mα,β (τ, ξ) ≈ Ω
with
|ξ|
Z
R
1
β
|τ |1− α
1
∞
|τ −
1
and H(τ ) =
β
(1+|τ |)1+ α
τ
)
|ξ|α
1
1
Aα,β (τ ) =
Let G(τ ) =
A (
α−β α,β
β
η|1− α
β
(1 + |η|)1+ α
(5.41)
dη.
(5.42)
, so that Aα,β = G ∗ H. Since G ∈ L1 + L∞ (R)
and H ∈ L ∩ L (R), we have that H ∗ G is continuous, positive and bounded, so that we
have, for |τ | ≤ 2, Aα,β (τ ) ≈ Ω(1). For |τ | > 2, we write:
• H ∗ G(τ ) ≥
•
R |τ |/2
•
R
−|τ |/2
2
|τ |
1− αβ R
1
−1
H(η) dη
G(τ − η)H(η) dη ≤
G(τ − η)H(η) dη ≤
|η|>|τ |/2
so that Aα,β (τ ) ≈ Ω
1
β
|τ |1− α
2
|τ |
1− αβ
∥H∥1
1
R
|η|>|τ |/2
1
β
β
|τ −η|1− α |η|1+ α
dη = C |τ1| ≤ C
1
|τ |
1− αβ
.
Now, the end of the proof is easy. Using Kalton and Verbitsky’s theorem (Theorem
Theorem 5.2), we begin with the inequality (5.13): from (5.32), we find that EKα,β = EKα,β
(with equivalence of norms).We now endow R×Rn with the quasi-metric ρ̃α,β ((t, x), (s, y)) =
1
(Kα,β (t − s, x − y))− n+β and apply again Kalton and Verbitsky’s theorem. We find that
V α,β = M(W̃ α,β 7→ L2 ) whith W̃ α,β = Kα,β ∗ L2 Taking the Fourier transform in time and
space variables, we see that
!
1
1− β ,α−β
−1
α,β
2
W̃
= Ft,x
= Ḣt,x α
β L
|ξ|α−β + |τ |1− α
1− β ,α−β
and thus V α,β = M(Ḣt,x α
5.5
7→ L2 ). This ends the proof.3
Dominating Functions for the Navier–Stokes Integral
Equations
In this section, we are going to solve Equation (5.3) through simple estimates on the
⃗0 = U
⃗ and we define inductively U
⃗ n+1 as
associated Picard iterates: we start from U
⃗ n=1 = U
⃗ −
U
Z tX
3
⃗ n ds.
∂j O(ν(t − s)) :: Un,j U
0 j=1
3 In
1− β ,α−β
the first edition of this book [319], we concluded that W α,β = Ḣt,x α
The correct statement has been given in [321].
, but this seems dubious.
A Capacitary Approach of the Navier–Stokes Integral Equations
91
Our starting point is the estimate
|
Z tX
3
⃗ ds|
∂j O(ν(t − s)) :: Vj W
0 j=1
Z tZ
≤ C0
0
1
⃗ (s, y)| |W
⃗ (s, y)| ds dy.
|V
ν 2 (t − s)2 + |x − y|4
Definition 5.3.
For 0 < T ≤ +∞, a function Ω(t, x) belongs to the set Γν,T of dominating functions for the
Navier–Stokes equations on (0, T ) if, for all 0 < t < T and all x ∈ R3 ,
Z tZ
1
4C0
Ω2 (s, y) ds dy ≤ Ω(t, x)
(5.43)
2
2
ν (t − s) + |x − y|4
0
Similarly, a function Ω(t, x) belongs to the set Γν of dominating functions for the Navier–
Stokes equations if, for all t ∈ R and all x ∈ R3 ,
ZZ
1
4C0
Ω2 (s, y) ds dy ≤ Ω(t, x)
(5.44)
2
2
4
R×Rn ν (t − s) + |x − y|
Of course, we have Γν ⊂ Γν,+∞ .
Dominating function may be used to establish the existence of solutions to the integral
Navier–Stokes equations:
Navier–Stokes equations and dominating functions
Theorem 5.4.
⃗ (t, x)| ≤ Ω(t, x) with Ω ∈ Γν,T , then the equation
If, for all 0 < t < T and x ∈ R3 , |U
Z tX
3
⃗ −
⃗u = U
∂j O(ν(t − s)) :: uj ⃗u ds
0 j=1
has a solution ⃗u on (0, T ) × R3 such that |⃗u(t, x)| ≤ 2Ω(t, x).
⃗ (t, x)| for 0 < t < T , and
Proof. We define Ω0 (t, x) as Ω0 (t, x) = |U
Z tZ
1
Ωn+1 (t, x) = Ω0 (t, x) + C0
Ω (s, y)2 dy ds.
2 (t − s)2 + |x − y|4 n
ν
3
0
R
By induction on n, we find that Ωn (t, x) ≤ 2Ω(t, x). Thus, the non-decreasing sequence
Ωn (t, x) converge to Ω∞ (t, x) which satisfies Ω∞ ≤ 2Ω and
Z tZ
1
Ω∞ (t, x) = Ω0 (t, x) + C0
Ω (s, y)2 dy ds.
2
2
4 ∞
0
R3 ν (t − s) + |x − y|
⃗n =U
⃗ n+1 − U
⃗ n , we have
For W
Z tX
3
⃗
⃗ n + Wn,j U
⃗ n + Wn,j W
⃗ n ds|
|Wn+1 | = |
∂j O(ν(t − s)) :: Un,j W
0 j=1
Z tZ
≤ C0
0
ν 2 (t
−
1
⃗ n (s, y)|2 + 2|U
⃗ n (s, y)|)|W
⃗ n (s, y)| ds dy
(|W
+ |x − y|4
s)2
92
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ n (t, x)| ≤ Ωn+1 (t, x) − Ωn (t, x). Thus, U
⃗ n converges
By induction on n, we find that |W
almost everywhere.
What we did was just applying Proposition 5.2 to the kernel
Kν (t − s, x − y) = C0 1t−s>0
to solve
ZZ
Ω∞ = Ω0 +
(0,T )×R3
1
ν 2 (t − s)2 + |x − y|4
Kν (t − s, x − y)Ω2∞ (y) dy.
This proposition associates a Banach space EKν ,T to Kν , and the sufficient condition we find
⃗ to get a solution to the Navier–Stokes integral equations on (0, T ) × R3 is |U
⃗ | ∈ EK ,T
on U
ν
⃗
and ∥|U |∥EKν ,T ≤ 1/4. Note that the space EKν ,T does not depend on ν, different values of
ν give equivalent norms.
Recall that
⃗ (t, x) = Wνt ∗ ⃗u0 −
U
Z tX
3
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G ds
0 j=1
so that
⃗ (t, x)| ≤ |Wνt ∗ ⃗u0 | + C0
|U
Z tZ
ν 2 (t
0
with
−
1
⃗
|f⃗ ∗ ∇G|
ds dy
+ |x − y|4
s)2
3 X
3
X
⃗
|f⃗ ∗ ∇G|
=(
|fi ∗ ∂j G|2 )1/2 .
i=1 j=1
Navier–Stokes equations and EKν ,T spaces
Corollary 5.2.
Let
⃗ (t, x) = Wνt ∗ ⃗u0 −
U
Z tX
3
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G ds.
0 j=1
If
• |Wνt ∗ ⃗u0 | ∈ EKν ,T and ∥|Wνt ∗ ⃗u0 |∥EKν ,T ≤ 18
q
q
⃗
⃗
•
|f⃗ ∗ ∇G|
∈ EKν ,T and ∥ |f⃗ ∗ ∇G|∥
EKν ,T ≤
1
√
2 2
R
⃗ − t P3 ∂j O(ν(t − s)) :: uj ⃗u ds has a solution ⃗u on
then the equation ⃗u = U
j=1
0
(0, T ) × R3 such that ∥|⃗u| ∥EKν ,T ≤ 12 .
Similarly, the kernek Kν = C0 ν 2 (t−s)21+|x−y|4 induces a norm ∥ ∥Kν on the space V 2,1 (R×
1
,1
2
R3 ) = M(Ḣt,x
7→ L2 ), and we have:
A Capacitary Approach of the Navier–Stokes Integral Equations
93
Navier–Stokes equations and the multiplier space
Corollary 5.3.
Let
⃗ (t, x) = Wνt ∗ ⃗u0 −
U
Z tX
3
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G ds.
0 j=1
Let 0 < T ≤ +∞.If
• 10<t<T |Wνt ∗ ⃗u0 | ∈ V 2,1 (R × R3 ) and ∥10<t<T |Wνt ∗ ⃗u0 |∥Kν ≤ 81
q
q
⃗
⃗
• 10<t<T |f⃗ ∗ ∇G|
∈ V 2,1 (R × R3 ) and ∥10<t<T |f⃗ ∗ ∇G|∥
Kν ≤
1
√
2 2
then the equation
⃗ −
⃗u = U
Z tX
3
∂j O(ν(t − s)) :: uj ⃗u ds
0 j=1
has a solution ⃗u on (0, +T ) × R3 such that 10<t<T |⃗u| ∈ V 2,1 (R × R3 ) and
∥10<t<T |⃗u| ∥Kν ≤ 12 .
⃗ may be computed as
Remark: note that U
⃗ = Wνt ∗ ⃗u0 +
U
Z tX
3
∂j O(ν(t − s)) :: F⃗j ds
(5.45)
0 j=1
where F⃗j satisfies
f⃗ =
3
X
∂j F⃗j
(5.46)
j=1
q
P3
⃗
One then replaces estimates on |f⃗ ∗ ∇G|
by similar estimates on ( j=1 |F⃗j |2 )1/4. Conditions
expressed on F⃗j are easier to deal with than for f⃗.
5.6
Oseen’s Theorem and Dominating Functions
In this section, we partly reprove Theorem 4.9 in the light of Theorem 5.4.
Lemma 5.5. There exists constants ϵ0 , ϵ1 and ϵ2 such that, for all t ∈ R, all x ∈ R3 and
all ν > 0, we have
ZZ
1
1
1
1
p
dy ds ≤ ϵ0 p
(5.47)
2 (t − s)2 + |x − y|4
2
ν
ν
3
( ν|s| + |y|)
ν|t| + |x|
R×R
Z
Wνt (x − y)
R3
1
1
dy ≤ ϵ1 p
|y|
ν|t| + |x|
(5.48)
94
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
Z
R3
p
ν|t|
1
1
dy ≤ ϵ2 p
(νt)2 + |x − y|4 |y|2
( ν|t| + |x|)2
(5.49)
Proof. See formula (4.30) and Corollary 4.1.
Combining this lemma with Theorem 5.4, we then find:
Rough data (global existence)
Theorem 5.5. There exists a constant η0 > 0 (which does not depend on ν) such that
the function
ν
Gν (t, x) = η0 p
ν|t| + |x|
belongs to V 2,1 (R × R3 ) with ∥Gν ∥Kν ≤ 1. Moreover, there exists a positive constant η1
(which does not depend on ν) such that, if ⃗u0 satisfies:
ν
|x|
(5.50)
p
ν 2 ν|t|
νt2 + |x|4
(5.51)
|⃗u0 (x)| ≤ η1
P3
and f⃗ = j=1 ∂j F⃗j satisfies
|f⃗(t, x)| ≤ η1
or, for j = 1, . . . , 3,
|F⃗j | ≤ η1
ν2
ν|t| + |x|2
(5.52)
then there exists a unique solution ⃗u of
⃗u = Wνt ∗ ⃗u0 +
Z tX
3
∂j O(ν(t − s)) :: F⃗j − uj ⃗u ds
(5.53)
0 j=1
on [0, +∞) × R3 such that:
|⃗u(t, x)| ≤
1
Gν (t, x)
2
(5.54)
Remark: Inequality (5.50) may be viewed as a localization of the smallness condition on
the Reynolds number of the fluid. If U is the characteristic velocity of the fluid, L the
characteristic length and ν the kinematic viscosity, the Reynolds number is Re = UνL ; here,
we have a condition on the pointwise estimate of |⃗u0 (x)||x|
.
ν
5.7
Functional Spaces and Multipliers
We shall be interested in this section in the following functional spaces:
A Capacitary Approach of the Navier–Stokes Integral Equations
95
Definition 5.4.
Xν,T is the space of distributions u0 ∈ S ′ (R3 ) such that
10<t<T Wνt ∗ u0 ∈ EKν ,T
and Xν is the space of distributions u0 such that
10<t Wνt ∗ u0 ∈ V 1,2 (R × R3 ).
Remark: The space Xν has been introduced by Lemarié-Rieusset in 2013 in a conference
organized by Warwick University in Venice [320] as a near optimal space for solving the
Navier–Stokes equations; the same conclusion has been reached independently by Dao and
Nguyen in 2017 [143].
We shall discuss some examples of subspaces E of Xν , characterized by u0 ∈ E ⇔
10<t Wνt ∗ u0 ∈ E, where E is a subspace of V 1,2 (R × R3 ).
ˆ Example 1: A classical example is the space Lpt Lqx with
Z Z
(
E = Lpt Lqx = {F Lebesgue measurable/
+ 3q = 1 and 3 < q < +∞:
2
p
p
|F (t, x)|q dx) q dt < +∞}.
R3
R
In order to check that Lpt Lqx ⊂ V 1,2 (R×R3 ), we just write Lpt Lqx ⊂ Ṁ min(p,q),5 (R×R3 ),
with 2 < min(p, q) ≤ 5 and then apply the Fefferman–Phong inequality. Indeed, we
have, if p ≤ q
ZZ
|u(s, y)|p ds dy ≤
t+R2
Z
[t−R2 ,t+R2 ]×B(x,R)
t−R2
≤
p
∥u(s, .)∥pq |B(x, R)|1− q ds
p
C∥u∥pLp Lqx R3(1− q )
t
= C∥u∥pLp Lqx R5−p .
t
If q ≤ p, we have
ZZ
t+R2
Z
q
|u(s, y)| ds dy ≤
[t−R2 ,t+R2 ]×B(x,R)
t−R2
∥u(s, .)∥qq ds
q
≤∥u∥qLp Lqx (2R2 )1− p = C∥u∥qLp Lqx R5−q .
t
t
Solutions in Lp Lq were first described in 1972 by Fabes, Jones and Rivière [168]. The
corresponding initial values belong to a homogeneous Besov space [36, 313, 475]:
−2
10<t Wνt ∗ u0 ∈ Lpt Lqx ⇔ u0 ∈ Ḃq,pp
ˆ Example 2: The same proof works when one changes the order of integration in t
and in x and considers Lqx Lpt with p2 + 3q = 1 and 3 < q < +∞:
E = Lqx Lpt = {F Lebesgue measurable/
Z
R3
Z
q
( |F (t, x)|p dt) p dx < +∞}.
R
The corresponding initial values belong to a homogeneous Triebel–Lizorkin space [36,
475]:
−2
10<t Wνt ∗ u0 ∈ Lqx Lpt ⇔ u0 ∈ Ḟq,pp
96
The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ Example 3: When q = 3, the limiting cases of Example 1 and Example 2 corresponds
3
3 ∞
to L∞
t Lx and Lx Lt , which we define [313] as
3
L∞
t Lx = {F (t, x) Lebesgue measurable/ ∥ ∥F (t, x)∥L3 (dx) ∥L∞ (dt) < +∞}
and
L3x L∞
t = {F (t, x) Lebesgue measurable/ ∥ ∥F (t, x)∥L∞ (dt) ∥L3 (dx) < +∞}.
We have
∞ 3
3,5
L3x L∞
(R × R3 )
t ⊂ Lt Lx ⊂ Ṁ
The corresponding initial values then belong to L3 :
∞ 3
3
10<t Wνt ∗ u0 ∈ L3x L∞
t ⇔ 10<t Wνt ∗ u0 ∈ Lt Lx ⇔ u0 ∈ L
This is based on the inequality |Wνt ∗ u(x)| ≤ Mu0 (x), where Mu0 is the Hardy–
Littlewood maximal function of u0 . The idea of using the maximal function in order
to estimate the integrals in the Navier–Stokes problem goes back to Calderón in 1993
[78].
ˆ Example 4: A variation on example 1 is the case of
1
E = {F (t, x) Lebesgue measurable/ sup |t| p ∥F (t, x)∥Lq (dx) < +∞}
t∈R
with
2
p
+
3
q
= 1 and 3 < q < +∞ (and where supt∈R is taken as the essential
supremum). Indeed, let 2 < r < min(p, q); we shall see that E ⊂ Ṁ r,5 (R × R3 ). We
just write
ZZ
Z
r
t+R2
|u(s, y)| ds dy ≤
[t−R2 ,t+R2 ]×B(x,R)
t−R2
r
−p
≤ C∥|s|
r
∥u(s, .)∥rq |B(x, R)|1− q ds
1
r
r
∥Ṁ 1, pr R2(1− p ) (sup |s| p ∥u∥q )r R3(1− q )
s∈R
r
−p
= C∥|s|
1
∥Ṁ 1, pr (sup |s| p ∥u∥q )r R5−r .
s∈R
Solutions in this space E were first described in 1995 by Cannone [81]. The corresponding initial values belong to a homogeneous Besov space [36, 81, 313, 475]:
−2
1
p
sup t p ∥Wνt ∗ u0 ∥q < +∞ ⇔ u0 ∈ Ḃq,∞
0<t
ˆ Example 5: Of course, the same proof works when one changes the order of integration in t and in x and consider the space
1
E = {F (t, x) Lebesgue measurable/ ∥ sup |t| p |F (t, x)|∥Lq (dx) < +∞}
t∈R
with
2
p
+
3
q
= 1 and 3 < q < +∞ (and where supt∈R is taken as the essential
supremum). Let us consider again 2 < r < min(p, q); we shall see that E ⊂ Ṁ r,5
A Capacitary Approach of the Navier–Stokes Integral Equations
97
(R × R3 ). We just write
ZZ
|u(s, y)|r ds dy
[t−R2 ,t+R2 ]×B(x,R)
Z
r
r
≤
B(x,r)
1
∥|s|− p ∥Ṁ 1, pr (2R)2(1− p ) (sup |s| p |u(s, y)|)r dy
s∈R
r
−p
≤ C∥|s|
∥Ṁ 1, pr R
r
2(1− p
)
r
−p
= C∥|s|
1
r
∥ sup |s| p |u(s, y)|∥rq R3(1− q )
s∈R
∥
1
p
Ṁ 1, r
∥ sup |s| p |u(s, y)|∥rq R5−r .
s∈R
The corresponding initial values belong to a homogeneous Triebel–Lizorkin space [36,
475]:
−2
1
p
sup t p |Wνt ∗ u0 | ∈ Lq ⇔ u0 ∈ Ḟq,∞
0<t
1
ˆ Example 6: In example 5, the elements of E satisfy |F (t, x)| ≤ t− p G(x), with
G ∈ Lq . More generally, one may look at the dominating functions which satisfy
|F (t, x)| = tα G(x) with 0 < α < 21 . We have
ZZ
Kν (t − s, x − y)s−2α G2 (y) ds dy ≤
Z
Z
C0
1
−2α
s
ds
G2 (y) dy
sup
2
2
ν (t − s) + 1
|x − y|2+2α
t∈R
so that we look for a dominating function G for the kernel of the Riesz transform
I1−2α , hence G ∈ V 1−2α (R3 ).
This leads us to consider
1
E = {F (t, x) Lebesgue measurable/ ∥ sup |t| p |F (t, x)|∥V 3/q (dx < +∞}
t∈R
2
p
3
q
= 1 and 3 < q < +∞ (and where supt∈R is taken as the essential
with +
supremum).
The corresponding initial values belong to a homogeneous Triebel–Lizorkin space
−2/p
ḞV 3/q ,∞ (based on the multiplier space V 3/q ), which has not yet been defined in the
literature. Besov spaces on multiplier spaces were introduced by Lemarié-Rieusset in
2002 [313]. Besov spaces and Triebel–Lizorkin spaces based on Morrey–Campanato
spaces were defined by Kozono and Yamazaki in 1994 [279] and are extensively studied
by Sickel, Yang and Yuan in [436].
ˆ Example 7: If, in Example 6, we take p = +∞, we are led to consider
E = {F (t, x) Lebesgue measurable/ ∥ sup |F (t, x)|∥V 1 (dx < +∞}
t∈R
(where supt∈R is taken as the essential supremum).
The corresponding initial values then belong to V 1 (R3 ):
sup |Wνt ∗ u0 | ∈ V 1 (R3 ) ⇔ u0 ∈ V 1 (R3 )
t>0
This is based once again on the inequality |Wνt ∗ u(x)| ≤ Mu0 (x), where Mu0 is the
Hardy–Littlewood maximal function of u0 . The boundedness of the maximal function
on the multiplier space V 1 has been proven by Maz’ya and Verbitsky [358].
98
The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ Example 8: If we take, in Example 6, q = +∞, we meet a slight disappointment.
Indeed, if we look for a dominating function F (t, x) = H(t), we find that
Z
ZZ
Z
1
C0
p
H 2 (s) ds
Kν (t − s, x − y)H 2 (s) ds dy =
4
1 + |y|
ν|t − s|
but the only dominating function H for the kernel √
1
ν|t−s|
is the null function.
Thus, we must work on a bounded time interval and look for a dominating function
HT such that:
Z t
1
p
H 2 (s) ds ≤ CT H(t).
(5.55)
for 0 < t < T,
ν|t − s|
0
The inequality (5.55) has a solution
√
ν
Hν (t) = √
.
t ln( eT
t )
−1(ln)
1
The condition sup0<t<T t 2 ln( eT
t )|Wνt ∗ u0 (x)| < +∞ is equivalent to u0 ∈ B∞,∞ ,
−1
. Such initial values for the Navier–Stokes
a space close to the Besov space B∞,∞
equations have been studied by Yoneda [510].
ˆ Example 9: From all the previous examples, one can see that it is worthwhile to
consider the space
E = Ṁ p,5 (R × R3 )
with 2 < p ≤ 5.
We thus must characterize the space E of distributions u0 such that 1t>0 Wνt ∗ u0
belong to Ṁ p,5 (R × R3 ).
If u0 belongs to E, then obviously u0 (x − x0 ) (x0 ∈ R3 ) belongs to E with the same
norm, and λu0 (λx) (λ > 0) belongs to E with the same norm. Thus, we find that
−1
−1
E ⊂ Ḃ∞,∞
[313] and |Wνt ∗ u0 (x)| ≤ C √1νt ∥u0 ∥E . Conversely, if u0 ∈ Ḃ∞,∞
, if R > 0,
3
2
2
2
if x ∈ R and if |t| > 2R , let QR (t, x) = [t − R , t + R ] × B(x, R); we have
ZZ
Z
p
|1s>0 Wνs ∗ u0 (y)|p ds dy ≤ C∥u0 ∥pḂ −1 R3
|1s>0 s− 2 | ds
∞,∞
QR (t,x)
[t−R2 ,t+R2 ]
with |s| ≥ 21 |t| ≥ R2 , hence
ZZ
|1s>0 Wνs ∗ u0 (y)|p ds dy ≤ C∥u0 ∥pḂ −1 R5−p .
∞,∞
QR (t,x)
On the other hand, if |t| < 2R2 , we have QR (t, x) ⊂ Q√3R (0, x). Thus we see that u0
belongs to E if and only if we have
√
t|Wνt ∗ u0 (x)| < +∞
(5.56)
sup
t>0,x∈R3
and
sup
R>0,x∈R3
1
R5−p
ZZ
[0,R2 ]×B(x,R)
|Wνs ∗ u0 (y)|p ds dy < +∞.
(5.57)
A Capacitary Approach of the Navier–Stokes Integral Equations
99
From (5.56) and p > 2, we have
Z +∞
|Wνs ∗ u0 (y)|p ds ≤ C∥u0 ∥pḂ −1 R2−p .
∞,∞
R2
Thus, we have that u0 belongs to E if and only if we have
Z
Z +∞
1
sup
(
|Wνs ∗ u0 (y)|p ds) dy < +∞.
5−p
R>0,x∈R3 R
B(x,R) 0
(5.58)
R +∞
1
This is equivalent to the fact that ( 0 |Wνs ∗ u0 (y)|p ds) p belongs to Ṁ p,q (R3 ) with
2
3
p + q = 1.
Thus, we have obtained: for 2 < p ≤ 5,
−2
p
10<t Wνt ∗ u0 ∈ Ṁ p,5 (R × R3 ) ⇔ u0 ∈ ḞṀ p,q
,p
−2
p
is a Triebel–Lizorkin-type space based on Morrey
with p2 + 3q = 1. The space ḞṀ p,q
,p
spaces instead of Lebesgue spaces. In the notations of the book [436], this is the space
1
−2,p
− q1
Ḟp,pp
.
ˆ Example 10: One may try to explore the space Ep of the distributions that satisfy
(5.57) with 1 ≤ p ≤ 2.
For p = 2, one obtains the space BM O−1 . We shall see in Chapter 9 that the Navier–
Stokes equations may be solved for a small data in BM O−1 (this is the theorem
of Koch and Tataru [266]). However, one must use a new tool, using cancellation
properties of the convolution kernels that occur in the Oseen tensor, and no longer
deal only with absolute values.
−1
, for which the formalism of capacFor p < 2, we obtain the Besov space Ḃ∞,∞
itary inequalities is clearly not working (see the cheap Navier–Stokes equation of
Montgomery–Smith [369]).
ˆ Example 11: If we want to take into account the results of Kozono and Yamazaki
−2
[279], we should look for an initial value in ḂṀ p1,q ,∞ with p2 + 3q = 1. This is the
larger space in the scale considered by Kozono and Yamazaki. In particular, we have
−2
−2
−2
−2
−2
p
⊂
the inclusions Ḃq,pp ⊂ ḂṀ p1,q ,∞ (Example 1), Ḟq,pp ⊂ ḂṀ p1,q ,∞ (Example 2), Ḃq,∞
−2
−2
−2
−2/p
−2
p
ḂṀ p1,q ,∞ (Example 4), Ḟq,∞
⊂ ḂṀ p1,q ,∞ (Example 5), ḞV 3/q ,∞ ⊂ ḂṀ p1,q ,∞ (Example
−2
1
−2,p
− q1
p
6) and ḞṀ p,q
= Ḟp,pp
,p
−2
⊂ ḂṀ p1,q ,∞ (Example 9).
−2
The condition u0 ∈ ḂṀ p1,q ,∞ is equivalent to
√
1
sup t p ∥Wνt ∗ u0 (x)∥Ṁ 1,q (dx) < +∞ and sup t∥Wνt ∗ u0 (x)∥L∞ (dx) < +∞.
t>0
0<t
This gives, for all 0 < θ < 1,
θ
sup t p +
t>0
1−θ
2
∥Wνt ∗ u0 (x)∥
1 q
Ṁ θ , θ
< +∞.
For θ < 21 , we obtain that 1t>0 Wνt ∗ u0 (x) belongs to Ṁ r,5 (R × R3 ) for max(θ, pθ +
1−θ
1
1
2 ) < r < 2 (with the same proof as Example 4).
100
The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ Example 12: A variation on example 9 is the case of parabolic Morrey spaces in
mixed norms considered by Krylov for the heat equation [282, 281, 322]: defining
Qr(t, x) = (t − r2 , t + r2 ) × B(x, r),
E = {F (t, x) measurable/
sup
r>0,t>0,x∈R3
2
r q +3q−1 ∥F (t, x)∥Lpt Lqx (Qr (t,x)) < +∞}
or
E = {F (t, x) measurable/
sup
r>0,t>0,x∈R3
with
Ṁ
p,5
2
p
+
3
q
2
r q +3q−1 ∥F (t, x)∥Lqx Lpt (Qr (t,x)) < +∞}
> 1 and 2 < p, q < +∞. [When p = q, we find again the Morrrey space
(R × R3 ) of example 9.]
Chapter 6
The Differential and the Integral
Navier–Stokes Equations
In Chapter 4, we have seen classical solutions of the Navier–Stokes equations: the solution ⃗u
was C 2 in space variable, and C 1 in time variable, and the pressure was C 1 in space variable,
so that all the derivatives in the Navier–Stokes equations were classical derivatives. In
Chapter 5, we considered measurable solutions of the integral equations derived from the
Navier–Stokes equations, and we did not assume any differentiability on the solutions.
In the following chapters, we will study solutions in the sense of distributions of the
differential equations, such as Kato’s mild solutions or Leray’s weak solutions.
In this chapter, we shall discuss the relations between the differential equations and the
integral equations, where the pressure has been eliminated through the Leray projection
operator. In the absence of external forces, such discussion has been developed by Furioli,
Lemarié-Rieusset and Terraneo in [187] (see also Lemarié-Rieusset [313] and Dubois [158])1 .
The elimination of the pressure has also been discussed in 2011 by Tao in [461] and in 2020
by Bradshaw and Tsai [58] and Fernández-Dalgo and Lemarié-Rieusset [174]
6.1
Very Weak Solutions for the Navier–Stokes Equations
We now consider the Navier–Stokes equations:
⃗ u + f⃗ − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
(6.1)
Although they were derived under the assumptions that ⃗u was a regular vector field with
small derivatives, we shall consider solutions in the sense of distributions. In order to relax
⃗ u in the form div(⃗u ⊗ ⃗u).
regularity assumptions on ⃗u, it is better to write the term (⃗u.∇)⃗
In order to be able to define div(⃗u ⊗ ⃗u) as a distribution on (0, T ) × R3 , we shall assume
that ⃗u is locally square integrable on (0, T ) × R3 .
Another problem is to be able to define the initial value of ⃗u. This will be usually done
by an integrability assumption on ∂t ⃗u up to time t = 0:
Definition 6.1.
Let T > 0, 1 ≤ p ≤ ∞, and σ ∈ R. The local spaces (Lpt H σ )loc are the spaces of distributions
u ∈ D′ ((0, T ) × R3 ) such that, for every 0 < T0 < T and every φ ∈ D(R3 ),
φu ∈ Lpt ((0, T0 ), H σ ).
1 This discussion goes back to the 70’s, with, for instance, the paper of Fabes, Jones and Rivière [168]
in the case of Lpt Lqx mild solutions.
DOI: 10.1201/9781003042594-6
101
102
The Navier–Stokes Problem in the 21st Century (2nd edition)
The following lemmas will allow us to define the value of a time-dependent distribution
⃗u at a given time t:
Lemma 6.1.
Let v be a distribution on (0, T ) × R3 such that, for some σ ∈ R, v ∈ (L1t H σ )loc . We define
Rt
V = 0 v(s, .) ds. We have the following properties:
σ
• V ∈ (L∞
t H )loc .
• t ∈ (0, T ) 7→ V (t, .) is continuous from (0, T ) to D′ (R3 ), limt→0 V (t, .) = 0 in D′ (R3 ).
• ∂t V = v in D′ ((0, T ) × R3 ).
Proof. To define V as a distribution can be done locally: we define φV for φ ∈ D(R3 ). Let
w = φv on (0, T0 ) × R3 . We extend w to R × R3 by defining w = 0 if t < 0 or t > T0 .
Then, w ∈ L1 (R, H σ ). By standard arguments (truncation and regularization), one sees
that D(R × R3 ) is dense in L1 (R, H σ ). The map
Z
3
w ∈ D(R × R ) 7→ F (w) =
t
w(s, .) ds ∈ Cb (R, H 1 )
0
is bounded for the norm ∥w∥L1 H 1 . Thus, it can be extended to L1 H 1 with values in
Cb (R, H 1 ). As Cb (R, H 1 ) is continuously embedded in D′ , we find that ∂t F (w) = w in
D′ , as it is obvious if w ∈ D.
Corollary 6.1.
Let u be a distribution on (0, T ) × R3 such that, for some σ0 , σ1 ∈ R, u ∈ (L1t H σ0 )loc and,
Rt
∂t u ∈ (L1t H σ1 )loc . We define U = 0 ∂t u(s, .) ds. We have the following properties:
σ
• U ∈ (L∞
t H )loc .
• t ∈ (0, T ) 7→ U (t, .) is continuous from (0, T ) to D′ (R3 ), limt→0 U (t, .) = 0 in D′ (R3 ).
• ∂t U = ∂t u in D′ ((0, T ) × R3 ).
min(σ1 ,σ2 )
• There exists a u0 ∈ Hloc
(R3 ) such that u = U + 1 ⊗ u0
• Representing u as u = U + 1 ⊗ u0 , we find that t ∈ (0, T ) 7→ u(t, .) = U (t, .) + u0 is
continuous from (0, T ) to D′ (R3 ), and limt→0 u(t, .) = u0 in D′ (R3 ).
We shall often use a lemma on energy estimates:
Energy estimates
Lemma 6.2.
Let u be a distribution on (0, T ) × R3 such that u ∈ L2 ((0, T ), H 1 (R3 )) and ∂t u ∈
L2 ((0, T ), H −1 (R3 )). Then u has representant such that u ∈ C([0, T ], L2 ) and
∥u(t, .)∥22
=
∥u0 ∥22
Z
t
⟨∂t u(s, .)|u(s, .)⟩H −1 ,H 1 ds.
+2
0
The Differential and the Integral Navier–Stokes Equations
103
Proof. We may extend u to (−T, T ) by defining u(t, x) = u(−t, x) for t < 0. Then, u ∈
L2 ((−T, T ), H 1 (R3 )) and ∂t u ∈ L2 ((−T, T ), H 1 (R3 )). If T0 < T and if θ is asmooth function
on R which is compactly supported in (−T, T ) and is equal to 1 on a neighborhood of
[−T0 , T0 ], then θu ∈ L2 (R, H 1 (R3 )) and ∂t (θu) ∈ L2 (R, H 1 (R3 )). By standard arguments
(truncation and regularization), one sees that D(R × R3 ) is dense in the space
E = {u ∈ L2 (R, H 1 (R3 )) / ∂t (θu) ∈ L2 (R, H 1 (R3 ))}.
We may then define the trace of u ∈ E for time t = t0 by extending to E the map
u ∈ D 7→ u(t0 , .) ∈ L2 :
Z t0
∥u(t0 , .)∥22 = 2
⟨∂t u(s, .)|u(s, .)⟩H −1 ,H 1 ds.
∞
Similarly, the bilinear form
Z
t1
(u, v) 7→ B(u, v) =
⟨∂t u(s, .)|v(s, .)⟩H −1 ,H 1 ds
t0
is bounded on E × E and B(u, u) = ∥u(t1 , .)∥22 − ∥u(t0 , .)∥22 on D.
We then have the concept of a very weak solution:
Very weak solution
Definition 6.2.
A very weak solution ⃗u of equations (6.1) on (0, T ) × R3 , for data ⃗u0 ∈ D′ (R3 ) with
div ⃗u0 = 0 and f⃗ ∈ D′ ((0, T ) × R3 ) is a distribution vector field ⃗u(t, x) ∈ D′ ((0, T ) × R3 )
such that:
• div ⃗u = 0
• ⃗u is locally square integrable on (0, T ) × R3
• the map t ∈ (0, T ) 7→ ⃗u(t, .) is continuous from (0, T ) to D′ (R3 ) and
limt→0+ ⃗u(t, .) = ⃗u0
• for all φ
⃗ ∈ D((0, T ) × R3 ) with div φ
⃗ = 0, we have
⟨∂t ⃗u − ν∆⃗u + div(⃗u ⊗ ⃗u) − f⃗|⃗
φ⟩D′ ,D = 0
(6.2)
⃗ is implicitly defined by Equation (6.2): if H
⃗ = −∂t ⃗u +ν∆⃗u −div(⃗u ⊗⃗u)+ f⃗,
The term ∇p
⃗ = 0. We then conclude with the following classical lemma:
then (6.2) implies that curl H
Lemma 6.3.
⃗ is a time-dependent distribution vector field on (0, T ) × R3 such that curl H
⃗ = 0, then
If H
′
3
⃗
⃗
there exists a distribution p ∈ D ((0, T ) × R ) such that H = ∇p.
R
Proof. Let ω ∈ D(R) with ω(s) ds = 1. Define h the distribution on (0, T ) × R3 defined
by
Z +∞
Z
⟨h|φ⟩D′ D = ⟨H1 |
φ(t, y1 , x2 , x3 ) − ( φ(t, z1 , x2 , x3 ) dz1 ) ω(y1 ) dy1 ⟩D′ ,D
x1
R
104
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ = H
⃗ − ∇h.
⃗
⃗ = 0 and K1 = 0. Thus ∂1 K2 =
We have ∂1 h = H1 Let K
We have curl K
⃗ does not depend on x1 . We now write in a
∂2 K1 = 0, and similarly ∂1 K3 = 0. Thus K
similar way K2 = ∂2 k, where k does not depend on x1 :
Z +∞
Z
⟨k|φ⟩D′ D = ⟨K2 |
φ(t, x1 , y2 , x3 ) − ( φ(t, x1 , z2 , x3 ) dz2 ) ω(y2 ) dy2 ⟩D′ ,D
x2
R
⃗ =K
⃗ − ∇k.
⃗ We have curl L
⃗ = 0 and L1 = L2 = 0. Thus L depends only on t and x3 .
Let L
We write L = ∂3 l, where the distribution l depends only on t and x3 , and we conclude by
taking p = h + k + l + q, where q is any distribution on (0, T ) × R3 which depends only on
t.
6.2
Heat Equation
As for the case of classical solutions, the study of solutions for the Navier–Stokes equations will be dealt with by studying fisrt the Stokes equations and the heat equation. In
this section, we begin with basic lemmas for the heat equation.
We first define general spaces where we shall study the heat equation:
Definition 6.3 (Distribution spaces for the heat equations).
The space L1 ((0, T ) × R3 ) is the space of distributions F on (0, T ) × R3 that can be
written for some k ∈ N0 and some K ∈ N0 as
F =
X
∂ α Fα with Fα ∈ L1 ((0, T ), L1 (
|α|≤k
dx
)).
(1 + |x|)K
Similarly, the space Λ1 (R3 ) is the space of distributions u on R3 that can be written for
some k ∈ N0 and some K ∈ N0 as
u=
X
∂ α uα with uα ∈ L1 (
|α|≤k
dx
).
(1 + |x|)K
The space L∞ ((0, T ) × R3 ) is the space of distributions F on (0, T ) × R3 that can
be written for some k ∈ N0 and some K ∈ N0 as
F =
X
∂ α Fα with Fα ∈ L∞ ((0, T ), L1 (
|α|≤k
dx
)).
(1 + |x|)K
Heat equation
Proposition 6.1.
Let ν > 0, 0 < T < ∞, u0 ∈ Λ1 (R3 ) and f ∈ L1 ((0, T ) × R3 ). Then the equation
∂t u = ν∆u + f
(6.3)
u|t=0 = u0
The Differential and the Integral Navier–Stokes Equations
105
has a unique solution u ∈ L∞ ((0, T ) × R3 ). Moreover,
Z
t
Wν(t−s) ∗ f (s, .) ds.
u = Wνt ∗ u0 +
(6.4)
0
Proof. Let us first remark that if
u ∈ L∞ ((0, T ) × R3 ), u =
X
∂ α uα , uα ∈ L∞ ((0, T ), L1 (
|α|≤k
X
f ∈ L1 ((0, T ) × R3 ), f =
∂ α fα , fα ∈ L1 ((0, T ), L1 (
|α|≤k
dx
)),
(1 + |x|)K
dx
)),
(1 + |x|)K
and if ∂t u = ν∆u + f and T < +∞, then ∂t u ∈ (L1t H −4−k )loc so that Lemma 6.1 applies
and t 7→ u(t, .) is continuous on (0, T ) and has a limit when t = 0, and thus the initial value
u0 is well defined.
Existence: Due to the linearity of the equation, we discuss existence of the solution by
considering the cases f = 0, u0 = ∂ α u0,α and u0 = 0, f = ∂ α fα .2
dx
dx
1
1
First, we consider the cases u0 ∈ L1 ( (1+|x|)
K ) and f ∈ L ((0, T ), L ( (1+|x|)K )). We may,
dx
of course, assume that K ≥ 4. If v ∈ L1 ( (1+|x|)
K ) and 0 < t < T , we have
Z Z
( Wνt (x − y)|v(y)| dy)
dx
=
(1 + |x|)K
dx
≤C
Wνt (x − y)
(1 + |x|)K
Z
Z Z
( Wνt (x − y)
dx
)f (y) dy
(1 + |x|)K
with
Z
(K−3)/2
(νt)(K−3)/2
dx
′ 1 + (νt)
√
≤
C
.
(1 + |y|)K
( νt + |x − y|)K (1 + |x|K
This gives
∥Wνt ∗ u0 ∥L1 (
dx
(1+|x|)K
)
≤ C(1 + (νT )(K−3)/2 )∥u0 ∥L1 (
dx
(1+|x|)K
)
and
Z
∥
t
Wν(t−s) ∗ f (s, .) ds∥L1 (
0
dx
(1+|x|)K
)
≤ C(1 + (νT )(K−3)/2 )∥f ∥L1 ((0,t),L1 (
dx
(1+|x|)K
).
Rt
Thus, U = Wνt ∗ u0 and V = 0 Wν(t−s) ∗ f (s, .) ds belong to L∞ ((0, T ) × R3 ). Moreover,
∂t U = ν∆U in D′ , limt→0 U (t, .) = u0 and ∂t V = ν∆V + f , limt→0 V (t, .) = 0 (check it
when u0 ∈ D(R3 ) and f ∈ D((0, T ) × R3 ) and conclude by a density argument).
The cases of u0 = ∂ α u0,α and of f = ∂ α fα are then straightforward, as Wνt ∗ u0 =
α
∂ (Wνt ∗ u0,α ) and
Z t
Z t
Wν(t−s) ∗ f (s, .) ds = ∂ α (
Wν(t−s) ∗ fα (s, .) ds).
0
0
2 In the first edition of this book [319], we studied the case u ∈ B α
1 α
0
∞,∞ anf f ∈ L B∞,∞ . Remark that,
R
dx
α
α
if v ∈ B∞,∞
≤
C∥v∥
, then v = v1 + ∆N v2 , with 2N > α, (|v1 | + |v2 ∥) (1+|x|)
B∞,∞ .
4
106
The Navier–Stokes Problem in the 21st Century (2nd edition)
Uniqueness: Let u ∈ L∞ ((0, T ) × R3 ) be such that ∂t u = ν∆u and u|t=0 = 0. Let
φ ∈ D(R3 ), and let v = e−|x| (φ ∗ u). We have v ∈ L2 ((0, T ), H 1 ) and ∂t v ∈ L2 ((0, T ), H −1 ),
so that (by Lemma 6.2) we may write
∥v(t, .)∥22
=
Z tZ
∥v(0, .)∥22
+2
v ∂t v dx ds.
0
We have v(0, .) = 0 by the assumption u(0, .) = 0. Moreover, we have
Z
e−2|x| (φ ∗ u)ν∆(φ ∗ u) dx
Z
Z
x ⃗
−2|x| ⃗
2
· ∇(φ ∗ u)|2 dx
=−ν e
|∇(φ ∗ u)| dx + 2ν e−2|x| (φ ∗ u)
|x|
Z
≤ ν e−2|x| |φ ∗ u|2 dx.
We have
∥v(t, .)∥22 ≤ 2ν
Z
t
∥v(s, .)∥22 ds
0
and thus v = 0. As φ ∗ u = 0 for all φ ∈ D, we find that u = 0 (taking φ an approximation
of identity).
Corollary 6.2.
Let u0 ∈ Λ1 (R3 ) and f ∈ L1 ((0, T ) × R3 ). Let u ∈ L∞ ((0, T ) × R3 ) be the solution of
∂t u = ν∆u + f
u|t=0 = u0
If div ⃗u0 = 0 and div f⃗ = 0, then div ⃗u = 0.
Proof. If v = div ⃗u, we have v ∈ L∞ ((0, T )×R3 ), ∂t v = ν∆v and v|t=0 = 0. Thus, v = 0.
6.3
The Leray Projection Operator
Assume that ⃗u is a very weak solution of
⃗
∂t ⃗u = ν∆⃗u − div(⃗u ⊗ ⃗u) + f⃗ − ∇p
div ⃗u = 0
(6.5)
⃗u|t=0 = ⃗u0
The vector field f⃗ − div⃗u ⊗ ⃗u is decomposed into the sum of a divergence-free vector field
⃗
∂t ⃗u − ∆⃗u and a curl-free vector field ∇p.
⃗ where F⃗ is solenoidal (i.e.
The decomposition of a vector field F⃗0 into F⃗ = F⃗ + H,
⃗
divergence free) and H is irrotational (i.e. curl free) is not unique: if ψ is harmonic (i.e.
⃗
⃗ + ∇ψ).
⃗
∆ψ = 0), then we have another decomposition F⃗0 = (F⃗ − ∇ψ)
+ (H
To exclude
The Differential and the Integral Navier–Stokes Equations
107
⃗ be equal to 0 at infinity, in which
harmonic corrrections, we may require, if possible, that H
case F⃗ will be called the Leray projection of F⃗0 :
Leray projection operator
Definition 6.4.
Let F⃗0 be a distribution vector field on R3 such that F⃗0 ∈ S ′ . If there exists a vector
⃗ ∈ S ′ such that
field H
⃗ is curl free: ∇
⃗ ∧H
⃗ =0
• H
⃗ is divergence free: div H
⃗ = divF⃗0
• F⃗0 − H
⃗ = 0 in S ′
• limt→+∞ et∆ H
⃗ is unique and F⃗ = F⃗0 − H
⃗ is called the Leray projection of F⃗0 .
then H
⃗
⃗
We shall write F = PF0 .
⃗ is easily checked: if ∇
⃗ ∧H
⃗ = 0 and div H
⃗ = 0, then ∆H
⃗ = ∇div
⃗
⃗ −
Uniqueness of H
H
⃗ ∇∧
⃗ H)
⃗ = 0, so that H
⃗ is harmonic; if moreover H
⃗ ∈ S ′ , then the support of the Fourier
∇∧(
⃗ is included in {0} so that H
⃗ is a polynomial; if moreover limt→+∞ et∆ H
⃗ =0
transform of H
in S ′ , then H = 0. We have straightforward examples of Leray projections:
• Obviously, if F⃗0 ∈ S ′ , then if div F⃗0 = 0, we have P(F⃗0 ) = F⃗0 .
• Let L2σ be the space of divergence-free square integrable vector fields. If F⃗0 ∈ L2 , then
PF⃗0 is the orthogonal projection of F⃗0 on L2σ .
• in Theorem 4.4, we described the Leray projection of a regular and localized vector
field F⃗0
• if F⃗0 is compactly supported, then we may use the Green function G =
a fundamental solution of −∆G = δ and define PF⃗0 as
1
4π|x|
which is
⃗ ∗ div F⃗0 .
PF⃗0 = F⃗0 + ∇G
⃗ ∗ div F⃗0 is a smooth function that is O(|x|−3 ) at
Outside of the support of F⃗0 , ∇G
infinity.
Since
⃗ ∧ (∇
⃗ ∧ F⃗0 ) = ∇
⃗ ∧ (∇
⃗ ∧ PF⃗0 ) = −∆PF⃗0 ,
∇
we find that, formally, we have
⃗
⃗
∇
∇
PF⃗0 = √
∧ (√
∧ F⃗0 ).
−∆
−∆
This gives a direct way to compute PF⃗0 :
Proposition 6.2.
Let E ⊂ S ′ be a Banach space of distributions such that:
108
The Navier–Stokes Problem in the 21st Century (2nd edition)
• the Riesz transforms
∂
√ j
−∆
operate boundedly on E
• the elements of E vanish at infinity: for f ∈ E, limt→+∞ et∆ f = 0 in S ′ . Then,
if
⃗
⃗ = √∇
R
−∆
is the vectorial operator defined by the Riesz transforms, and if F⃗0 is vector
field with components in E, then the Leray projection of F⃗0 is well-defined and
we have
⃗ ∧ (R
⃗ ∧ F⃗0 ).
PF⃗0 = R
An example of such Banach space E is the Besov-like space Ḃ 0,α defined by the
Littlewood-Paley decomposition3 as
X
X
f ∈ Ḃ 1,α ⇔ f ∈ S ′ , f =
∆j f in S ′ ,
min(1, 2αj )∥∆j f ∥∞ < +∞.
j∈Z
j∈Z
In particular, the Leray projection of F⃗0 is well defined if F⃗0 ∈ Lp (1 ≤ p < +∞), or
s
s
F⃗0 ∈ Bp,q
(s ∈ R, 1 ≤ p < +∞, 1 ≤ q ≤ +∞), or F⃗0 ∈ Ḃ∞,q
(s < 0, 1 ≤ q ≤ +∞) or
0
⃗
F0 ∈ Ḃ∞,1 .
We now introduce another class of vector fields for which the Leray projection is well
defined:
Proposition 6.3.
1
Let k ∈ N0 . Let F⃗1 be a distribution vector field on R3 such that F⃗1 ∈ L1 ( (1+|x|)
3+k dx).
α⃗
⃗
⃗
If F0 = ∂ F1 with |α| = k, then the Leray projection of F0 is well defined and may be
computed as
⃗ R G) ∗ div F⃗0
PF⃗0 = F⃗0 + lim ∇(θ
R→+∞
where θ ∈ D is equal to 1 on a neighborhood of 0.
Proof. Let us remark that the function ∂i ∂j ∂ α ((1 − θ1 )G) is controlled by
|∂i ∂j ∂ α ((1 − θ1 )G)(x)| ≤ C
3 Recall
1
.
(1 + |x|)3+k
that the Littlewood–Paley decomposition of a tempered distribution if the equality
f = SN f +
+∞
X
∆j f
j=0
where SN f = F −1 (φ( 2ξN )fˆ) (with φ ∈ D equal to 1. for |ξ| < 12 and to 0 fot |ξ| > 1) and ∆j f = Sj+1 f −Sj f .
If SN f → 0 in S ′ as N → −∞, we have the homogeneous Littlewood–Paley decomposition
X
f =
∆j f.
j∈Z
The Differential and the Integral Navier–Stokes Equations
109
We have a similar control for ∂j ∂k ∂ α ((1 − θR )G), with R > 1,:
|∂i ∂j ∂ α ((1 − θR )G)(x)| ≤ C1{|x|>γR}
1
(1 + |x|)3+k
(where C and γ > 0 don’t depend on R). Moreover, we have
Z
1
1
1
dy ≤ C
(1 + |x|)3+k (1 + |x − y|)3+k
(1 + |y|)3+k
if k ≥ 1, while, if ϵ > 0,
Z
1
1
1
dy ≤ Cϵ
.
3+ϵ
3
(1 + |x|)
(1 + |x − y|)
(1 + |y|)3
⃗ R G) ∗ div F⃗0 is well defined as
Thus, limR→+∞ ∇(θ
⃗ R G) ∗ div F⃗0 = ∇(θ
⃗ 1 G) ∗ div F⃗0 +
lim ∇(θ
3
X
R→+∞
⃗
∂i ∂ α ∇((1
− θ1 )G) ∗ F⃗1,i .
i=1
We write
⃗ R G) = −δ + ∆θR G + 2∇θ
⃗ R · ∇G
⃗
div ∇(θ
with
⃗ R · ∇G)(x)|
⃗
|∂i ∂ α (∆θR G + 2∇θ
≤
C
1
.
R (1 + |x|)3+k
This gives that
⃗ R · ∇G)
⃗
lim (∆θR G + 2∇θ
∗ div F⃗0 = 0
R→+∞
and thus
⃗ R G) ∗ div F⃗0 = − div F⃗0 .
div lim ∇(θ
R→+∞
Finally, we write
⃗ R G) ∗ div F⃗0 ) =
et∆ ( lim ∇(θ
R→+∞
with
3
X
⃗ t∆ G ∗ F1,i
∂i ∂ α ∇e
i=1
⃗ t∆ G(x)| ≤ C √ 1
|∂i ∂ α ∇e
.
( t + |x|)3+k
By dominated convergence, we find that, for ϵ > 0,
Z
1
⃗ R G) ∗ div F⃗0 )| dx = 0.
lim
|et∆ ( lim ∇(θ
t→+∞
R→+∞
(1 + |x|)3+k+ϵ
Let us remark that many usual spaces on which the Riesz transforms operate are included
1
in L1 ( (1+|x|)
3 dx):
• Lebesgue spaces (E = Lp with 1 < p < +∞): let q =
1
that Lp ⊂ L1 ( (1+|x|)
3 dx).
p
p−1 ;
we have
1
(1+|x|)3
∈ Lq ,so
110
The Navier–Stokes Problem in the 21st Century (2nd edition)
R
• Morrey spaces (E = Ṁ p,q with 1 < p ≤ q < +∞): we have B(0,2j ) |f (x)| dx ≤
3
C∥f ∥Ṁ p,q 2j(3− q ) and thus
Z
|f (x)|
1
dx ≤
(1 + |x|)3
Z
|f (x)| dx +
B(0,1)
+∞
X
2−3j
Z
|f (x)| dx < +∞.
B(0,2j+1 )
j=0
• weighted Lebesgue spaces (E = Lp (w dx) with 1 < p < +∞ and w ∈ Ap (the
p
; we have
Muckenhoupt class of weights)): let q = p−1
Z
Z
q
|f (x)| dx ≤ ∥f ∥Lp (w dx) (
w− p dx)1/q .
B(0,2j )
B(0,2j )
q
q
w− p ∈ Aq , and thus there exists r < q such that w− p ∈ Ar [448]. Thus, we have
Z
Z
q
q
1
1
w− p dx) r (
w p(r−1) dx)1− r ≤ C23j
(
B(0,2j )
B(0,2j )
and we find
 rq

Z
C
B(0,2j )
and thus
Z
|f (x)|
r
 23j q
|f (x)| dx ≤ ∥f ∥Lp (w dx)  R
q
1
( B(0,1 w p(r−1) dx)1− r
1
dx ≤
(1 + |x|)3
Z
|f (x)| dx +
B(0,1)
+∞
X
−3j
Z
|f (x)| dx < +∞.
2
j=0
B(0,2j+1 )
We shall need to express, if possible, the Leray projection of time-dependent distributions. We first define distributions that vanish at infinity:
Definition 6.5.
Let F be a distribution on (0, T ) × R3 . We say that F vanishes at infinity if
• F belongs to (L1 H σ )loc for some σ ∈ R
• for every θ ∈ D((0, T )), Fθ
limτ →+∞ eτ ∆ Fθ = 0 in S ′ .
=
R
θ(t)F (t, .) dt belongs to S ′ and
Leray projection operator II
Definition 6.6.
Let F⃗0 be a distribution vector field on (0, T ) × R3 . We say that F⃗0 admits a Leray
projection if there exists a distribution vector field F⃗ such that,
• F⃗0 and F⃗ belong to (L1 H σ )loc for some σ ∈ R
• F⃗0 − F⃗ vanishes at infinity
The Differential and the Integral Navier–Stokes Equations
111
• for almost every t, F⃗ (t, .) = P(F⃗0 (t, .))
Then, F⃗ is unique and is called the Leray projection of F⃗0 .
We shall write F⃗ = PF⃗0 .
6.4
Stokes Equations
In this section, we are going to consider the Stokes equations as a preliminary step to
the study of Navier–Stokes equations. The Cauchy problem for Stokes equations reads as

⃗ + f⃗
 ∂t ⃗u =
ν∆⃗u − ∇p
(6.6)
div ⃗u = 0

⃗u|t=0 = ⃗u0
We then have a decomposition of f⃗ into the sum of a divergence free vector field ∂t ⃗u − ∆⃗u
⃗ This suggests, if f⃗ admits a Leray projection Pf⃗, to rather
and a curl free vector field ∇p.
study the Stokes equations defined as

 ∂t ⃗u =
ν∆⃗u + Pf⃗
(6.7)
div ⃗u = 0

⃗u|t=0 = ⃗u0
Lemma 6.4.
Let f⃗ admit a Leray projection Pf .
⃗ =
(A) If ⃗u is solution of equations (6.7), then ⃗u is solution of equations (6.6) with ∇p
f⃗ − Pf⃗.
(B) If ⃗u is solution of equations (6.6) and if ⃗u and f⃗ vanish at infinity, then ⃗u is solution
⃗ = f⃗ − Pf⃗.
of equations (6.7) and ∇p
Proof. Let ⃗u be solution of equations (6.6) and assume that ⃗u and f⃗ vanish at infinity. We
⃗ + f⃗ = Pf⃗ − ∇q,
⃗ and we want to prove that ∇q
⃗ = 0. We take θ ∈ D((−ϵ, ϵ)) and
write ∇p
consider, for t ∈ (ϵ, T − ϵ),
Z
Qθ,t = θ(t − s)(Pf⃗(s, .) + ν∆⃗u(s, .) − ∂t ⃗u(s, .)) ds
Z
Z
Z
= θ(t − s)Pf⃗(s, .) ds + ν∆ θ(t − s)⃗u(s, .) ds + θ′ (t − s)⃗u(s, .) ds.
We have div Qθ,t = curl Qθ,t = 0, while Qθ,t ∈ S ′ and limτ →+∞ eτ ∆ Qθ,t = 0. Thus, we have
⃗ = 0, and considering an approximation of identity, ∇q
⃗ = 0.
Qθ,t = 0. This gives θ ∗ ∇q
Thus, equations (6.6) and (6.7) are almost equivalent. We can even simplify equations
(6.7) by dropping the requirement that ⃗u is divergence free, and thus reducing equations
(6.7) to the heat equation:

∂t ⃗u = ν∆⃗u + Pf⃗
(6.8)

⃗u|t=0 = ⃗u0
112
The Navier–Stokes Problem in the 21st Century (2nd edition)
Lemma 6.5.
Let f⃗ admit a Leray projection Pf ∈ L1 ((0, T ) × R3 ) and let ⃗u0 ∈ Λ1 (R3 ) with div ⃗u0 = 0.
Let ⃗u ∈ L∞ ((0, T ) be the solution of the heat equation (6.8). Then ⃗u is solution of the Stokes
equations (6.7).
Proof. This is a consequence of Corollary 6.2.
Definition 6.7 (Distribution space for the Stokes equations).
The space L10 ((0, T ) × R3 ) is the space of distributions F on (0, T ) × R3 that can be
written for some k ∈ N0 and some σ ∈ R as
X
F = Gσ +
∂ α Fα
|α|≤k
with
Gσ ∈ L1 ((0, T ), Ḃ 0,σ ) and Fα ∈ L1 ((0, T ), L1 (
dx
)).
(1 + |x|)3+|α|
Stokes equations
Proposition 6.4.
Let ν > 0, 0 < T < ∞. Let
⃗u0 ∈ Λ1 (R3 ) with div ⃗u0 = 0 and f⃗ ∈ L10 ((0, T ) × R3 ).
Then the vector field Pf⃗ is well defined, Pf⃗ belongs to L1 ((0, T ) × R3 ) and the equation

∂t ⃗u = ν∆⃗u + Pf⃗
(6.9)

⃗u|t=0 = ⃗u0
has a unique solution ⃗u ∈ L∞ ((0, T ) × R3 ). This solution is given by the formula
t
Z
Wν(t−s) ∗ Pf⃗(s, .) ds.
⃗u = Wνt ∗ ⃗u0 +
(6.10)
0
Proof. We only need to check that Pf⃗ is well defined and belongs to L1 ((0, T ) × R3 ). If
dx
⃗gσ ∈ L1 ((0, T ), Ḃ 0,σ ), then P⃗gσ ∈ L1 ((0, T ), Ḃ 0,σ ). If f⃗α ∈ L1 ((0, T ), L1 ( (1+|x|)
3+|α| )), then
P(∂ α f⃗α ) = ∂ α f⃗α +
3
3
X
X
⃗
⃗
(∂i ∂ α ∇((1
− θ)G)) ∗ f⃗α,i +
∂i ∂ α ∇((θG)
∗ f⃗α,i )
i=1
i=1
⃗
where θ ∈ D(R3 ) is equal to 1 on a neighborhood of 0. We have (∂i ∂ α ∇((1
− θ)G)) ∗ f⃗α,i ∈
dx
dx
1
1
α⃗
1
⃗
L ((0, T ), L ( (1+|x|)3+|α| )) if |α| > 0, (∂i ∂ ∇((1 − θ)G)) ∗ fα,i ∈ L ((0, T ), L1 ( (1+|x|)
4+|α| ))
dx
1
1
1
⃗
⃗
if |α| = 0, and (θG) ∗ fα,i ∈ L ((0, T ), L (
3+|α| )). Hence, Pf belongs to L ((0, T ) ×
(1+|x|)
R3 )
The Differential and the Integral Navier–Stokes Equations
6.5
113
Oseen Equations
From Proposition 6.4, we see that we have (almost) equivalence between the differential
and the integral formulation of the Navier–Stokes equations in the formalism described by
Oseen’s tensor, provided that the term ⃗u ⊗ ⃗u may be defined and integrated4 :
Oseen’s equations
Theorem 6.1.
Let ν > 0, 0 < T < ∞. Let
⃗u0 ∈ Λ1 (R3 ) with div ⃗u0 = 0 and f⃗ ∈ L10 ((0, T ) × R3 ).
dx
Then for a vector field ⃗u such that ⃗u ∈ L2 ((0, T ), L2 ( (1+|x|)
4 )) the following assertions
are equivalent:
• (A) on (0, T ) × R3 , ⃗u is a solution of the differential equation
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
⃗u|t=0 = ⃗u0
(6.11)
• (B) on (0, T ) × R3 , ⃗u is a solution of the integral equation
Z t
⃗u =Wνt ∗ ⃗u0 +
Wν(t−s) ∗ Pf⃗(s, .) ds
0
−
Z tX
3
(6.12)
∂j O(ν(t − s)) :: uj ⃗u ds
0 j=1
6.6
Mild Solutions for the Navier–Stokes Equations
Due to Lemma 6.3, we see that a very weak solution is solution in D′ ((0, T ) × R3 of the
problem
⃗
∂t ⃗u = ν∆⃗u − div(⃗u ⊗ ⃗u) + f⃗ − ∇p
div ⃗u = 0
⃗u|t=0 = ⃗u0
(6.13)
which is a slight modification of Problem (6.1).
⃗ u for a very weak solution, as
Of course, we cannot in general rewrite div(⃗u ⊗ ⃗u) as (⃗u.∇)⃗
the latter expression cannot be defined as a distribution when ⃗u is only assumed to be locally
square integrable. This will be possible if we assume a little amount of differentiability on
⃗u:
4 In the first edition of this book [319], we studied the case ⃗
u ∈ (L2 L2 )uloc . Remark that (L2 L2 )uloc ⊂
dx
L2 ((0, T ), L2 ( (1+|x|)
)).
4
114
The Navier–Stokes Problem in the 21st Century (2nd edition)
Weakly regular very weak solution
Definition 6.8.
A very weak solution ⃗u of equations (6.1) on (0, T ) × R3 is said to be weakly regular
if there exists p ∈ [2, ∞) and r ∈ [2, ∞) such that, for every compact subset K of
p
r
⃗ ⊗ ⃗u ∈ Ltr−1 Lxp−1 .
(0, T ) × R3 , we have 1K (t, x) ⃗u ∈ Lrt Lpx and 1K (t, x) ∇
⃗ u in D′ .
In that case, we have div(⃗u ⊗ ⃗u) = (⃗u.∇)⃗
Due to Theorem 6.1, we may introduce the class of solutions such that moreover the
pressure p is determined by the Leray projection operator:
Oseen solution
Definition 6.9.
Let ν > 0, 0 < T < ∞. Let
⃗u0 ∈ Λ1 (R3 ) with div ⃗u0 = 0 and f⃗ ∈ L10 ((0, T ) × R3 ).
An Oseen solution ⃗u of Equations (6.1) on (0, T ) × R3 , for initial value ⃗u0 and forcing
term f⃗ is a very weak solution ⃗u(t, x) ∈ D′ ((0, T ) × R3 ) such that moreover:
dx
• ⃗u ∈ L2 ((0, T ), L2 ( (1+|x|)
4 ))
⃗ − f⃗ = P(div(⃗u ⊗ ⃗u) − f⃗)
• div(⃗u ⊗ ⃗u) + ∇p
In the paper by Furioli, Terraneo and Lemarié-Rieusset [186] and the book by LemariéRieusset [313], a criterion was given on ⃗u and f⃗ to ensure that a very weak solution is indeed
an Oseen solution. Their criterion was stated in terms of uniform local square integrability:
Definition 6.10 (The space (L2 L2 )uloc ).
A distribution u on (0, T ) × R3 is said to be uniformly locally square integrable if u is locally
square integrable and if
sZ Z
T
∥u∥(L2 L2 )uloc = sup
|u(s, x)|2 dx ds < +∞.
x0 ∈R3
0
B(x0 ,1)
Proposition 6.5.
Let ν > 0, 0 < T < ∞. Let
⃗u0 ∈ Λ1 (R3 ) with div ⃗u0 = 0 and f⃗ ∈ L10 ((0, T ) × R3 ).
Let ⃗u be a very weak solution of equations (6.1) on (0, T ) × R3 . If ⃗u belongs to
dx
L2 ((0, T ), L2 ( (1+|x|)
u belongs to the closure of D((0, T ) × R3 ) in (L2 L2 )uloc , then
3 )) or if ⃗
the very weak solution ⃗u is an Oseen solution.
Proof. Just check that ⃗u vanishes at infinity and apply Lemma 6.4
A special case of Oseen solutions are solutions that may be determined by Picard’s
iteration method: the mild solutions that belong to adapted spaces.
The Differential and the Integral Navier–Stokes Equations
115
Definition 6.11 (Adapted space).
A Banach space E of distribution vector fields on (0, T ) × R3 is called an adapted space for
the Navier–Stokes equations if
• E is continuously embedded into (L2 L2 )uloc , i.e.
Z tZ
|f⃗(s, x)|2 dx ds ≤ C∥f⃗∥2E .
sup
x0 ∈R3
0
B(x0 ,1)
• for every ⃗v and w
⃗ in E, the solution ⃗z of the Stokes problem

⃗
∂t ⃗z = ν∆⃗z + P div(⃗v ⊗ w)
⃗z|t=0 = 0

still belongs to E and ∥⃗z∥E ≤ C0 ∥⃗v ∥E ∥w∥
⃗ E for a positive constant C0 which depends
only on E (and ν). Thus the bilinear operator Bν : (⃗v , w)
⃗ 7→ Bν (⃗v , w)
⃗ = ⃗z is bounded
on E.
The operator norm of Bν will be called the Oseen constant of E (and will be denoted as
ΩE,ν ):
ΩE,ν =
sup
∥Bν (⃗v , w)∥
⃗ E.
(6.14)
∥⃗
v ∥E ≤1, ∥w/|
⃗ E ≤1
⃗0 −
If E is an adapted space, with Oseen constant ΩE,ν , then a solution ⃗u of ⃗u = U
Bν (⃗u, ⃗u), where
Z t
⃗ 0 =Wνt ∗ ⃗u0 +
U
Wν(t−s) ∗ Pf⃗(s, .) ds
0
Bν (⃗u, ⃗v ) =
Z tX
3
∂j O(ν(t − s)) :: uj ⃗v ds
0 j=1
⃗ 0 ∥E ≤
may be found by Picard’s iteration method as soon as ∥U
in the closed ball B = {⃗u ∈ E / ∥⃗u∥E ≤ 2Ω1E,ν }.
1
4ΩE,ν
and will be unique
Mild solution
Definition 6.12.
An Oseen solution ⃗u of Equations (6.1) on (0, T ) × R3 , for initial value ⃗u0 and forcing
term f⃗ is a mild solution if there exists an adapted Banach space E of distribution
vector fields on (0, T ) × R3 such that:
Rt
• Wνt ∗ ⃗u0 ∈ E and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ E
Rt
• ∥Wνt ∗ ⃗u0 + 0 Wν(t−s) ∗ Pf⃗(s, .) ds∥E ≤ 4Ω1E,ν
and if ⃗u is the unique solution of Equations (6.1) such that ∥⃗u∥E ≤
1
2ΩE,ν .
In the book [313], the Navier–Stokes problem with null external force is considered in a
large collection of adapted spaces; those mild solutions are smooth on (0, T )×R3 , due to the
116
The Navier–Stokes Problem in the 21st Century (2nd edition)
regularization properties of the heat kernel. In presence of singular forces, mild solutions
might fail to be weakly regular in the sense of Definition 6.8: if ⃗u is a Landau solution,
i.e., a steady solution of ∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u)) = 0 with f⃗ = c0 (0, 0, δ(x)) (where δ is
the Dirac mass)5 and if c0 is small enough, then ⃗u is a mild solution with initial value
3,∞
⃗ u is
⃗u0 = ⃗u and forcing term f⃗ (in the adapted space L∞
u.∇)⃗
t Lx ) but the product (⃗
not locally integrable, as it is homogeneous of degree −3. [Let us remark however that
f⃗ ∈
/ L10 ((0, T ) × R3 ).]
6.7
Suitable Solutions for the Navier–Stokes Equations
Another important class of weak solutions is based on quadratic energy estimates. In
that case, we must assume not only that ⃗u is square integrable, but we must make a similar
⃗ ⊗ ⃗u.
assumption on ∇
Weak solution
Definition 6.13.
A weak solution ⃗u of Equations (6.1) on (0, T ) × R3 is an Oseen solution such that
dx
• ⃗u ∈ L∞ ((0, T ), L2 ( (1+|x|)
4 ))
⃗ ⊗ ⃗u ∈ L2 ((0, T ), L2 ( dx 4 ))
• ∇
(1+|x|)
associated to an initial value ⃗u0 and to a forcing term f⃗ such that
dx
• ⃗u0 ∈ L2 ( (1+|x|)
u0 = 0
4 ) with div ⃗
dx
• f⃗ = div F, where the tensor F is such that F ∈ L2 ((0, T ), L2 ( (1+|x|)
4 )).
⃗ u, but it is not
Of course, a weak solution is weakly regular, so that div(⃗u ⊗ ⃗u) = (⃗u.∇)⃗
|⃗
u|2
regular enough to grant that ∂t ( 2 ) = ⃗u.∂t ⃗u. While ⃗u.∆⃗u and ⃗u.f⃗ are well defined, we
⃗ u nor ⃗u.∇p
⃗ as distributions. If ⃗u and p were regular enough, we could
cannot define ⃗u.(⃗u.∇)⃗
write (since div ⃗u = 0)
∂t (
|⃗u|2
⃗ u − ⃗u.∇p
⃗ + ⃗u.f⃗
) =ν⃗u.∆⃗u − ⃗u.(⃗u.∇)⃗
2
|⃗u|2
|⃗u|2
2
⃗
=ν∆(
) − ν|∇ ⊗ ⃗u| − div (p +
)⃗u + ⃗u.f⃗
2
2
(6.15)
We do not need much regularity to give meaning to the second line of equality (6.15):
Lemma 6.6.
If ⃗u is a weak solution of Equations (6.1) on (0, T )×R3 (in the sense of Definition 6.13), then
one can choose p (which is defined up to a time-dependent additive factor q(t) ∈ D′ ((0, T )))
3/2
so that ⃗u is locally L4t L3x and p locally L2t Lx on (0, T ) × R3 .
5 See
Theorem 10.13.
The Differential and the Integral Navier–Stokes Equations
117
2
2 1
2 6
4 3
Proof. ⃗u is locally L∞
t Lx and Lt Hx ⊂ Lt Lx , thus locally Lt Lx . Moreover, we know (from
Proposition 6.3) that
XX
XX
⃗ =
⃗ i ∂j (Gθ) ∗ (Fi,j − ui uj ) +
⃗ i ∂j (G(1 − θ)) ∗ (Fi,j − ui uj ).
∇p
∇∂
∇∂
i
j
i
j
[Recall that θ ∈ D is equal to 1 on a neighborhood of 0.]
Let Ki,j = ∂i ∂j (G(1 − θ)). We define p as
XX
p(t, x) =
∂i ∂j (Gθ) ∗ (Fi,j − ui uj )
i
+
j
XXZ
i
(Ki,j (x − y) − Ki,j (−y))(Fi,j (t, y) − ui (t, y)uj (t, y)) dy
j
=A(t, x) + B(t, x).
Assume that θ = 1 on B(0, 1) and θ = 0 outside from ball B(0, 2). On (0, T ) × B(0, R), we
have
XX
A(t, x) =
∂i ∂j (Gθ) ∗ (θ4R (Fi,j − ui uj ))
i
where θ4R (x) =
x
).
θ( 4R
j
We have, for x ̸= 0,
|∂i ∂j (Gθ)(x)| ≤ C
1
|x|3
and
|∇∂i ∂j (Gθ)(x)| ≤ C
1
.
|x|4
Moreover, the Fourier transform of ∂i ∂j (Gθ) is bounded, as ∂i ∂j (Gθ) = θ∂i ∂j G+(∂i θ)∂j G+
(∂j θ)∂i G + (∂i ∂j θ)G; the second, the third and the fourth terms are integrable, while the
Fourier transform of ∂i ∂j G is bounded and the Fourier transform of θ is integrable. Thus,
convolution with ∂i ∂j (Gθ) is Calderón–Zygmund operator and is bounded on every Lp ,
1 < p < +∞ [215]. As ⃗u is locallly L4 L3 and F is locally L2 L2 , we find that A is L2 L3/2 on
(0, T ) × B(0, R).
On the other hand, if |x| < R, R ≥ 1, we have for every y
|Ki,j (x − y) − Ki,j (−y))| ≤ 2∥Ki,j ∥∞ < +∞
and, when |y| > 2R,
|Ki,j (x − y) − Ki,j (−y))| ≤ 2
R
.
|y|4
dx
2 ∞
As Fi,j and ui uj belong to L2 ((0, T ), L1 ( (1+|x|)
(hence is
4 ) ), we find that B is in Lt Lx
L2 L3/2 ) on (0, T ) × B(0, R).
Thus, for a weak solution, we may define the distribution
|⃗u|2
|⃗u|2
|⃗u|2
2
⃗
µ⃗u = ν∆(
) − ν|∇ ⊗ ⃗u| − div (p +
)⃗u + ⃗u.f⃗ − ∂t (
).
2
2
2
We have a semi-continuity result for the map ⃗u 7→ µ⃗u :
(6.16)
118
The Navier–Stokes Problem in the 21st Century (2nd edition)
Convergence of weak solutions
Theorem 6.2.
Let (⃗un )n∈N be a sequence of weak solutions of Equations (6.1) on (0, T ) × R3 (with
initial value ⃗u0,n and forcing term f⃗n = div Fn ) such that
• supn∈N ∥⃗un ∥L∞ ((0,T ),L2 (
dx
(1+|x|)4
⃗ ⊗ ⃗un ∥ 2
• supn∈N ∥∇
L ((0,T ),L2 (
• supn∈N ∥Fn ∥L2 ((0,T ),L2 (
))
<∞
dx
(1+|x|)4
dx
(1+|x|)4
))
))
<∞
< +∞
and assume that ⃗un converges to ⃗u in D′ ((0, T ) × R3 ) and that Fn converges strongly
to F in L2loc ((0, T ) × R3 ). Then:
• ⃗u is a weak solution of the Navier–Stokes equations
• for every ϕ ∈ D((0, T ) × R3 ) such that ϕ ≥ 0, we have
⟨µ⃗u |ϕ⟩D′ ,D ≥ lim sup ⟨µ⃗un |ϕ⟩D′ ,D
(6.17)
n→+∞
Proof. If Φ ∈ D((0, T )×R3 , we find that Φ ⃗un converges in D′ to Φ ⃗u; moreover, the sequence
−3/2
Φ ⃗un is bounded in L2t Hx1 and the sequence ∂t (Φ ⃗un ) is bounded in L2 Hx
(by Lemma
2/7
6.6). Thus, the sequence Φ ⃗un is bounded in the Sobolev space H (R × R3 ): just write
2/7
(1 + τ 2 + ξ 2 )2/7 ≤ (1 + τ 2 )(1 + ξ 2 )−3/2
(1 + ξ 2 )5/7 .
Since the functions Φ ⃗un are all supported in the same compact set (the support of Φ),
Rellich’s theorem gives that Φ⃗un converges strongly to Φ⃗un in L2t L2x . Since Φun converges
weakly in L2t Hx1 , hence in L2t L6x , we see that it converges strongly in L4t L3x .
Using again Lemma 6.6, we find that pn converges *-weakly to p. Thus, we find that ⃗u is
solution to the Navier–Stokes equations. Moreover, inthe terms defining
µ⃗un , one
has the fol
2
2
lowing convergences in D′ : ∆(|⃗un |2 ) → ∆(|⃗u|2 ), div (pn + |⃗un2 | )⃗un → div (p + |⃗u2| )⃗u ,
⃗un .f⃗n → ⃗u.f⃗ and ∂t (|⃗un |2 ) → ∂t (|⃗u|2 ). The only term for which we do√not have convergence
⃗ ⊗ ⃗un |2 . But if Φ ∈ D is a non-negative function, we have that Φ(∇
⃗ ⊗ ⃗un ) converges
is |∇
√
2 2
⃗
weakly to Φ(∇ ⊗ ⃗u) in Lt Lx , hence by the Banach–Steinhaus theorem we get that
ZZ
ZZ
√
⃗ ⊗ ⃗u|2 dt dx = ∥ Φ(∇
⃗ ⊗ ⃗u)∥22 ≤ lim inf
⃗ ⊗ ⃗un |2 dt dx.
Φ|∇
Φ|∇
The theorem is proved.
Thus, if we assume that the ⃗un and pn are regular enough to satisfy equality (6.15) (so
that µ⃗un = 0), we find that the limit ⃗u satisfies µ⃗u ≥ 0, i.e., that the distribution µ⃗u is
associated to a non-negative locally finite measure m⃗u :
Z
⟨µ⃗u |ϕ⟩D′ ,D =
ϕ(t, x) dm⃗u .
(0,T )×R3
The Differential and the Integral Navier–Stokes Equations
119
Such a solution is called a suitable solution:
Suitable solution
Definition 6.14.
A weak solution ⃗u of Equations (6.1) on (0, T ) × R3 is called a suitable solution if
µ⃗u ≥ 0, i.e., if it satisfies in D′ the local energy inequality
2
|⃗u|2
|⃗u|2
⃗ ⊗ ⃗u|2 − div (p + |⃗u| )⃗u + ⃗u.f⃗
∂t (
) ≤ ν∆(
) − ν|∇
(6.18)
2
2
2
2
2 1
An interesting case is when we have global estimates: assume that ⃗u ∈ L∞
t Lx ∩Lt Hx and
3/2
F ∈ L2t L2x (in which case p ∈ L2t L2x +L2t Lx ), and that limt→0 ∥⃗u(, t.)−⃗u0 ∥2 = 0. Integrating
inequality (6.18) against Φ(t, x) = θϵ (t)φ2 (x/R), where φ ∈ D(R3 ) satisfies φ(x) = 1 on
t−t0 +2ϵ
B(0, 1), and where θϵ (t) = α( t−ϵ
) with α a smooth non-decreasing function
ϵ ) − α(
ϵ
on R such that α(s) = 0 when s ≤ 1 and α(s) = 1 for s ≥ 2, R > 0, 0 < ϵ < t0 /3, we find
that:
Z
Z T
Z
1 T ′
. 2
ν
x
−
θ (t)∥⃗u(t, .)φ( )∥2 dt ≤
θϵ (t)( |⃗u(t, x)|2 ∆(φ2 )( ) dx) dt
2 0 ϵ
R
2R2 0
R
Z T
. ⃗
−ν
θϵ (t)∥φ( )(∇
⊗ ⃗u)∥22 dt
R
0
Z
Z
2 T
x
1
x
⃗
+
θϵ (t)( φ( )(p + |⃗u|2 )⃗u · ∇φ(
) dx) dt
R 0
R
2
R
Z T
Z
x ⃗
−
θϵ (t)( φ2 ( ) (∇
⊗ ⃗u) · F dx) dt
R
0
Z
Z
2 T
x ⃗
x
−
θϵ (t)( φ( ) (∇φ(
) ⊗ ⃗u) · F dx) dt
R 0
R
R
Letting R go to ∞, we get
−
1
2
Z
T
θϵ′ (t)∥⃗u(t, .)∥22 dt ≤ −ν
0
T
Z
⃗ ⊗ ⃗u∥22 dt+
θϵ (t)∥∇
0
T
Z
θϵ (t)⟨⃗u|f⃗⟩H 1 ,H −1 dt
0
If t0 is a Lebesgue point of t 7→ ∥⃗u(t, .)∥22 , we find that
∥⃗u(t0 , .)∥22
≤
∥⃗u0 ∥22
Z
− 2ν
t0
⃗ ⊗ ⃗u∥22 dt + 2
∥∇
0
Z
t0
⟨⃗u|f⃗⟩H 1 ,H −1 dt.
(6.19)
0
This inequality is thus satisfied for almost every t0 , and even for every t0 as t 7→ ⃗u(t, .)
is weakly continuous from [0, T ) to L2 , so that t 7→ ∥⃗u(t, .)∥22 is semi-continuous. This
inequality is called the Leray energy inequality6 .
6 A similar proof gives the strong Leray inequality: for every Lebesgue point t of t 7→ ∥⃗
u(t, .)∥22 and
0
for every t ∈ [t0 , T ), we have:
Z t
Z t
⃗ ⊗⃗
∥⃗
u(t, .)∥2 ≤ ∥⃗
u(t0 , .)∥22 − 2ν
∥∇
u∥22 ds + 2
⟨⃗
u|f⃗⟩H 1 ,H −1 ds.
t0
t0
120
The Navier–Stokes Problem in the 21st Century (2nd edition)
This global inequality might be satisfied even if the local inequality is not fulfilled. For
instance, solutions constructed by a Galerkin method are known to satisfy (6.19), but we
do not know whether they are suitable (see the discussion by Biryuk, Craig and Ibrahim in
[43]).
Leray weak solution
Definition 6.15.
A weak solution ⃗u of Equations (6.1) on (0, T ) × R3 is called a Leray weak solution if
it satisfies
2
2 1
• ⃗u ∈ L∞
t Lx ∩ Lt Ḣx
• f⃗ ∈ L2t Hx−1
• for every t ∈ (0, T ),
∥⃗u(t, .)∥2 ≤
∥⃗u0 ∥22
Z
− 2ν
0
t
⃗ ⊗ ⃗u∥22 ds + 2
∥∇
Z
0
t
⟨⃗u|f⃗⟩H 1 ,H −1 ds.
Chapter 7
Mild Solutions in Lebesgue or Sobolev Spaces
7.1
Kato’s Mild Solutions
The search for solutions to the Navier–Stokes equations has known three eras. The first
one was based on explicit formulas for hydrodynamic potentials, given by Lorentz (1896
[342]) and Oseen (1911 [384]), and further used by Leray in his seminal work introducing
weak solutions (1934 [328]). Then, in the fifties, a second approach was developed by Hopf
[238] and Ladyzhenskaya [262], based on the Faedo–Galerkin approximation method who
turned the partial differential equations into the study of an ordinary differential equation
in a finite-dimensional space.
The third period began in the mid-sixties, when the theory of accretive operators was
developed, leading to the theory of semi-groups of operators. The problem was to find
solution for non-linear equations of the type
d
 dt u = Lu + f (t, u)
(7.1)

u(0) = u0
where L was an unbounded operator on a Banach space E. The solution began by studying
the properties of the semi-group U (t) = etL . Then the equation (7.1) was turned into an
integral equation (due to Duhamel’s formula)
Z
t
U (t − s)f (s, u(s)) ds
u(t) = U (t)u0 +
(7.2)
0
The study of the properties of the integral term could then allow the use of Banach’s
contraction principle. The solutions obtained by this formalism were called mild solutions
by Browder (1964 [68]) and Kato (1965 [253]).
Recall that we consider the Cauchy initial value problem for the Navier–Stokes equations
(with reduced (unknown) pressure p, reduced force density f⃗ and kinematic viscosity ν > 0):
given a divergence-free vector field ⃗u0 on R3 and a force f⃗ on (0, +∞) × R3 , find a positive
T and regular functions ⃗u and p on [0, T ] × R3 solutions to
⃗ u + f⃗ − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
(7.3)
We have reformulated this problem into an integral equation: find ⃗u such that
⃗u = Wνt ∗ ⃗u0 −
Z tX
3
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G + uj ⃗u ds
(7.4)
0 j=1
DOI: 10.1201/9781003042594-7
121
122
The Navier–Stokes Problem in the 21st Century (2nd edition)
or, equivalently,
t
Z
Wν(t−s) ∗ P(−f⃗ + div ⃗u ⊗ ⃗u) ds
⃗u = Wνt ∗ ⃗u0 −
(7.5)
0
Throughout this chapter and the following one, we are going to study mild solutions of
the Navier–Stokes equations. More precisely, we shall seek to exhibit Banach spaces YT of
Lebesgue measurable functions F (t, x) defined on [0, T ] × R3 so that the operator
Z t
⃗
⃗
⃗ ds
B(F , G) =
Wν(t−s) ∗ P div(F⃗ ⊗ G)
(7.6)
0
is a bounded bilinear operator on YT3 × YT3 :
⃗ Y ≤ CY ∥F⃗ ∥Y ∥G∥
⃗ Y .
∥B(F⃗ , G)∥
T
T
T
T
(7.7)
Then, starting with the initial data ⃗u0 and the force f⃗, if
Z t
⃗
U (t, x) = Wνt ∗ ⃗u0 −
Wν(t−s) ∗ Pf⃗ ds
0
is a measurable function of t and x such that
⃗ ∥Y <
∥U
T
1
,
4CYT
⃗ − B(⃗u, ⃗u), defined on (0, T ) × R3 , such
we will find a solution ⃗u(t, x) of the equation ⃗u = U
1
that ∥⃗u∥YT < 2CY .
T
The next step will then be to identify Banach spaces XT of distributions on R3 and ZT
on (0, T ) × R3 such that ⃗u0 ∈ (XT )3 ⇔ ( or ⇒)10<t<T Wνt ∗ ⃗u0 ∈ (YT )3 and f⃗ ∈ (ZT )3 ⇒
Rt
Wν(t−s) ∗ Pf⃗ ds ∈ (YT )3 .
0
7.2
Local Solutions in the Hilbertian Setting
The simplest framework where to look for mild solutions to the Navier–Stokes equations
is for initial values in the Sobolev space H 1 (R3 ) and forces in L2 ((0, T ), L2 ). This has been
done in 1964 by Fujita and Kato [185]. Before stating their results, we begin with three easy
lemmas on Sobolev spaces.
We recall that the Sobolev space H s , s ∈ R, is defined as the space of tempered distributions f such that the Fourier transform fˆ of f is locally integrable and satisfies:
s
Z
1
∥f ∥H s =
(1 + |ξ|2 )s |fˆ(ξ)|2 dξ < +∞.
(2π)3
Similarly, when s < 3/2, the homogeneous Sobolev space Ḣ s , s ∈ R, is defined as the space
of tempered distributions f such that the Fourier transform fˆ of f is locally integrable and
satisfies:
s
Z
1
∥f ∥Ḣ s =
|ξ|2s |fˆ(ξ)|2 dξ < +∞.
(2π)3
If 0 ≤ s < 3/2, we have the embedding Ḣ s (R3 ) ⊂ Lp with
1
p
=
1
2
− 3s .
Mild Solutions in Lebesgue or Sobolev Spaces
123
For s ≥ 3/2, we will not use the space Ḣ s (which is no longer a space of distrributions),
but we will use the notation ∥ ∥Ḣ s , when dealing with tempered distributions f such that
the Fourier transform fˆ of f is locally integrable and satisfies:
s
Z
1
∥f ∥Ḣ s =
|ξ|2s |fˆ(ξ)|2 dξ < +∞.
(2π)3
In particular, we will never work in a space Ḣ s , s ≥ 3/2, but we may work in spaces
Ḣ σ1 ∩ Ḣ σ2 , where σ1 < 3/2. For instance, for s ≥ 0, H s = L2 ∩ Ḣ s .
Lemma 7.1.
If u0 ∈ L2 , then Wνt ∗ u0 ∈ C([0, +∞), L2 ) with
sup ∥Wνt ∗ u0 ∥2 = ∥u0 ∥2 .
(7.8)
t>0
Moreover, Wνt ∗ u0 ∈ L2 ((0, +∞), Ḣ 1 ) with
1
∥Wνt ∗ u0 ∥L2 Ḣ 1 = √ ∥u0 ∥2
2ν
(7.9)
Proof. To check that Wνt ∗ u0 ∈ C([0, +∞), L2 ), just use the spatial Fourier transform
2
Fx (Wνt ∗ u0 )(ξ) = e−νt|ξ| û0 (ξ).
To check that it belongs to L2 ((0, +∞), Ḣ 1 ) (where Ḣ 1 is the homogeneous Sobolev space),
just write:
Z +∞
Z +∞ Z
2
1
∥û0 ∥22
2
⃗
∥∇(W
∗
u
)∥
dt
=
|ξ|2 e−2νt|ξ| |û0 (ξ)|2 dξ dt =
νt
0 2
3
(2π) 0
(2π)3 2ν
0
Lemma 7.2.
Rt
If f ∈ L2 (0, +∞), L2 ) and F (t, x) = 0 Wν(t−s) ∗f (s, .) ds, then F belongs to C([0, +∞), Ḣ 1 )
and we have
⃗ (t, .)∥2 ≤ √1 ∥f ∥L2 L2 .
∥∇F
(7.10)
2ν
Moreover, F ∈ L2 ((0, +∞), Ḣ 2 ) and we have
∥∆F ∥L2 L2 ≤
1
∥f ∥L2 L2
ν
Proof. Just write:
√
⃗ (t, .)∥2 = ∥ −∆F (t, .)∥2
∥∇F
Z Z t
√
= sup | (
−∆(Wν(t−s) ∗ f (s, .) ds)u0 (x) dx|
∥u0 ∥2 =1
0
t
Z Z
=
sup |
∥u0 ∥2 =1
≤
√
f (s, x) −∆ Wν(t−s) ∗ u0 (x) ds dx|
0
sup ∥f ∥L2 L2 ∥Wν(t−s) ∗ u0 ∥L2 Ḣ 1
∥u0 ∥2 =1
1
≤ √ ∥f ∥L2 L2
2ν
(7.11)
124
The Navier–Stokes Problem in the 21st Century (2nd edition)
From this inequality, and from the density of D((0, +∞) × R3 ), we find that F belongs to
C([0, +∞), L2 ) with F (0, .) = 0. We may extend f and F to t < 0 by taking f = F = 0.
We then have in the distributional sense that
∂t F = ν∆F + f
(7.12)
If G = ∆F , we find, taking the time-space Fourier transform on R × R3 , that
|ξ|2
fˆ(τ, ξ)
iτ + ν|ξ|2
Ĝ(τ, ξ) = −
and finally (by Plancherel inequality)
∥∆F ∥L2 L2 ≤
1
∥f ∥L2 L2 .
ν
Lemma 7.3.
Let 0 < δ < 3/2 and s ≥ 0. Then, if u and v belong to H s (R3 ) ∩ Ḣ δ , we have that
3
uv ∈ H s+δ− 2 and
∥uv∥
3
H s+δ− 2
≤ Cs,δ (∥u∥Ḣ δ ∥v∥H s + ∥v∥Ḣ δ ∥u∥H s )
(7.13)
The same estimate holds for homogeneous norms:
∥uv∥
3
Ḣ s+δ− 2
≤ Cs,δ (∥u∥Ḣ δ ∥v∥Ḣ s + ∥v∥Ḣ δ ∥u∥Ḣ s )
(7.14)
Proof. If s + δ − 32 ≤ 0, we use thrice the Sobolev embedding inequality: 0 ≤ s < 3/2, hence
6
6
3
6
H s ⊂ L 3−2s ; 0 < δ < 3/2, hence Ḣ δ ⊂ L 3−2δ ; 0 ≤ 3/2 − s − δ, hence H 2 −s−δ ⊂ L 2(s+δ) .
The conclusion follows from the Hölder inequality:
Z
6
6
6
| uvw dx| ≤ ∥u∥ 3−2s
∥v∥ 3−2δ
∥w∥ 2(s+δ)
≤ C∥u∥H s ∥v∥Ḣ δ ∥w∥ 32 −s−δ .
H
If s + δ − 32 > 0, we use the Plancherel formula and compute the norms of the Fourier
transforms. Let
Z
Z
2 s+δ− 32
I = (1 + |ξ| )
| û(ξ − η)v̂(η) dη|2 dξ ≤ 2(I1 + I2 )
where
Z
2 s+δ− 32
(1 + |ξ| )
I1 =
Z
û(ξ − η)v̂(η) dη|2 dξ
|
|η|<|ξ−η|
and
Z
I2 =
3
(1 + |ξ|2 )s+δ− 2 |
Z
û(γ)v̂(ξ − γ) dγ|2 dξ.
|γ|<|ξ−γ|
We have
I1 ≤
∥v∥2Ḣ δ
≤ ∥v∥2Ḣ δ
≤
ZZ
3
(1 + |ξ|2 )s+δ− 2 |η|−2δ |û(ξ − η)|2 dη dξ
|η|<|ξ−η|
ZZ
3
(1 + 2|ξ − η|2 )s+δ− 2 |η|−2δ |û(ξ − η)|2 dη dξ
|η|<|ξ−η|
2
Cs,δ ∥v∥Ḣ δ ∥u∥2H s
I2 provides a symmetric control by Cs,δ ∥u∥2Ḣ δ ∥v∥2H s . The case of homogeneous norms is
similar. The lemma is proved.
Mild Solutions in Lebesgue or Sobolev Spaces
125
We may now state and prove the result of Fujita and Kato:
Navier–Stokes equations and Sobolev spaces: local solutions
Theorem 7.1.
If ⃗u0 ∈ (H 1 (R3 ))3 and f⃗ ∈ L2 ((0, T ), (L2 (R3 )3 ), then there exists a T0 ∈ (0, T ) and
a mild solution ⃗u of Equation (7.4) on (0, T0 ) × R3 such that ⃗u ∈ C([0, T0 ], (H 1 )3 ) ∩
L2 ((0, T0 ), (H 2 )3 ).
⃗ be defined as
Proof. Let U
⃗ (t, x) = Wνt ∗ ⃗u0 −
U
Z
t
Wν(t−s) ∗ Pf⃗(s, .) ds.
0
We have:
1
∥Wνt ∗ ⃗u0 ∥L∞ H 1 ≤ ∥⃗u0 ∥H 1 and ∥Wνt ∗ ⃗u0 ∥L2 Ḣ 2 ≤ √ ∥⃗u0 ∥H 1 .
2ν
Similarly, we may extend f⃗ beyond T by f⃗ = 0 when t > T . We find, for F⃗ =
Pf⃗(s, .) ds,
1
1
∥F⃗ ∥L∞ Ḣ 1 ≤ √ ∥f⃗∥L2 L2 , ∥F⃗ ∥L2 Ḣ 2 ≤ ∥f⃗∥L2 L2 ,
ν
2ν
Rt
0
Wν(t−s) ∗
while we have on (0, T0 ) × R3 ,
1
∥F⃗ ∥L∞ L2 ≤ CT02 ∥f⃗∥L2 L2 .
⃗ belongs to C([0, T0 ], (H 1 )3 ) ∩
Thus, we find that, for any positive T0 ≤ T , U
2 3
L ((0, T0 ), (H ) ).
⃗ ∈ (YT )3 . Moreover,
We now take YT0 = C([0, T0 ], H 1 ) ∩ L2 ((0, T0 ), H 2 ). We have U
0
3
3
B is bounded on (YT0 ) × (YT0 ) . Indeed, we have, for ⃗u in (YT0 )3 , the inclusion ⃗u ∈
1/4
L4 H 3/2 ∩ L4 Ḣ 1 with the inequality ∥⃗u∥L4 Ḣ 1 ≤ T0 ∥⃗u∥YT0 ; thus we get from Lemma 7.3
3
that, for ⃗u and ⃗v in (YT0 ) ,
2
1/4
∥⃗u ⊗ ⃗v ∥L2 ((0,T0 ),H 1 ) ≤ CT0 ∥⃗u∥YT0 ∥⃗v ∥Y T0 .
We thus get P div(⃗u ⊗ ⃗v ) ∈ L2 L2 ∩ L2 Ḣ −1 . Using Lemma 7.2, we find that
√
so that
1/4
ν∥B(⃗u, ⃗v )∥L∞ H 1 + ν∥B(⃗u, ⃗v )∥L2 Ḣ 2 ≤ CT0 ∥⃗u∥YT0 ∥⃗v ∥YT0
(7.15)
1
1/4 1
1/2
∥B(⃗u, ⃗v )∥YT0 ≤ C∥⃗u∥YT0 ∥⃗v ∥YT0 T0 ( √ (1 + T0 ) + )
ν
ν
(7.16)
Thus, for T0 small enough
T0 ≤ min(1, Cν
1
(∥⃗u0 ∥H 1 + ∥f⃗∥L2 L2 )4
),
we may find a solution for the Navier–Stokes equations ⃗u ∈ (YT0 )3 .
(7.17)
126
The Navier–Stokes Problem in the 21st Century (2nd edition)
The proof of the theorem gives, as a corollary, uniqueness of mild solutions:
Proposition 7.1.
If ⃗u0 ∈ (H 1 (R3 ))3 and f⃗ ∈ L2 ((0, T ), (L2 (R3 )3 ), if ⃗u and ⃗v are two mild solutions of
Equation (7.4) on (0, T )×R3 such that ⃗u and ⃗v belong to C([0, T ), (H 1 )3 )∩L2 ((0, T ), (H 2 )3 ),
then ⃗u = ⃗v .
Proof. If w
⃗ = ⃗u −⃗v , then w
⃗ = B(⃗v , ⃗v ) − B(⃗u, ⃗u) = −B(w,
⃗ ⃗v ) − B(⃗u, w).
⃗ We have w(0,
⃗ .) = 0.
Let T ∗ ∈ [0, T ] the maximal time such that w
⃗ = 0 on [0, T ∗ ) × R3 . If T ∗ < T , and if
T0 ∈ (T ∗ , T ), we find that
∥w
⃗ ⊗ ⃗v + ⃗u ⊗ w∥
⃗ L2 ((0,T0 ),H 1 ) ≤ C(T0 − T ∗ )1/4 ∥w∥
⃗ YT0 (∥⃗u∥YT0 + ∥⃗v ∥Y T0 ).
Thus, we get
1
1
1/2
∥w∥
⃗ YT0 ≤ C(∥⃗u∥YT0 + ∥⃗v ∥YT0 )(T0 − T ∗ )1/4 ( √ (1 + T0 ) + )∥w∥
⃗ Y T0
ν
ν
For T0 − T ∗ small enough, we get ∥w∥
⃗ YT0 = 0, hence w
⃗ = 0 on [0, T0 ) × R3 , which is absurd
since T ∗ is maximal. Thus, T ∗ = T .
7.3
Global Solutions in the Hilbertian Setting
Kato and Fujita moreover gave a criterion for the existence of global solutions:
Navier–Stokes equations and Sobolev spaces: global solutions
Theorem 7.2.
Let T∞ ∈ (0, +∞]. Let ⃗u0 ∈ (H 1 (R3 ))3 with div ⃗u0 = 0 and f⃗ be defined on (0, T∞ )
be such that f⃗ ∈ L2 ((0, T ), (L2 (R3 )3 ) for all T < T∞ . Let TMAX be the maximal time
where one can find a mild solution ⃗u of Equation (7.4) on (0, TMAX ) × R3 such that,
for all T < TMAX , ⃗u belongs to C([0, T ], (H 1 )3 ) ∩ L2 ((0, T ), (H 2 )3 ).
• If TMAX < T∞ , then sup0<t<TMAX ∥⃗u(t, .)∥H 1 = +∞.
RT
• If TMAX < T∞ , then 0 MAX ∥⃗u(s, .)∥2Ḣ 3/2 ds = +∞.
• There exists a positive constant ϵ0 (independent of T∞ , ν, ⃗u0 and f⃗), such that,
RT
if ∥⃗u0 ∥Ḣ 1/2 < ϵ0 ν and 0 ∞ ∥f⃗(s, .)∥2 − 1 ds < ϵ20 ν 3 , then TMAX = T∞ .
Ḣ
2
Proof. If T < TMAX < T1 < T∞ , we have a solution ⃗u on [0, T ] of
Z t
Z t
⃗u = Wνt ∗ ⃗u0 +
Wν(t−s) Pf⃗(s, .) ds −
Wν(t−s) ∗ P div(⃗u(s, .) ⊗ ⃗u(s, .)) ds
0
0
and a solution ⃗u on [T, T + δ] of
Z t
Z t
⃗u = Wν(t−T ) ∗ ⃗u(T, .) +
Wν(t−s) Pf⃗(s, .) ds −
Wν(t−s) ∗ P div(⃗u(s, .) ⊗ ⃗u(s, .)) ds
T
T
Mild Solutions in Lebesgue or Sobolev Spaces
1
with δ = min(T1 −T, 1, Cν (∥⃗u(T,.)∥
4
⃗
H 1 +∥f ∥L2 ((T ,T1 ),L2 ) )
127
). This gives a mild solution on (0, T +
δ), and, due to the maximality of TMAX , we must have T + δ ≤ TMAX . This is possible only
if lim inf T →T − ∥⃗u(T, .)∥H 1 = +∞. This proves the first point of the theorem.
MAX
If ⃗u is a mild solution on (0, TMAX ), we have as well that
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u) + Pf⃗
so that, for all T < TMAX , ⃗u ∈ L2 ((0, T ), H 2 ) and ∂t ⃗u ∈ L2 ((0, T ), L2 ). Thus, we find:
Z
Z
Z
Z
d
⃗ ⊗ ⃗u|2 dx + 2 ⃗u.f⃗ dx
|⃗u(t, x)|2 dx = 2 ⃗u.∂t ⃗udx = −2ν |∇
(7.18)
dt
since ⃗u is divergence-free so that
Z
Z
⃗ u dx
⃗u.P div(⃗u ⊗ ⃗u) dx = ⃗u.(⃗u.∇)⃗
and
Z
I=
⃗ u dx = −
⃗u.(⃗u.∇)⃗
Z
⃗ u + |⃗u|2 div ⃗u dx = −I = 0.
⃗u.(⃗u.∇)⃗
Thus, the L2 norm of ⃗u remains bounded:
Z
∥⃗u(t, .)∥2 ≤ ∥⃗u0 ∥2 +
t
∥f⃗(s, .)∥2 ds.
(7.19)
0
Similarly, we have:
d
dt
Z
3 Z
X
⃗ ⊗ ⃗u(t, x)|2 dx = 2
|∇
∂i ⃗u.∂i ∂t ⃗udx
i=1
Z
= −2ν
2
|∆⃗u| dx − 2
3 Z
X
⃗ u) dx − 2
∂i ⃗u.∂i ((⃗u.∇)⃗
Z
⃗ u dx − 2
∂i ⃗u.((∂i ⃗u).∇)⃗
Z
∆⃗u.f⃗ dx
(7.20)
i=1
Z
= −2ν
|∆⃗u|2 dx − 2
3 Z
X
∆⃗u.f⃗ dx
i=1
since
Z
J=
⃗ i ⃗u dx = −
∂i ⃗u.(⃗u.∇)∂
Z
⃗ i ⃗u + |∂i ⃗u|2 div ⃗u dx = −J = 0.
∂i ⃗u.(⃗u.∇)∂
We then get
d
dt
Z
⃗ ⊗ ⃗u(t, x)|2 dx ≤ −2ν∥∆⃗u∥22 + 2
|∇
Thus, ∥⃗u∥2Ḣ 1
theorem.
3
X
⃗ ⊗ ⃗u∥2 ∥∂i ⃗u∥3 ∥∂i ⃗u∥6 + 2∥∆⃗u∥2 ∥f⃗∥2
∥∇
i=1
(7.21)
⃗ ⊗ ⃗u∥2 ∥∥∆⃗u∥2 ∥⃗u∥ 3/2 + 1 ∥f⃗∥22
≤ −ν∥∆⃗u∥22 + C∥∇
Ḣ
ν
C ⃗
1 ⃗ 2
2
2
≤
∥∇ ⊗ ⃗u∥2 ∥⃗u∥Ḣ 3/2 + ∥f ∥2 .
4ν
ν
Rt
R
2
C
t
≤ (∥⃗u0 ∥2Ḣ 1 + ν1 0 ∥f⃗∥22 )e 4ν 0 ∥⃗u(s,.)∥Ḣ 3/2 ds . This proves the second point of the
128
The Navier–Stokes Problem in the 21st Century (2nd edition)
Finally, we may write
Z
Z
d
1/4
2
|(−∆) ⃗u(t, x)| dx = 2 ((−∆)1/2 ⃗u).∂t ⃗udx
dt
Z
Z
= −2ν |(−∆)3/4 ⃗u|2 dx − 2 ((−∆)1/2 ⃗u). div(⃗u. ⊗ ⃗u) dx
Z
+ 2 ((−∆)1/2 )⃗u.f⃗ dx
≤ −ν∥⃗u∥2Ḣ 3/2 + C∥⃗u∥Ḣ 3/2 ∥⃗u ⊗ ⃗u∥Ḣ 1/2 +
(7.22)
1 ⃗ 2
∥f ∥ − 1
Ḣ 2
ν
Using Lemma 7.3, we get
Z
d
1
|(−∆)1/4 ⃗u(t, x)|2 dx ≤ (C0 ∥⃗u∥Ḣ 1/2 − ν)∥⃗u∥2Ḣ 3/2 + ∥f⃗∥2 − 1
Ḣ 2
dt
ν
(7.23)
Thus, if
∥⃗u0 ∥Ḣ 1/2 ≤
we find that ∥⃗u∥Ḣ 1/2 ≤
1
(1 − √ )ν
2
√ν ,
2C0
Z
ν
and
2C0
Z
0
T∞
∥f⃗∥2
1
Ḣ − 2
ν3
,
4C02
(7.24)
and finally
TMAX
0
ds ≤
∥⃗u∥2Ḣ 3/2 ds ≤ ∥⃗u0 ∥2Ḣ 1/2 +
1
ν
Z
0
TMAX
∥f⃗∥2
1
Ḣ − 2
ds.
(7.25)
Thus, the third point is proved.
7.4
Sobolev Spaces
Fujita and Kato’s theorems may be extended to the case of Sobolev spaces H s with
s > 1/2 (see Chemin [105]).
Navier–Stokes equations and Sobolev spaces
Theorem 7.3.
(A) If ⃗u0 ∈ (H s (R3 ))3 and f⃗ ∈ L2 ((0, T ), (H s−1 (R3 )3 ) with s > 1/2, then there
exists a T0 ∈ (0, T ) and a unique mild solution ⃗u of Equation (7.4) on (0, T0 ) ×
R3 such that ⃗u ∈ C([0, T0 ], (H s )3 ) ∩ L2 ((0, T0 ), (H s+1 )3 ).
(B) Let s > 1/2. Let T∞ ∈ (0, +∞]. Let ⃗u0 ∈ (H s (R3 )3 with div ⃗u0 = 0 and
f⃗ be defined on (0, T∞ ) be such that f⃗ ∈ L2 ((0, T ), (H s−1 (R3 )3 ) for all T <
T∞ . Let TMAX be the maximal time where one can find a mild solution ⃗u of
Equation (7.4) on (0, TMAX ) × R3 such that, for all T < TMAX , ⃗u belongs to
C([0, T ], (H s )3 ) ∩ L2 ((0, T ), (H s+1 )3 ).
• If TMAX < T∞ , then sup0<t<TMAX ∥⃗u(t, .)∥H s = +∞.
RT
• If TMAX < T∞ , then 0 MAX ∥⃗u(s, .)∥2Ḣ 3/2 ds = +∞.
Mild Solutions in Lebesgue or Sobolev Spaces
129
• There exists a positive constant ϵ0 (independent of T∞ , ν, ⃗u0 and f⃗), such that,
RT
if ∥⃗u0 ∥Ḣ 1/2 < ϵ0 ν and 0 ∞ ∥f⃗(s, .)∥2 − 1 ds < ϵ20 ν 3 , then TMAX = T∞ .
Ḣ
2
Proof. The proof of (A) is similar to the proof of Theorem 7.1. We check easily that, for
⃗ belongs to C([0, T0 ], (H s )3 ) ∩ L2 ((0, T0 ), (H s+1 )3 ), since
any positive T0 ≤ T , U
1
∥Wνt ∗ ⃗u0 ∥L∞ H s ≤ ∥⃗u0 ∥H s , ∥Wνt ∗ ⃗u0 ∥L2 Ḣ s+1 ≤ √ ∥⃗u0 ∥H s
2ν
1
1
⃗ F⃗ ∥L∞ H s−1 ≤ √ ∥f⃗∥L2 H s−1 , ∥∆F⃗ ∥ 2 s−1 ≤ ∥f⃗∥L2 H s−1 ,
∥∇
L Ḣ
ν
2ν
1
∥F⃗ ∥L∞ L2 ≤ CT02 ∥f⃗∥L2 L2 .
We take YT0 = C([0, T0 ], H s ) ∩ L2 ((0, T0 ), H s+1 ). We find again that YT0 ⊂
1
1/4
L ((0, T0 ), H s+ 2 ) ∩ L4 ((0, T0 ), Ḣ 1 ): if s ≥ 1, H s ⊂ Ḣ 1 and ∥u∥L4 Ḣ 1 ≤ T0 ∥u∥YT0 ; if
4
2
(2s−1)/4
1/2 < s < 1, we have YT0 ⊂ L 1−s Ḣ 1 , hence ∥u∥L4 Ḣ 1 ≤ T0
∥u∥YT0 . This gives (if
T0 ≤ 1)
min(1,2s−1)/4
∥⃗u ⊗ ⃗v ∥L2 ((0,T0 ),H s ) ≤ CT0
∥⃗u∥YT0 ∥⃗v ∥YT0
so that
min(2s−1,1)/4
∥B(⃗u, ⃗v )∥YT0 ≤ C∥⃗u∥YT0 ∥⃗v ∥YT0 T0
1
1
1/2
( √ (1 + T0 ) + )
ν
ν
(7.26)
Thus, for T0 small enough, we may find a solution for the Navier–Stokes equations ⃗u ∈
(YT0 )3 .
The proof of (B) is similar to the proof of Theorem 7.2. The only difference is on the
study of the role of the norm of ⃗u in L2 Ḣ 3/2 , as we may no longer use Leibnitz’s rule on
derivatives.
First, we have H s−1 ⊂ H −3/2 , hence
Z
d
2
2
∥⃗u∥2 = −2ν∥⃗u∥Ḣ 1 + 2 ⃗u.f⃗ dx
dt
≤ 2∥⃗u∥H 3/2 ∥f⃗∥H s−1
≤ ∥⃗u∥22 + ∥⃗u∥2Ḣ 3/2 + ∥f⃗∥2H s−1
and
∥⃗u∥22 ≤ ∥⃗u0 ∥22 et +
Z
0
t
et−τ (∥⃗u∥2Ḣ 3/2 + ∥f⃗∥2H s−1 ) dτ.
We have moreover:
Z
Z
d
s/2
2
|(−∆) ⃗u(t, x)| dx = 2 (−∆)s/2 ⃗u.(−∆)s/2 ∂t ⃗udx
dt
Z
= −2ν |(−∆)(s+1)/2 ⃗u|2 dx
Z
⃗ u) dx
− 2 (−∆)s/2 ⃗u.(−∆)s/2 ((⃗u.∇)⃗
Z
− 2 (−∆)s/2 ⃗u.(−∆)s/2 f⃗ dx
(7.27)
130
The Navier–Stokes Problem in the 21st Century (2nd edition)
We then get
d
∥⃗u∥2Ḣ s ≤ −2ν∥⃗u∥2Ḣ s+1 + ∥⃗u∥H s+1 ∥f⃗∥H s−1
dt
Z
+ 2|
⃗ u) dx|
(−∆)s/2 ⃗u.(−∆)s/2 ((⃗u.∇)⃗
1
≤ −ν∥⃗u∥2Ḣ s+1 + ∥f⃗∥2H s−1 + 2∥⃗u∥2 ∥f⃗∥H s−1
ν
Z
s/2
⃗ u) dx|.
+ 2| (−∆) ⃗u.(−∆)s/2 ((⃗u.∇)⃗
We will prove in the next section (Section 7.5) that
Z
⃗ u) dx| ≤ C∥⃗u∥ s ∥⃗u∥ s+1 ∥⃗u∥ 3/2 .
| (−∆)s/2 ⃗u.(−∆)s/2 ((⃗u.∇)⃗
Ḣ
Ḣ
Ḣ
(7.28)
This gives
C2
1
d
∥⃗u∥2Ḣ s ≤
∥⃗u∥2Ḣ s ∥⃗u∥2Ḣ 3/2 . + ∥f⃗∥2H s−1 + 2∥⃗u∥2 ∥f⃗∥H s−1
dt
4ν
ν
and
C2
Rt
2
∥⃗u∥2Ḣ s ≤∥⃗u0 ∥2Ḣ s e 4ν 0 ∥⃗u∥Ḣ 3/2 dτ
Z t 2R
t
2
C
1
+
e 4ν τ ∥⃗u∥Ḣ 3/2 dσ ( ∥f⃗∥2H s−1 + 2∥⃗u∥2 ∥f⃗∥H s−1 ) dτ
ν
0
Thus, the theorem is proved.
The case s = 1/2 is similar, except that the existence time is no more controlled by a
power of the norm of the initial value:
Navier–Stokes equations and the critical Sobolev space
Theorem 7.4.
(A) If ⃗u0 ∈ (H 1/2 (R3 ))3 and f⃗ ∈ L2 ((0, T ), (H −1/2 (R3 )3 ), then there exists a T0 ∈
(0, T ) and a unique mild solution ⃗u of Equation (7.4) on (0, T0 ) × R3 such that
⃗u ∈ C([0, T0 ], (H 1/2 )3 ) ∩ L2 ((0, T0 ), (H 3/2 )3 ).
(B) Let ⃗u0 ∈ (H 1/2 (R3 ))3 with div ⃗u0 = 0 and f⃗ be defined on (0, +∞) be such that
f⃗ ∈ L2 ((0, +∞), (Ḣ −1/2 (R3 )3 ). There exists a positive constant ϵ0 (indepenR +∞
dent of ν, ⃗u0 and f⃗), such that, if ∥⃗u0 ∥Ḣ 1/2 < ϵ0 ν and 0 ∥f⃗(s, .)∥2 − 1 ds <
ϵ20 ν 3 , then the mild solution is defined on (0, +∞).
Ḣ
2
⃗ belongs to C([0, T ], (H 1/2 )3 ) ∩
Proof. We check easily that, for any finite T , U
2
3/2 3
2 −1/2
⃗
⃗ belongs to Cb ([0, +∞), (Ḣ 1/2 )3 )∩
L ((0, T ), (H ) ). If T = +∞, and f ∈ L Ḣ
, then U
2
3/2 3
L ((0, +∞), (Ḣ ) ). Indeed, if we split f⃗ in f⃗ = f⃗L + f⃗H with f⃗L ∈ L2 L2 and
f⃗H ∈ L2 Ḣ −1/2 , we may use the following estimates for the high-frequency f⃗H :
Z t
1
∥
Wν(t−s) ∗ Pf⃗H ds∥L∞ ((0,T0 ),Ḣ 1/2 ) ≤ C √ ∥f⃗H ∥L2 ((0,T ),Ḣ −1/2 )
0
ν
0
Mild Solutions in Lebesgue or Sobolev Spaces
131
t
Z
1
Wν(t−s) ∗ Pf⃗H ds∥L2 ((0,T0 ),Ḣ 1/2 ) ≤ C ∥f⃗H ∥L2 ((0,T ),Ḣ −1/2 )
0
ν
0
Z t
Z t
∥
Wν(t−s) ∗ Pf⃗H ds∥L2 (R3 ) ≤
∥Wν(t−s) ∗ Pf⃗H ∥2 ds
0
0
Z t
1
≤C
∥f⃗H ∥Ḣ −1/2 ds
(ν(t
−
s))1/4
0
∥
≤
4C t3/4 ⃗
√ ∥fH ∥L2 ((0,T ),Ḣ −1/2 )
0
3
ν
and the following estimates for the low-frequency f⃗L :
Z t
Z t
∥
Wν(t−s) ∗ Pf⃗L ds∥L2 (R3 ) ≤
∥Wν(t−s) ∗ Pf⃗L ∥2 ds
0
0
Z t
≤
∥f⃗L ∥2 ds
0
√
≤ t∥f⃗H ∥L2 ((0,T ),L2 )
0
Z
∥
0
t
t
Z
Wν(t−s) ∗ Pf⃗L ds∥Ḣ 1/2 (R3 ) ≤
∥Wν(t−s) ∗ Pf⃗L ∥Ḣ 1/2 ds
0
Z
t
≤C
0
1
∥f⃗L ∥2 ds
(ν(t − s))1/4
4C t3/4 ⃗
√ ∥fH ∥L2 ((0,T ),L2 )
≤
0
3
ν
Z
∥
0
t
t
Z
Wν(t−s) ∗ Pf⃗L ds∥L2 ((0,T0 ),Ḣ 3/2 ) ≤∥
∥Wν(t−s) ∗ Pf⃗L ∥Ḣ 3/2 ds∥L2 ((0,T0 ))
0
Z
≤C∥
t
1
∥f⃗L ∥2 ds∥L2 ((0,T0 ))
(ν(t − s))3/4
0
1/4
T
≤4C 03/4 ∥f⃗H ∥L2 ((0,T ),L2 )
0
ν
We take YT0 = L4 ((0, T0 ), Ḣ 1 ). We find
∥⃗u ⊗ ⃗v ∥L2 ((0,T0 ),Ḣ 1/2 ) ≤ C∥⃗u∥YT0 ∥⃗v ∥Y T0
so that
1/2
∥B(⃗u, ⃗v )∥YT0 ≤C∥B(⃗u, ⃗v )∥L∞ ((0,T
0 ),Ḣ
≤C ′ ∥⃗u∥YT0 ∥⃗v ∥YT0
1
ν
1/2
1/2 )
∥B(⃗u, ⃗v )∥L2 ((0,T
0 ),Ḣ
3/2 )
(7.29)
.
3/4
⃗ ∥Y small enough (i.e., ⃗u0 small enough in Ḣ 1/2 and f⃗ small enough in L2 Ḣ −1/2
Thus, for ∥U
T0
with T0 = +∞, or T0 small enough depending on ⃗u0 and f⃗), we may find a solution for the
Navier–Stokes equations ⃗u ∈ (YT0 )3 .
For this solution ⃗u, B(⃗u, ⃗u) belongs to Cb ([0, T0 ), (Ḣ 1/2 )3 ) ∩ L2 ((0, T0 ), (Ḣ 3/2 )3 ) and, for
any finite T1 < T0 , to C([0, T1 ], (L2 )3 ).
132
7.5
The Navier–Stokes Problem in the 21st Century (2nd edition)
A Commutator Estimate
In this section, we prove that, for ⃗u ∈ (Ḣ 1/2 ∩ Ḣ s+1 )3 with div ⃗u = 0, we have the
inequality (7.28). Indeed, we have
Z
s/2
⃗
(−∆)s/2 ⃗u. ⃗u.∇(−∆)
⃗u dx = 0
Hence, we have
Z
⃗ u) dx
(−∆)s/2 ⃗u.(−∆)s/2 ((⃗u.∇)⃗
Z
s/2
=
(−∆)
⃗u.
3 X
(−∆)s/2 (ui ∂i ⃗u) − ui (−∆)s/2 ∂i ⃗u dx
i=1
and thus
Z
3
X
⃗ u) dx| ≤ ∥⃗u∥ s
∥(−∆)s/2 (ui ∂i ⃗u) − ui (−∆)s/2 ∂i ⃗u∥2
| (−∆)s/2 ⃗u.(−∆)s/2 ((⃗u.∇)⃗
Ḣ
i=1
We then end the proof of (7.28) with the following commutator estimate [105]:
Proposition 7.2.
Let s > 1/2. Then we have
∥(−∆)s/2 (uv) − u(−∆)s/2 v∥2 ≤ C(∥u∥Ḣ 3/2 ∥v∥Ḣ s + ∥u∥Ḣ s+1 ∥v∥Ḣ 1/2 )
Proof. We compute the norm of the Fourier transform; let
Z Z
2
I=
û(ξ − η)v̂(η) (|ξ|s − |η|s ) dη dξ ≤ 2(I1 + I2 )
where
2
Z Z
s
s
û(ξ − η)v̂(η) (|ξ| − |η| )dη
I1 =
dξ
|η|<2|ξ−η|
and
2
Z Z
s
s
û(γ)v̂(ξ − γ) (|ξ| − |ξ − γ| )dγ
I2 =
dξ.
|γ|< 21 |ξ−γ|
We have
I1 ≤
∥v∥2Ḣ 1/2
ZZ
≤ 32s ∥v∥2Ḣ 1/2
=
2
||ξ|s − |η|s | |η|−1 |û(ξ − η)|2 dη dξ
|η|<2|ξ−η|
ZZ
|ξ − η|2s |η|−1 |û(ξ − η)|2 dη dξ
|η|<2|ξ−η|
2s
2
C3 ∥v∥Ḣ 1/2 ∥u∥2Ḣ s+1
and
I2 ≤
∥u∥2Ḣ 3/2
2
ZZ
≤ Cs ∥u|2Ḣ 3/2
|γ|< 12 |ξ−γ|
Z
|γ|< 12 |ξ−γ|
= C Cs ∥u∥2Ḣ 3/2 ∥v∥2Ḣ s .
||ξ|s − |ξ − γ|s |
|v̂(ξ − γ)|2 dγ dξ
|γ|3
|ξ − γ|2s−2 |γ|−1 |v̂(ξ − γ)|2 dη dξ
(7.30)
Mild Solutions in Lebesgue or Sobolev Spaces
7.6
133
Lebesgue Spaces
Another simple framework where to look for mild solutions to the Navier–Stokes equations is for initial values in the Sobolev space Lp (R3 ) and forces in Lr ((0, T ), Lq ) with
2
3
3
r + q < 2 + p . This has been done in 1984 by Kato [255]. (see also Cannone and Planchon
for the discussion on external forces [85]).
The case p > 3 is very easy.
Proposition 7.3.
Let 3 < p < +∞. Then:
• The bilinear operator B defined as
Z t
⃗ =
⃗ ds
B(F⃗ , G)
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is continuous on YT = C([0, T ], (Lp )3 ) for every finite T :
3
1
⃗ p ≤ Cp (νT ) 2 − 2p
sup ∥B(F⃗ , G)∥
0<t<T
1
⃗ .)∥p
sup ∥F⃗ (t, .)∥p sup ∥G(t,
ν 0<t<T
0<t<T
(7.31)
⃗
where the constant Cp depends only on p (and not on T , ν, F⃗ nor G).
Rt
• If ⃗u0 ∈ (Lp (R3 )3 and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C([0, T ], (Lp (R3 )3 ), then there exists
a T0 ∈ (0, T ) and a mild solution ⃗u of Equation (7.4) on (0, T0 ) × R3 such that
⃗u ∈ C([0, T0 ], (Lp )3 ).
• If f⃗ ∈ Lr ((0, T ), (Lq )3 ) with 1 < q < p and 2r + 3q < 2 + p3 , or q = p and r ≥ 1, or if
3p
3
3
sup0<t<T tβ ∥f⃗∥q < +∞ and limt→0 tβ ∥f⃗∥q = 0 with 2p+3
< q < p and β = 1− 2q
+ 2p
,
Rt
p
3 3
⃗
then 0 Wν(t−s) ∗ Pf (s, .) ds ∈ C([0, T ], (L (R ) )
Proof. Indeed, we use the estimate on the size of ∂j O(νt) that is derived from Theorem 4.6
and Corollary 4.2 and write the inequality:
Z tZ
1
⃗ ≤ C0
⃗ y)| ds dy
|B(F⃗ , G)|
|F⃗ (s, y)| |G(s,
(7.32)
2 (t − s)2 + |x − y|4
ν
0
We get the inequality
⃗ p ≤ C0
∥B(F⃗ , G)∥
t
Z
Z
∥
0
1
⃗ y)| dy∥p ds
|F⃗ (s, y)| |G(s,
ν 2 (t − s)2 + |x − y|4
t
Z
≤ C0
∥
0
Z
= Cp
t
ν 2 (t
1
⃗ .)∥p ds
∥ p ∥F⃗ (s, .)∥p ∥G(s,
− s)2 + |x|4 p−1
1
1
0
3
(ν|t − s|) 2 + 2p
(7.33)
⃗ .)∥p ds
∥F⃗ (s, .)∥p ∥G(s,
so that (for p > 3)
3
1
3
⃗ p ≤ Cp T 12 − 2p
⃗ .)∥p
sup ∥B(F⃗ , G)∥
ν − 2 − 2p sup ∥F⃗ (t, .)∥p sup ∥G(t,
0<t<T
0<t<T
0<t<T
⃗
where the constant Cp depends only on p (and not on T , ν, F⃗ nor G).
(7.34)
134
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ belongs to
Thus, we may find a mild solution in C([0, T0 ], (Lp )3 ), as soon as U
C([0, T0 ], (Lp )3 ) and T0 is small enough.
Rt
In order to check that F⃗ = 0 Wν(t−s) ∗ Pf⃗ ds belongs to C([0, T0 ], (Lp )3 ), we may write,
if 1 < q ≤ p,
1
∥Wν(t−s) ∗ Pf⃗∥p ≤ C∥f⃗∥q
3 1
1
(ν(t − s)) 2 ( q − p )
Thus, if f⃗ ∈ Lrt Lqx with
1
3 1 1
( − )<1−
2 q
p
r
we get
1
3 1
1
∥F⃗ (t, .)∥p ≤ Cν t1− r − 2 ( q − p ) ∥f⃗∥Lr Lq .
(7.35)
The same estimate is valid in the case q = p and r = 1. We may assume r < +∞; from
(7.35) and the density of test functions in Lr Lq , we find that F⃗ belongs to C([0, T0 ], (Lp )3 ).
Similarly, we have (for 1 < q ≤ p)
∥Wν(t−s) ∗ Pf⃗∥p ≤ C(sβ ∥f⃗∥q )
1
3
1
1
(ν(t − s)) 2 ( q − p ) sβ
Thus, if β + 32 ( 1q − p1 ) = 1, we find that
∥F⃗ (t, .)∥p ≤ Cν sup sβ ∥f⃗(s, .)∥q .
(7.36)
0<s<t
Regularity of the heat kernel shows that F⃗ belongs to C((0, T0 ], (Lp )3 ). The continuity at
t = 0 is then ensured by (7.36) and the assumption that lims→0 sβ ∥f⃗(s, .)∥q = 0.
The critical case p = 3 is not as simple, as the bilinear operator B is no longer bounded
on C([0, T ], (L3 )3 ). The cancellation properties of solenoidal vector fields do not provide
enough compensation, as it has been proved by Oru [383] that B is not bounded on the
smaller space {⃗u ∈ C([0, T ], (L3 )3 ) / div ⃗u = 0}. The solution proposed by Weissler [498] is
then to use the smoothing properties of the heat kernel: if u0 ∈ L3 , then for any positive σ
we have
∥(−∆)σ Wνt ∗ u0 ∥3 ≤ Cσ (νt)−σ ∥u0 ∥3 .
(7.37)
Moreover, since for a regular function u0 (such that (−∆)σ u0 ∈ L3 ), we have ∥(−∆)σ Wνt ∗
u0 ∥3 = O(1), we find that, for u0 ∈ L3 ,
lim tσ ∥(−∆)σ Wνt ∗ u0 ∥3 = 0.
t→0+
(7.38)
Kato’s solution [255] was even simpler: his proof uses only direct estimations on the absolute
values of the integrands1 , beginning with the estimate, for any q > 3,
3
1
∥Wνt ∗ u0 ∥q ≤ Cq (νt) 2q − 2 ∥u0 ∥3 .
1 This
is this approach that we have followed in a systematic way in Chapter 5.
Mild Solutions in Lebesgue or Sobolev Spaces
135
Navier–Stokes equations and the critical Lebesgue space
Theorem 7.5.
• Let σ ∈ (0, 1/2]. The bilinear operator B defined as
⃗ =
B(F⃗ , G)
t
Z
⃗ ds
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is continuous on
YT,σ = {⃗u /
sup tσ/2 ∥⃗u∥Ḣ σ < +∞ and lim tσ/2 ∥⃗u∥Ḣ σ = 0}
0<t<T
t→0
3
3
(with ∥⃗u∥Ḣ σ = ∥(−∆)σ/2 ⃗u∥3 ) for every finite or infinite T ∈ (0, +∞]:
3
σ
σ
⃗
⃗
⃗
2
2
sup t ∥B(F⃗ , G)∥
Ḣ σ ≤ Cσ,ν sup t ∥F (t, .)∥Ḣ σ sup t ∥G(t, .)∥Ḣ σ
σ
2
0<t<T
3
0<t<T
3
0<t<T
3
(7.39)
⃗
where the constant Cσ,ν does not depend on T , F⃗ nor G.
3
• Similarly, for q ∈ (3, +∞) and β = 12 − 2q
, the bilinear operator B is continuous
on
ỸT,q = {⃗u / sup tβ ∥⃗u∥q < +∞ and lim tβ ∥⃗u∥q = 0}
t→0
0<t<T
for every finite or infinite T ∈ (0, +∞]:
⃗ q ≤ Cν,q sup tβ ∥F⃗ (t, .)∥q sup tβ ∥G(t,
⃗ .)∥q
sup tβ ∥B(F⃗ , G)∥
0<t<T
0<t<T
(7.40)
0<t<T
Rt
• If ⃗u0 ∈ (L3 (R3 )3 and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C([0, T ], (L3 (R3 )3 ) ∩ YT,σ ,
then there exists a T0 ∈ (0, T ) and a mild solution ⃗u of Equation (7.4) on
(0, T0 ) × R3 such that ⃗u ∈ C([0, T0 ], (L3 )3 ) ∩ YT0 ,σ . (A similar result holds for
C([0, T ], (L3 (R3 )3 ) ∩ ỸT,q ).
Rt
• If ⃗u0 ∈ (L3 (R3 )3 is small enough and 0 Wν(t−s) ∗Pf⃗(s, .) ds is small enough in
Y+∞,σ , then this mild solution is defined on (0, +∞): we have a global solution
in C([0, +∞)(L3 )3 ) ∩ Y+∞,σ . (A similar result holds for C([0, +∞), (L3 (R3 )3 ) ∩
Ỹ+∞,q ).
• If sup0<t<T tγ ∥f⃗∥p < +∞ and limt→0 tγ ∥f⃗∥p = 0 with 1 < p < 3 and γ =
Rt
3
3
3
3 3
⃗
2 − 2p , then 0 Wν(t−s) ∗ Pf (s, .) ds ∈ C([0, T ], (L (R ) ) ∩ YT,σ ∩ ỸT,q for
3
σ ∈ (0, 1 − γ) and q ∈ (3, γ ).
Remark: The restriction σ ≤ 1/2 in the first point of the theorem may be changed into
σ < 1. But in that case the proof would be more technical, as we shoud use, instead of the
Sobolev embedding inequalities, the Littlewood–Paley decomposition [15, 313] and Bony’s
paraproduct operators [51].
136
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proof. We find from the Sobolev embedding inequalities that, for 0 < σ < 1, ∥u∥q ≤
1
1−σ
q = 3 . If
Ḣ3σ , we have
Cσ ∥u∥Ḣ σ , with
3
for u and v in
3( 13 − q2 )
σ ≤ 1/2, we have 3 < q ≤ 6 and thus Lq/2 ⊂ Ḣ3
, hence,
∥uv∥Ḣ 2σ−1 ≤ C∥uv∥q/2 ≤ C ′ ∥u∥Ḣ σ ∥v∥Ḣ σ .
3
3
3
We thus find
t
Z
⃗
∥B(F⃗ , G)∥
Ḣ σ ≤ C
3
0
1
1−σ
1
2+ 2
(t − s)
ds ⃗
⃗ Y = C ′ t− σ2 ∥F⃗ ∥Y ∥G∥
⃗ Y .
∥F ∥YT ,σ ∥G∥
T ,σ
T ,σ
T ,σ
σ
s
Thus, B is bounded on YT,σ .
In the case of YT,σ , we used the regularizing property of the heat kernel. If we use only
the size of the heat kernel (or more precisely of the kernel of Wν(t−s) ∗ P div), we shall work
with Lebesgue norms and use the Young inequality on convolution between Lq/2 and Lr
with 1r + 2q − 1 = 1q , to get
⃗ q ≤C
∥B(F⃗ , G)∥
Z
0
t
1
(t − s)
3(r−1)
1
2+
2r
ds ⃗
⃗ Y = C ′ t−β ∥F⃗ ∥
⃗
∥F ∥ỸT ,q ∥G∥
ỸT ,q ∥G∥ỸT ,q .
T̃ ,q
s2β
3
Since 12 + 3(r−1)
= 1 − ( 2r
− 1) = 1 − β with 0 < β < 1, we find that B is bounded on ỸT,q .
2r
Rt
⃗ = Wνt ∗ ⃗u0 + Wν(t−s) ∗ Pf⃗(s, .) ds belongs to the space C([0, T ], (L3 (R3 )3 ) ∩ YT,σ ,
If U
0
⃗ ∥Y
we have limT0 →0 ∥U
= 0, thus, for T0 small enough, we may find a solution ⃗u in YT0 ,σ
T0 ,σ
⃗
of ⃗u = U − B(⃗u, ⃗u). We must check that B(⃗u, ⃗u) belongs to C([0, T ], (L3 (R3 )3 ). We discuss
only the case of ⃗u ∈ ỸT,q with 3 < q ≤ 6 (since, for 0 < σ < 1, 1r = 1−σ
3 and q = min(r, 6), we
⃗ ∈ C([0, T ], (L3 (R3 )3 ) ∩ YT,σ ⊂ C([0, T ], (L3 (R3 )3 ) ∩ ỸT,r ⊂ C([0, T ], (L3 (R3 )3 ) ∩ ỸT,q ).
have U
⃗ in ỸT,q with q ∈ (3, 6]) the L3 norm of B(F⃗ , G)
⃗ is
Of course, we know that, for F⃗ and G
bounded:
Z t
1
1
′ ⃗
⃗ 3≤C
⃗
⃗
∥B(F⃗ , G)∥
ds∥F⃗ ∥ỸT ,q ∥G∥
ỸT ,q = C ∥F ∥ỸT ,q ∥G∥ỸT ,q .
1−2β
s2β
0 (t − s)
⃗ 3 = 0, so that continuity at t = 0+ is obvious.
Moreover, we see that limt→0+ ∥B(F⃗ , G)∥
For proving continuity at time t > 0, we consider θ close to t: |t − θ| < 13 t; let η = |t − θ|.
⃗ = B(F⃗ , G),
⃗ we write, for s < min(t, θ),
For H
Z
Wν(t−s) − Wν(θ−s) =
θ
ν∆Wν(τ −s) dτ
t
so that
⃗ x) − H(θ,
⃗
H(t,
x) =
Z
θ
Z
Z
t
t
t−2η
⃗
⃗
∆Wν(τ −s) ∗ P div(F ⊗ G) ds dτ
0
+
t−2η
Z θ
−
t−2η
⃗ ds
Wν(t−s) ∗ P div(F⃗ ⊗ G)
⃗ ds
Wν(θ−s) ∗ P div(F⃗ ⊗ G)
Mild Solutions in Lebesgue or Sobolev Spaces
137
and
⃗ .) − H(θ,
⃗
∥H(t,
.)∥3 ≤ Cν,q
θ
Z
Z
1
ds
(τ − s)2−2β s2β
0
t
θ
Z
t/3
Z
t−2η
+ Cν,q
t/3
t
Z
t
+ Cν,q
t−2η
θ
!
1
ds
2−2β
(τ − s)
s2β
⃗
dτ ∥F⃗ ∥ỸT ,q ∥G∥
ỸT ,q
!
⃗
dτ ∥F⃗ ∥ỸT ,q ∥G∥
ỸT ,q
1
ds ⃗
⃗
∥F ∥ỸT ,q ∥G∥
ỸT ,q
1−2β
(t − s)
t2β
Z
1
ds ⃗
⃗
∥F ∥ỸT ,q ∥G∥
ỸT ,q
1−2β t2β
(θ
−
s)
t−2η
1
1
′
2β 1
⃗
⃗
≤ Cν,q ∥F ∥ỸT ,q ∥G∥ỸT ,q |t − θ| + |t − θ| 1−2β 2β + η 2β
t
η
t
t
+ Cν,q
|t − θ|
′
⃗
≤ 3 Cν,q
∥F⃗ ∥ỸT ,q ∥G∥
ỸT ,q
t2β
2β
.
In order to finish the proof, we consider the case of a force density f⃗ which satisfies
3
sup0<t<T tγ ∥f⃗∥p < +∞ and limt→0 tγ ∥f⃗∥p = 0 with 1 < p < 3 and γ = 23 − 2p
, and we
Rt
⃗
⃗
define F = Wν(t−s) ∗ Pf (s, .) ds. We have
0
∥(−∆)σ/2 F⃗ (t, .)∥3 ≤ Cν,γ,σ
Z
0
t
1
1
ds sup tγ ∥f⃗∥p
(t − s)1−γ−σ sγ
0<t<T
Thus, for 0 < σ < 1 − γ, and q =
we have F⃗ ∈ YT,σ ∩ ỸT,q . Moreover, F⃗ (t, .) is bounded
in L3 and limt→0+ ∥F⃗ (t, .)∥3 = 0, whereas, for t > 0 and |t − θ| < 13 t, we have
3
1−σ ,
γ
|t − θ|
.
∥F⃗ (t, .) − F⃗ (θ, .)∥3 ≤ Cν,γ sup sγ ∥f⃗∥p
tγ
0<s<T
Thus, the theorem is proved.
7.7
Maximal Functions
For small data, existence of global solutions in L3 may be proved in a simpler way. As
a matter of fact, while the bilinear operator B is not bounded on C([0, T ], (L3 )3 ) (for the
3
3 = sup
norm L∞
0<t<T ∥u(t, .)∥3 ), Calderón [78] and Cannone [81] showed that
t Lx : ∥u∥L∞
t Lx
B is bounded on the smaller space {⃗u ∈ C([0, T ], (L3 )3 ) / ess sup0<t<T |⃗u(t, x)| ∈ L3 } (for
the norm L3x L∞
= ∥ ess sup0<t<T |u(t, x)|∥3 ).
t : ∥u∥L3x L∞
t
Navier–Stokes equations and maximal functions
Theorem 7.6.
• Let 0 < T ≤ +∞. The bilinear operator B is continuous on
YT = {⃗u ∈ C([0, T ), (L3 )3 ) /
sup |⃗u(t, x)| ∈ L3 }
0<t<T
138
The Navier–Stokes Problem in the 21st Century (2nd edition)
• For ⃗u0 ∈ (L3 )3 , we have Wνt ∗ ⃗u0 ∈ YT with
sup |Wνt ∗ ⃗u0 (x)| ≤ M⃗u0 (x)
0<t<T
(where M⃗u0 is the Hardy–Littlewood maximal function of ⃗u0 ).
Rt
• If f⃗ ∈ L1 ((0, T ), (L3 )3 ), then 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ YT .
• Let ⃗u0 ∈ (L3 (R3 )3 and f⃗ ∈ L1 ((0, T ), (L3 )3 ). There exists a positive constant ϵ0 (independent of ν, T , ⃗u0 and f⃗), such that, if ∥⃗u0 ∥3 < ϵ0 ν and
RT
∥f⃗(s, .)∥3 ds < ϵ0 ν, then there exists a mild solution ⃗u of equation (7.4)
0
on (0, T ) × R3 such that ⃗u ∈ YT .
Remark: We have as well the following properties:
• If sup0<t<T tγ |Pf⃗(t, x)| ∈ Lp with 1 < p < 3 and γ =
Pf⃗(s, .) ds ∈ YT .
3
2
−
3
2p ,
then
Rt
0
Wν(t−s) ∗
1
. If, for j = 1, . . . , 3, sup0<t<T |f⃗(t, x)∗∂j G| ∈
• Let G be the Green function G(x) = 4π|x|
R
t
L3/2 , then
Wν(t−s) ∗ Pf⃗(s, .) ds ∈ YT .
0
3
u(t, x)| ≤ U (x) ∈ L3 , |⃗v (t, x)| ≤ V (x) ∈ L3 , we have
Proof. If ⃗u and ⃗v belong to (L3x L∞
t ) , |⃗
Z Z t
1
|B(⃗u, ⃗v )| ≤C (
ds) U (y)V (y) dy
2 |t − s|2 + |x − y|4
ν
Z 0
πC
1
≤
U (y)V (y) dy
2ν
|x − y|2
Since the Riesz potential I1 maps L3/2 to L3 , we find that
∥B(⃗u, ⃗v )∥L3x L∞
≤
t
C
∥⃗u∥L3x L∞
∥⃗v ∥L3x L∞
.
t
t
ν
(7.41)
Moreover, if ⃗uR = ⃗u(t, x)1U (x)<R , ⃗vR = ⃗v (t, x)1V (x)<R , and if T0 ≤ T is finite, we have:
limR→+∞ ∥⃗u − ⃗uR ∥L3x L∞
= limR→+∞ ∥⃗v − ⃗vR ∥L3x L∞
= 0, while ⃗uR and ⃗vR belong to the
t
t
space ỸT0 ,6 described in Theorem 7.5, so that B(⃗uR , ⃗vR ) ∈ C([0, T0 ], (L3 )3 ). By uniform
convergence, we find that B(⃗u, ⃗v ) ∈ C([0, T0 ], (L3 )3 ). Thus, B is bounded on YT .
Now, we recall a classical lemma (see [215] for instance):
Lemma 7.4.
If ω is a radially decreasing function on R3 and f a locally integrable function, then
Z
Z
1
|
ω(x − y)f (y) dy| ≤ ∥ω∥1 sup
|f (x − y)| dy
(7.42)
r>0 |B(0, r)| |y|<r
R3
or equivalently
|ω ∗ f | ≤ ∥ω∥1 Mf
where Mf is the Hardy–Littlewood maximal function of f .
(7.43)
Mild Solutions in Lebesgue or Sobolev Spaces
139
Using this lemma, it is obvious that, for ⃗u0 ∈ (L3 )3 , we have Wνt ∗ ⃗u0 ∈ YT :
|Wνt ∗ ⃗u0 (x)| ≤ M⃗u0 (x) ∈ L3
Similarly, if f⃗ ∈ L1t L3x , then MPf⃗(t,.) (x) = F (t, x) ∈ L1t L3x and F (x) =
L3 (R3 ); we have
Z t
Z t
|
Wν(t−s) ∗ Pf⃗(s, .) ds| ≤
F (s, x) ds ≤ F (x)
(7.44)
RT
0
F (t, x) dt ∈
(7.45)
0
0
so that
Z
∥
0
t
Wν(t−s) ∗ Pf⃗(s, .) ds∥L3x L∞
≤ C∥f⃗∥L1t L3x
t
(7.46)
Moreover, if f⃗ϵ = f⃗(t, x)1ϵ<|f⃗(t,x)|<1/ϵ , and if T0 ≤ T is finite, we have: limϵ→0+ ∥f⃗ −
Rt
f⃗ϵ ∥L1t L3x = 0, while f⃗ϵ satisfies the assumptions of Theorem 7.5, so that 0 Wν(t−s) ∗
Rt
Pf⃗ϵ (s, .) ds ∈ C([0, T0 ], (L3 )3 ). By uniform convergence, we find that 0 Wν(t−s) ∗Pf⃗(s, .) ds ∈
C([0, T0 ], (L3 )3 ). Thus, the theorem is proved.
7.8
Basic Lemmas on Real Interpolation Spaces
Real interpolation spaces [A0 , A1 ]θ,q have been introduced by Lions and Peetre [338],
but we shall use in the next sections only basic properties of those spaces (as described in
[36, 313]), mainly for the values q = 1 and q = ∞. We shall give here a definition of those
interpolation spaces and of their norms which is slightly different (but equivalent) to the
classical ones:
Definition 7.1.
Let A0 , A1 be two Banach spaces. The real interpolation spaces [A0 , A1 ]θ,1 and [A0 , A1 ]θ,∞
(0 < θ < 1) can be characterized through the following properties:
P
• f ∈ [A0 , A1 ]θ,1 if and only if it can be written in A0 + A1 as a sum j∈N λj fj with
P
θ
fj ∈ A0 ∩ A1 , ∥f ∥1−θ
A0 ∥f ∥A1 ≤ 1 and
j∈N |λj | < +∞. We define its norm in the
following way:
X
θ
∥f ∥[A0 ,A1 ]θ,1 = Pinf
|λj |∥fj ∥1−θ
(7.47)
A0 ∥fj ∥A1
f=
j∈N
λj fj
j∈N
• f ∈ [A0 , A1 ]θ,∞ if and only if f ∈ A0 + A1 and there exists a constant C such that for
every λ > 0 we may decompose f as f = fλ +gλ with fλ ∈ A0 , gλ ∈ A1 , ∥fλ ∥A0 ≤ Cλθ
and ∥gλ ∥A1 ≤ Cλθ−1 . We define its norm in the following way:
∥f ∥[A0 ,A1 ]θ,∞ = sup
inf
λ>0 f =fλ +gλ
λ−θ ∥fλ ∥A0 + λ1−θ ∥gλ ∥A1
Of course, for a Banach space E, we have [E, E]θ,1 = [E, E]θ,∞ = E.
We shall need the classical lemma [36]:
Lemma 7.5.
If A0 ∩ A1 is dense in A0 and in A1 , then [A0 , A1 ]′θ,1 = [A′0 , A′1 ]θ,∞ .
(7.48)
140
The Navier–Stokes Problem in the 21st Century (2nd edition)
We shall use in the following sections the following easy interpolation results:
Lemma 7.6.
For η = 1 or η = +∞, we write [A0 , A1 ]0,η = A0 and [A0 , A1 ]1,η = A1 . For γ ∈ {1, +∞},
0 ≤ τ0 < τ1 ≤ 1, 0 < θ < 1 and τ = (1 − θ)τ0 + θτ1 , if [A0 , A1 ]τ0 ,1 ⊂ B0 ⊂ [A0 , A1 ]τ0 ,∞ and
[A0 , A1 ]τ1 ,1 ⊂ B1 ⊂ [A0 , A1 ]τ1 ,∞ , we have
[B0 , B1 ]θ,γ = [A0 , A1 ]τ,γ
(7.49)
Proof. Case γ = ∞: If u ∈ [A0 , A1 ]τ,∞ , with norm C0 , we may write u = uj + vj with
−jτ
∥uj ∥AP
and ∥vj ∥A1P≤ C0 2j(1−τ ) . If wj = P
uj − uj+1 = vj+1 − vj , we have
0 ≤ C0 2
u0 = j≥0 wj in A0 and v0 = j<0 wj ∈ A1 . Thus, u = j∈Z wj . We have wj ∈ A0 ∩ A1 ⊂
[A0 , A1 ]τ0 ,1 ⊂ B0 with
τ0
j(τ0 (1−τ )−τ (1−τ0 )
0
∥wj ∥B0 ≤ C∥wj ∥1−τ
= CC0 2j(τ0 −τ1 )θ .
A0 ∥wj ∥A1 ≤ CC0 2
Similarly, we have
∥wj ∥B1 ≤ CC0 2j(τ0 −τ1 )(θ−1) .
This gives [A0 , A1 ]τ,∞ ⊂ [B0 , B1 ]θ,∞ .
Conversely, let u ∈ [B0 , B1 ]θ,∞ , with norm C0 . For λ > 0, we may write u = uλ + vλ
with ∥uλ ∥B0 ≤ C0 λ−θ and ∥vλ ∥B1 ≤ C0 λ1−θ . Since B0 ⊂ [A0 , A1 ]τ0 ,∞ , we may write
1
uλ = u0,λ + u1,λ with µ = λ τ0 −τ1 and
τ0
1
∥u0,λ ∥A0 ≤ ∥uλ ∥B0 µτ0 ≤ C0 λ−θ+ τ0 −τ1 = C0 λ−τ τ1 −τ0
and similarly
τ0 −1
1
∥u1,λ ∥A1 ≤ ∥uλ ∥B0 µ(τ0 −1) ≤ C0 λ−θ+ τ0 −τ1 = C0 λ(1−τ ) τ1 −τ0 ,
1
while vλ = v0,λ + v1,λ with ν = λ τ0 −τ1 and
τ1
1
∥v0,λ ∥A0 ≤ ∥vλ ∥B1 ν τ1 ≤ C0 λ1−θ+ τ0 −τ1 = C0 λ−τ τ1 −τ0
and similarly
τ1 −1
1
∥v1,λ ∥A1 ≤ ∥vλ ∥B1 λγ(τ1 −1) ≤ C0 λ1−θ+ τ0 −τ1 = C0 λ(1−τ ) τ1 −τ0 .
This gives [B0 , B1 ]θ,∞ ⊂ [A0 , A1 ]τ,∞ .
τ0
0
Case γ = 1: Let us consider u ∈ A0 ∩ A1 . We have ∥u∥B0 ≤ C∥u∥1−τ
A0 ∥u∥A1 , ∥u∥B1 ≤
τ1
1
C∥u∥1−τ
A0 ∥u∥A1 , hence
1−τ
θ
τ
∥u∥1−θ
B0 ∥u∥B1 ≤ C∥u∥A0 ∥u∥A1
and thus [A0 , A1 ]τ,1 ⊂ [B0 , B1 ]θ,1 .
Conversely, let u ∈ B0 ∩ B1 . We may decompose u into u = uj + vj with ∥uj ∥A0 ≤ 2−j
1
1
1−τ0
1−τ1
P
and ∥vj ∥A1 ≤ C min(∥u∥Bτ00 2j τ0 , ∥u∥Bτ11 2j τ1 ). Hence, we have u = j∈Z wj with
τ
τ0 j
τ
∥wj ∥1−τ
A0 ∥wj ∥A1 ≤ C min(∥u∥B0 2
τ −τ0
τ0
τ
, ∥u∥Bτ11 2j
τ −τ1
τ1
)
so that
∥u∥[A0 ,A1 ]τ,1 ≤ C
X
τ
min(∥u∥Bτ00 2j
τ −τ0
τ0
τ
, ∥u∥Bτ11 2j
j∈Z
Since θ =
τ −τ0
τ1 −τ0 ,
this gives [B0 , B1 ]θ,1 ⊂ [A0 , A1 ]τ,1 .
τ −τ1
τ1
τ −τ0
τ1 −τ
) ≤ C ′ ∥u∥Bτ11−τ0 ∥u∥Bτ10−τ0
Mild Solutions in Lebesgue or Sobolev Spaces
141
Lemma 7.7.
Let 0 < θ < 1, 0 < η < 1, θ + η ≤ 1.
(a) If T is a linear operator which is bounded from A0 to B0 and bounded from A1 to B1 ,
then it is bounded from [A0 , A1 ]θ,1 to [B0 , B1 ]θ,1 with operator norm
θ
∥T ∥L([A0 ,A1 ]θ,1 →[B0 ,B1 ]θ,1 ) ≤ ∥T ∥1−θ
L(A0 →B0 ) ∥T ∥L(A1 →B1 )
and similarly it is bounded from [A0 , A1 ]θ,∞ to [B0 , B1 ]θ,∞ with operator norm
θ
∥T ∥L([A0 ,A1 ]θ,∞ →[B0 ,B1 ]θ,∞ ) ≤ ∥T ∥1−θ
L(A0 →B0 ) ∥T ∥L(A1 →B1 )
(b) If T is a bilinear operator which is bounded from A0 × B0 to C0 and from A1 × B1 to
C1 , then it is bounded from [A0 , A1 ]θ,1 × [B0 , B1 ]θ,1 to [C0 , C1 ]θ,1 .
(c) If T is a bilinear operator which is bounded from A0 ×B0 to C0 , from A1 ×B0 to C1 and
from A0 ×B1 to C1 , then it is bounded from [A0 , A1 ]θ,∞ ×[B0 , B1 ]η,∞ to [C0 , C1 ]θ+η,∞ ,
from [A0 , A1 ]θ,1 × [B0 , B1 ]η,∞ to [C0 , C1 ]θ+η,1 and from [A0 , A1 ]θ,∞ × [B0 , B1 ]η,1 to
[C0 , C1 ]θ+η,1 .
Proof. (a) Assume that we have ∥T (u)∥B0 ≤ M0 ∥u∥A0 and ∥T (u)∥B1 ≤ M1 ∥u∥A1 . For
u ∈ A0 ∩ A1 , we have
1−θ
1−θ
θ
∥T (u)∥1−θ
M1θ ∥u∥1−θ
B0 ∥T (u)∥B1 ≤ M0
A0 ∥u∥A1
and we find that T maps [A0 , A1 ]θ,1 to [B0 , B1 ]θ,1 with operator norm Mθ ≤ M01−θ M1θ .
Similarly, if u ∈ [A0 , A1 ]θ,∞ with norm Nθ , we may decompose u for every λ > 0 (and
−θ
0
µ = λM
and ∥vλ ∥A1 ≤ N θ µ1−θ and we find
M1 ) into u = uλ + vλ with ∥uλ ∥A0 ≤ Nθ µ
that T (u) = T (uλ ) + T (vλ ) with
∥T (uλ )∥B0 ≤ M0 Nθ µ−θ = M01−θ M1θ Nθ λ−θ
and
∥T (vλ )∥B1 ≤ M1 Nθ µ1−θ = M01−θ M1θ Nθ λ1−θ
Thus, we find that T maps [A0 , A1 ]θ,∞ to [B0 , B1 ]θ,∞ with operator norm Mθ ≤
M01−θ M1θ .
(b) Assume that we have ∥T (u, v)∥C0 ≤ M0 ∥u∥A0 ∥v∥B0 and ∥T (u, v)∥C1
M1 ∥u∥A1 ∥v∥B1 . If u ∈ A0 ∩ A1 and v ∈ B0 ∩ B1 , we have
≤
1−θ
1−θ
θ
θ
θ
∥T (u, v)∥1−θ
M1θ ∥u∥1−θ
C0 ∥T (u, v)∥C1 ≤ M0
A0 ∥u∥A1 ∥v∥B0 ∥v∥B1
and we find that T maps [A0 , A1 ]θ,1 × [B0 , B1 ]θ,1 to [C0 , C1 ]θ,1 with operator norm
Mθ ≤ M01−θ M1θ .
(c) Assume that we have ∥T (u, v)∥C0 ≤ M0 ∥u∥A0 ∥v∥B0 , ∥T (u, v)∥C1 ≤ M1 ∥u∥A0 ∥v∥B1
and ∥T (u, v)∥C1 ≤ M2 ∥u∥A1 ∥v∥B0 . By interpolation, we find that T is bounded
from [A0 , A1 ]θ,1 × B0 to [C0 , C1 ]θ,1 with norm less or equal to M01−θ M2θ and from
[A0 , A1 ]θ,1 × [B0 , B1 ]1−θ,1 to C1 with norm less or equal to M01−θ M1θ . By interpolation, we find that, for 0 < η < 1 − θ, T is bounded from [A0 , A1 ]θ,1 × [B0 , B1 ]η,1 to
η
η
[C0 , C1 ]θ+η,1 with norm less or equal to C(M01−θ M2θ )1− 1−θ (M01−θ M1θ ) 1−θ . Interpolating between two values of η gives that T is bounded from [A0 , A1 ]θ,1 × [B0 , B1 ]η,∞
to [C0 , C1 ]θ+η,∞ ; interpolating between two values of θ gives that T is bounded
from [A0 , A1 ]θ,∞ × [B0 , B1 ]η,∞ to [C0 , C1 ]θ+η,∞ and from [A0 , A1 ]θ,1 × [B0 , B1 ]η,∞
to [C0 , C1 ]θ+η,1 .
142
The Navier–Stokes Problem in the 21st Century (2nd edition)
Example: Lorentz spaces.
In this example, we are interested in [L1 , L∞ ]1− p1 ,1 = Lp,1 and [L1 , L∞ ]1− p1 ,∞ = Lp,∞
for 1 < p < +∞ . Those interpolation spaces are Lorentz spaces [36] but we shall not use
the measure-theoretical definition of Lorentz spaces.
Another important identification of interpolation spaces for Lebesgue spaces is the
equality [L1 , L∞ ]1− p1 ,∞ = Lp,∗ , where Lp,∗ is the weak Lebesgue space introduced by
Marcinkiewicz. More precisely, we have:
Lemma 7.8.
Let 1 ≤ p0 < p1 ≤ +∞ and 0 < θ < 1. Then [Lp0 , Lp1 ]θ,∞ = Lp,∗ with
1
p
=
1−θ
p0
+
θ
p1 .
Proof. If f ∈ [Lp0 , Lp1 ]θ,∞ with norm M and λ > 0, we write f = fλ + gλ with ∥fλ ∥p0 ≤
p( p1 − p1 )
1
0 . We have
M µ−θ , ∥gλ ∥p1 ≤ M µ1−θ and µ = M
λ
|{x / |f (x)| > λ}| ≤ |{x / |fλ (x)| > λ/2}| + |{x / |gλ (x)| > λ/2}|
p0
p1
2
2
≤ M p0 µ−θp0
+ M p1 µ(1−θ)p1
λ
λ
1
= CM p p .
λ
1/p
Conversely, if f ∈ Lp,∗ , (∥f ∥p,∗ = supλ>0 λ (|{x / |f (x)| > λ})
< +∞) we write fλ =
f 1|f (x)|>λ and gλ = f − fλ . We have
X
X
p
∥fλ ∥p0 ≤
∥f 12j λ<|f (x)|≤2j+1 λ ∥p0 ≤
2(j+1) λ∥f ∥p,∗ (2j λ)− p0
j∈N
j∈N
p
=C∥f ∥p,∗ λθ p1 −p0
and
∥gλ ∥p0 ≤
X
∥f 12−(j+1) λ<|f (x)|≤2−j λ ∥p0 ≤
j∈N
X
p
2−j λ∥f ∥p,∗ (2−(j+1) λ)− p1
j∈N
p
= C∥f ∥p,∗ λ(θ−1) p1 −p0
Thus, the lemma is proved.
Consequences of those lemmas are the following ones:
ˆ Let 1 ≤ p0 < p1 ≤ +∞ and 0 < θ < 1. Then [Lp0 , Lp1 ]θ,1 = Lp,1 with
1
p
=
1−θ
p0
+
θ
p1 .
p
ˆ Forp 1 < p < ∞, the dual of Lp,1 is L p−1 ,∞ and the pointwise product maps Lp,1 ×
L p−1 ,∞ to L1 .
ˆ For 1 < p < ∞, and the pointwise product maps Lp,1 × L∞ to Lp,1 and Lp,∞ × L∞
to Lp,∞ .
p
< q < ∞, the pointwise product maps Lp,1 × Lq,∞ to Lr,1
ˆ For 1 < p < ∞ and p−1
p,∞
q,∞
r,∞
and L
×L
to L
with 1r = p1 + 1q .
Example: Besov spaces.
Let E be a Banach space such that:
ˆ E ⊂ S ′ (R3 )
ˆ E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E .
Mild Solutions in Lebesgue or Sobolev Spaces
143
s
We define the potential space HE
as the space of distributions f such that (Id−∆)s/2 f ∈ E
s/2
s
s
(with norm ∥f ∥HEs = ∥(Id − ∆) f ∥E ). Then, the Besov spaces BE,∞
and BE,1
are defined
in the following way: let s0 < s < s1 and θ ∈ (0, 1) such that s = (1 − θ)s0 + θs1 , then
s0
s1
s0
s1
s
s
s
[HE
, HE
]θ,∞ = BE,∞
and [HE
, HE
]θ,1 = BE,1
. Of course, we must check that BE,∞
and
s
BE,1 depend only on s and not on s0 and s1 .
Since (Id − ∆)σ/2 is an isomorphism between H s and H s−σ , we may assume with no
loss of generality that s1 < 0:
Lemma 7.9.
s0
s1
If s0 < s < s1 < 0, θ ∈ (0, 1) and s = (1 − θ)s0 + θs1 , then f ∈ [HE
, HE
]θ,∞ if and only if
s0
s1
|s|/2
sup0<t<1 t
∥Wt ∗ f ∥E < +∞. Moreover, the norm of f in [HE , HE ]θ,∞ is equivalent to
sup0<t<1 t|s|/2 ∥Wt ∗ f ∥E < +∞.
Proof. The proof is based on the following representation of (Id − ∆)−τ /2 when τ is a
positive real number:
Z +∞
1
dt
−τ /2
(Id − ∆)
f=
e−t Wt ∗ f tτ /2
(7.50)
Γ(τ /2) 0
t
(easily checked through the Fourier transform) and on the inequality
for σ > 0, sup min(1, t)σ/2 ∥(Id − ∆)σ/2 Wt ∥1 < +∞.
(7.51)
0<t
(7.51) is obvious for σ an even integer σ = 2N . For σ a general positive number, we write
σ = 2N + (σ − 2N ) = 2N − τ with 2N > σ and we use (7.50) to get
Z +∞
2N −σ ds
1
∥(Id − ∆)σ/2 Wt ∥1 ≤
e−s ∥(Id − ∆)2N Wt+s ∥1 s 2
2N −σ
s
Γ( 2 ) 0
Z +∞
2N −σ ds
≤C
e−s max(1, (t + s)−N ) s 2
.
s
0
If t > 1, we find
∥(Id − ∆)σ/2 Wt ∥1 ≤ C
+∞
Z
e−s s
2N −σ
2
0
ds
s
= C Γ(N − ).
s
2
If t ≤ 1, we have
∥(Id − ∆)σ/2 Wt ∥1 ≤ Ct−N
Z
t
s
2N −σ
2
0
Z
+C
t
1
ds
σ + C
s1+ 2
ds
s
Z +∞
1
e−s s
2N −σ
2
ds
s
≤ C ′ t−σ/2
Thus, (7.51) is proven.
s0
s1
Now, if f ∈ [HE
, HE
]θ,∞ with norm M , we decompose f into f = fλ + gλ with
∥(Id − ∆)s0 /2 fλ ∥E ≤ M λ−θ and ∥(Id − ∆)s1 /2 gλ ∥E ≤ M λ1−θ for λ = t
|s1 |−|s0 |
2
∥Wt ∗ f ∥E ≤ C(M λ−θ t−|s0 |/2 + M λ1−θ t−|s1 |/2 ) = 2CM t−
|s|
2
, and we get
.
Conversely, if sup0<t<1 t|s|/2 ∥Wt ∗ f ∥E = M < +∞, we use the identity (Id − ∆)s0 /2 f =
R +∞ −t
1
e Wt ∗ f t|s0 |/2 dt
Γ(|s0 |/2) 0
t , hence
Z +∞
1
dt
∥(Id − ∆)s0 /2 f ∥E ≤
M
e−t t|s0 |/2 max(1, t−|s|/2 ) < +∞.
Γ(|s0 |/2)
t
0
144
The Navier–Stokes Problem in the 21st Century (2nd edition)
s0
s0
s1
Thus, f ∈ HE
. In order to prove that f ∈ [HE
, HE
]θ,∞ , we must check that for all λ > 1,
−θ
we have f = fλ + gλ with ∥fλ ∥HEs0 ≤ CM λ and ∥gλ ∥HEs1 ≤ CM λ1−θ . Let A ∈ (0, 1). We
write f = uA + vA with
uA =
1
(Id − ∆)|s0 |/2
Γ(|s0 |/2)
and
vA =
1
(Id − ∆)|s0 |/2
Γ(|s0 |/2)
Z
Z
A
e−t Wt ∗ f t|s0 |/2
0
dt
t
+∞
e−t Wt/2 ∗ Wt/2 t ∗ f t|s0 |/2
A
dt
t
We have
∥uA ∥HEs0
1
M
≤
Γ(|s0 |/2)
Z
A
t
|s0 |−|s|
2
0
s−s0
dt
2M
=
A 2
t
Γ(|s0 |/2)(|s0 | − |s|)
and
Z
∥vA ∥HEs1 ≤ CM
+∞
A
≤ C ′M A
t
dt
t
e−t max(1, ( )(s0 −s1 )/2 ) max(1, ( )s/2 )t−s0 /2
2
2
t
s−s1
2
2
We conclude the proof by taking A = λ− s1 −s0 .
s
With Lemma 7.9, we have now a non-ambiguous definition of Besov spaces BE,1
and
We have the following properties for Besov spaces:
s
.
BE,∞
s
s
[36, 475].
and BLs p ,∞ = Bp,∞
ˆ for E = Lp , we find the usual Besov spaces BLs p ,1 = Bp,1
−s
s
ˆ If S is dense in E (so that E ′ ⊂ S ′ ), we have (BE,1
)′ = BE
′ ,∞ .
Example: homogeneous Besov spaces
Homogeneous Besov spaces may be defined in a similar way, at least for negative indexes,
replacing (Id − ∆)σ/2 with the fractional Laplacian operators (−∆)σ/2 .
Let again E be a Banach space such that:
ˆ E ⊂ S ′ (R3 )
ˆ E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E .
Then, for a positive σ, (−∆)σ/2 may be defined on E in the following way: if ω ∈ S is
such that its Fourier transform ω̂ is compactly supported and is identically equal to 1 on a
neighborhood of 0, then |ξ|σ (1 − ω̂(ξ) is a pointwise multiplier of S ′ (so that (−∆)σ/2 (δ − ω)
is a convolutor of S ′ , where δ is the Dirac mass at the origin) while |ξ|σ ω̂(ξ) is the Fourier
transform of a function (−∆)−σ/2 ω ∈ L1 . Then, (−∆)σ/2 is defined on E by:
(−∆)σ/2 f = ((−∆)σ/2 ω) ∗ f + ((−∆)σ/2 (δ − ω)) ∗ f
(7.52)
This definition does not depend on the choice of ω, and we have, for positive σ and τ ,
(−∆)σ/2 ((−∆)τ /2 f ) = (−∆)(σ+τ )/2 f .
Mild Solutions in Lebesgue or Sobolev Spaces
145
Definition 7.2.
Let E be a Banach space of tempered distributions that is stable under convolution with L1 .
−σ
We define, for positive σ, the Banach space ḢE
in the following way:
−σ
f ∈ ḢE
⇔ ∃g ∈ E f = (−∆)σ/2 g
and
∥f ∥Ḣ −σ =
E
inf
f =(−∆)σ/2 g
∥g∥E .
If g ∈ E satisfies (−∆)σ/2 g = 0, then ĝ is supported in {0}, hence g is a polynomial
function; moreover, we have
Z
|W ∗ g(x)| = |⟨W1/2 (−z)| W1/2 (x + z − y)g(y) dy⟩| ≤ C∥W1/2 (x + .) ∗ g∥E ≤ C ′ ∥g∥E
Since W ∗ g is a polynomial with the same degree as g, we find that g must be constant.
−σ
Thus, ḢE
is isomorphic to the Banach space E/(E ∩ R 1).
σ
for positive σ is not so direct. If there exists another Banach space
The definition of ḢE
F of tempered distributions that is stable under convolution with L1 and a positive βE such
that
sup tβE /2 ∥Wt ∗ f ∥F ≤ C∥f ∥E
t≥1
−σ/2
then we may define (−∆)
on E (with values in E + F ) for 0 < σ < β as:
Z +∞
dt
1
Wt ∗ f tσ/2
(−∆)−σ/2 f =
Γ(σ/2) 0
t
(7.53)
σ
We then define ḢE
in the following way:
σ
f ∈ ḢE
⇔ ∃g ∈ E f = (−∆)−σ/2 g
and
∥(−∆)−σ/2 g∥Ḣ σ = ∥g∥E .
E
0
= E.
Similarly, we define ḢE
τ
σ
.
isomorphism from ḢE to ḢE
If −∞ < σ < βE , −∞ < τ < βE then (−∆)(σ−τ )/2 is an
s
s
in the following way: let
and ḂE,1
We define the homogeneous Besov spaces ḂE,∞
s0
s1
s
s0 < s < s1 < βE and θ ∈ (0, 1) such that s = (1 − θ)s0 + θs1 , then [ḢE
, ḢE
]θ,∞ = ḂE,∞
s0
s1
s
s
s
and [ḢE , ḢE ]θ,1 = ḂE,1
. Of course, ḂE,∞
and ḂE,1
depend only on s and not on s0 and
s1 . In particular, we have
Lemma 7.10.
If s < 0, then
s
f ∈ ḂE,∞
if and only if sup t|s|/2 ∥Wt ∗ f ∥E < +∞.
0<t
s
Moreover, the norm of f in ḂE,∞
is equivalent to sup0<t t|s|/2 ∥Wt ∗ f ∥E .
s
and ḂLs p ,∞ =
For E = Lp , we find the usual homogeneous Besov spaces ḂLs p ,1 = Ḃp,1
3
s
Ḃp,∞
with βE = p [36, 475].
Lemma 7.11.
−s
s
If S is dense in E (so that E ′ ⊂ S ′ ), we have (ḂE,1
)′ = ḂE
′ ,∞ for −βE ′ < s < βE .
146
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proof. Let ω ∈ S be such that its Fourier transform ω̂ is compactly supported
and is identically equal to 1 on a neighborhood of 0. Let ∆j be the convos0
s1
lution operator with 23(j+1) ω(2j+1 x) − 23j ω(2j x). If f ∈ ḢE
∩ ḢE
, we have
P
∥∆j f ∥E ≤ C min(2−js0 ∥f ∥Ḣ s0 , 2−js1 ∥f ∥Ḣ s1 ). From this, we get that j∈Z 2js ∥∆j f ∥E ≤
s1 −s
s1 −s0
s
ḢE0
C∥f ∥
s−s0
s1 −s0
s
ḢE1
∥f ∥
E
E
2
. Finally, we get
s
f ∈ ḂE,1
⇔f =
X
∆j f with
j∈Z
X
2js ∥∆j f ∥E < +∞.
j∈Z
s
Hence, we get that T ∈ (ḂE,1
)′ if and only if it can be written as a *-weak convergent series
P
T = j∈Z ∆j Tj with supj∈Z ∥Tj ∥E ′ 2−js < +∞. We have ∥∆j Tj ∥Ḣ −s0 ≤ C2−js0 ∥T ∥E ′ and
−s
∥∆j Tj ∥Ḣ −s1 ≤ C2−js1 ∥ T ∥E ′ , so that T ∈ ḂE
′ ,∞ .
E′
E′
7.9
Uniqueness of L3 Solutions
The results of Kato [255] left an open question: uniqueness for mild solutions
in C([0, T ), (L3 )3 ). Given ⃗u0 ∈ (L3 )3 , we may not directly construct a solution in
C([0, T ), (L3 )3 ), since B is not continuous on C([0, T ), (L3 )3 ) (Oru [383]). Solutions are
always constructed in a smaller space (see Kato [255], Giga [209], Cannone [81], or Planchon [399]) and, thus, uniqueness was first granted only in the subspaces of C([0, T ), (L3 )3 )
where the iteration algorithm was convergent. In 1997, Furioli, Lemarié-Rieusset and Terraneo [186, 187] proved uniqueness in C([0, T ), (L3 )3 ):
Uniqueness of L3 solutions
Theorem 7.7.
If ⃗u and ⃗v are two solutions of the Equation (7.4) on (0, T ) × R3 with ⃗u0 ∈
Rt
(L3 )3 and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C([0, T ), (L3 (R3 ))3 ) so that ⃗u and ⃗v belong to
C([0, T ), (L3 (R3 ))3 ), then ⃗u = ⃗v .
Proof. In the two years following its proof by Furioli, Lemarié-Rieusset and Terraneo [186],
this theorem was reproved by many authors through various methods (Meyer [359], Monniaux [368], Lions and Masmoudi [340]) and was extended to the case of Morrey-Campanato
spaces by Furioli, Lemarié-Rieusset and Terraneo [187] and May [325, 353], as we shall see
in Section 8.3.
Here, we will sketch the proof of Furioli, Lemarié-Rieusset and Terraneo [186] and its
adaptation by Meyer [359] and Monniaux [368].
The first step of the proof is the reduction to prove local uniqueness. If we prove that
under the assumptions of Theorem 7.7 there exists some positive ϵ so that ⃗u = ⃗v on [0, ϵ],
we can end the proof in the following way:
ˆ let E = {τ ∈ [0, T ) / ∥⃗u(, .) − ⃗v (t, .)∥3 = 0 on [0, τ ]}; we have 0 ∈ E: ⃗u(0, .) = ⃗v (0, .);
2 This
is the homogeneous Littlewood–Paley decomposition of f .
Mild Solutions in Lebesgue or Sobolev Spaces
147
ˆ let τ ∗ = supτ ∈E τ ; if τ ∗ < T , then, by continuity, we have ⃗u(τ ∗ , .) = ⃗v (τ ∗ , .) in (L3 )3 ,
so that τ ∗ ∈ E; moreover, we have, on [0, T − τ ∗ ),
Z t
⃗u(t + τ ∗ , x) =Wνt ∗ ⃗u(τ ∗ , x) +
Wν(t−s) ∗ Pf⃗(s + τ ∗ , .) ds
0
Z t
−
Wν(t−s) ∗ P div(⃗u(s + τ ∗ , .) ⊗ ⃗u(s + τ ∗ , ; )) ds
0
Rt
and the same equation for ⃗v . Remark that 0 Wν(t−s) ∗ Pf⃗(s + τ ∗ , .) ds may be written
as
Z t+τ ∗
Z τ∗
Wν(t−s) ∗ Pf⃗(s, .) ds − Wνt ∗
Wν(τ ∗ −s) ∗ Pf⃗(s, .) ds
0
0
and thus fulfills hypotheses of Theorem 7.7. Thus, if we have local uniqueness, there
exists a positive ϵ such that ⃗u(s + τ ∗ , .) = ⃗v (s + τ ∗ , .) for 0 ≤ s ≤ ϵ, so that τ ∗ + ϵ ∈ E.
This is a contradiction with the definition of τ ∗ . Thus τ ∗ = T and ⃗u = ⃗v for all
t ∈ [0, T ).
The second step is to write the equation ⃗u = ⃗v as a fixed-point problem in w
⃗ = ⃗u −⃗v : we
⃗ −B(⃗u, ⃗u) and ⃗v = U
⃗ −B(⃗v , ⃗v ), so that, writing ⃗u = Wνt ∗⃗u0 −⃗u1 , ⃗v = Wνt ∗u0 −⃗v1 ,
have ⃗u = U
we get
w
⃗ = B(⃗u1 , w)
⃗ + B(w,
⃗ ⃗v1 ) − B(Wνt ∗ ⃗u0 , w)
⃗ − B(w,
⃗ Wνt ∗ ⃗u0 ).
(7.54)
The idea is that ⃗u1 and ⃗v1 are small in L3 norm: limt→0 ∥⃗u1 (t,√.)∥3 = limt→0 ∥⃗v1 (t, .)∥3 = 0,
while Wνt ∗ ⃗u0 is small in other norms (for instance, limt→0 t∥Wνt ∗ ⃗u0 ∥∞ = 0), so that
we may hope to find a contractive estimate to prove that w
⃗ = 0.
This contractive estimate is not to be hoped in terms of the norm of w
⃗ in
C([0, ϵ), (L3 (R3 ))3 , as we know that the bilinear operator B is not bounded on
C([0, ϵ), (L3 (R3 ))3 . Thus, the third step is to identify a norm on w
⃗ for which one has a
contractive estimate.
ˆ Besov norms: The proof of Furioli, Lemarié–Rieusset and Terraneo is based on basic
inequalities on Besov norms. In the following inequalities, the constants Ci , i = 1, . . . ,
depend on ν and p but not on τ :
R +∞
1. for 3/2 < p < 3 and 0 < η < 1 and φ ∈ D(R3 ), we have the inequality 0 ∥Wνt ∗
√
−∆φ∥Ḃ 1−η
ds ≤ C1 ∥φ∥Ḃ −ηp : it is enough to check that
p
p−1
,1
+∞
Z
∥Wνt ∗
0
p−1
√
,1
Z
−∆φ∥Ḃ 1−η
p
p−1
ds ≤ C0
,1
0
∥φ∥Ḃ −η−1
∥φ∥Ḃ −η+1
p
p
,1
,1
p−1
p−1
min(
,
) ds
3/2
1/2
s
s
+∞
r
= C1 ∥φ∥Ḃ −η−1
∥φ∥Ḃ −η+1
p
p
p−1
,1
p−1
,1
2. writing
Z
|
⃗ (t, x)).⃗
(⃗u(t, x) − U
φ(x) dx|
Z t
√
η−1 ∥Wνt ∗
≤ C2 ∥⃗u(s, .)⊗⃗u(s, .)∥Ḃp,∞
−∆φ∥Ḃ 1−η
p
0
p−1
ds
,1
148
The Navier–Stokes Problem in the 21st Century (2nd edition)
3
p
for 3/2 < p < 3 and η =
− 1, we get
⃗ (t, .)∥
sup ∥⃗u(t, .) − U
≤ C3 sup ∥⃗u(t, .) ⊗ ⃗u(t, .)∥
3 −1
p
Ḃp,∞
0<t<τ
3 −2
p
Ḃp,∞
0<t<τ
≤ C4 sup ∥⃗u(t, .) ⊗ ⃗u(t, .)∥3/2
0<t<τ
≤ C5 sup ∥⃗u(t, .)∥23
0<t<τ
⃗ (t, .)∥
3. similarly, sup0<t<τ ∥⃗v (t, .) − U
sup0<t<τ ∥w(t,
⃗ .)∥
3 −1
p
Ḃp,∞
4. for 3/2 < p < 3, η =
1
r
=
1−ϵ
3
3
p
3 −1
p
Ḃp,∞
≤ C5 sup0<t<τ ∥⃗v (t, .)∥23 , so that
< +∞
− 1 and 0 < ϵ < min(η, 1 − η), we have Ḣpη+ϵ ⊂ Lr with
and Ḣpη−ϵ ⊂ Lρ with
1
ρ
=
1+ϵ
3
s
(Sobolev embeddings) and, by duality of
σ
η−ϵ−1
the Sobolev embeddings, we have L ⊂ Ḣpη+ϵ−1 with 1s = 2−ϵ
3 and L ⊂ Ḣp
1
2+ϵ
3
with σ = 3 . thus, pointwise multiplication with a function in L maps Ḣpη+ϵ
η
η−1
to Ḣpη+ϵ−1 and Ḣpη−ϵ to Ḣpη−ϵ−1 , and by interpolation Ḃp,∞
to Ḃp,∞
.
5. for 3/2 < p < 3,
sup ∥B(⃗u1 , w)∥
⃗
3 −1
p
Ḃp,∞
0<t<τ
≤ C3 sup ∥⃗u1 (t, .) ⊗ w(t,
⃗ .)∥
3 −2
p
Ḃp,∞
0<t<τ
≤ C6 sup ∥⃗u1 (t, .)∥3 ∥w(t,
⃗ .)∥
3 −1
p
Ḃp,∞
0<t<τ
≤ C6 sup ∥⃗u1 (t, .)∥3 sup ∥w(t,
⃗ .)∥
0<t<τ
3 −1
p
Ḃp,∞
0<t<τ
and
sup ∥B(w,
⃗ ⃗v1 )∥
3 −1
p
Ḃp,∞
0<t<τ
≤ C6 sup ∥⃗v1 (t, .)∥3 sup ∥w(t,
⃗ .)∥
0<t<τ
3 −1
p
Ḃp,∞
0<t<τ
6. similarly, if 3 < r < η3 , we find (by interpolating Sobolev embedding inequalities)
η− 3
η
that pointwise multiplication with a function in Lr maps Ḃp,∞
to Ḃp,∞r .
7. for 3/2 < p < 3 and 3 < r <
∥B(Wνt ∗ ⃗u0 , w)(t,
⃗
.)∥
3p
3−p ,
we find
3 −1
p
Ḃp,∞
Z t
≤ C7
1
1
0
3
(t − s) 2 + 2r
∥(Wνs ∗ ⃗u0 ) ⊗ w(s,
⃗ .)∥
3 −1− 3
r
p
Ḃp,∞
1
ds
3
≤ C8 sup s 2 − 2r ∥Wνs ∗ ⃗u0 ∥r ∥w(s,
⃗ .)∥
3 −1
p
Ḃp,∞
0<s<t
8. Thus, we find
sup ∥w(t,
⃗ .)∥
0<t<τ
3 −1
p
Ḃp,∞
≤ C9 A(τ ) sup ∥w(t,
⃗ .)∥
3 −1
p
Ḃp,∞
0<t<τ
with
1
3
A(τ ) = sup ∥⃗u1 (t, .)∥3 + sup ∥⃗v1 (t, .)∥3 + sup t 2 − 2r ∥Wνt ∗ ⃗u0 ∥r .
0<t<τ
0<t<τ
0<t<τ
As we have limτ →0+ A(τ ) = 0, the theorem is proved.
Mild Solutions in Lebesgue or Sobolev Spaces
149
ˆ Lorentz norms: Meyer’s proof follows the lines of the proof of Furioli, LemariéRieusset and Terraneo, but deals with the Lorentz space Lp,∞ instead of the Besov
3
−1
p
space Ḃp,∞
:
1. for φ ∈ D(R3 ), we have the inequality
it is enough to check that
+∞
Z
∥Wνt ∗
R +∞
√
−∆φ∥L3,1 ds ≤ C0
0
0
√
∥Wνt ∗ −∆φ∥L3,1 ds ≤ C1 ∥φ∥L3/2,1 :
∥φ∥
+∞
Z
0
= C1
q
∥φ∥
3
L 2−3ϵ
∥φ∥
3
L 2+3ϵ
3ϵ
s1+ 2
min(
∥φ∥
,
3
L 2−3ϵ
3ϵ
s1− 2
) ds
3
L 2+3ϵ
2. from this inequality, we get
sup ∥B(⃗u1 , w)∥
⃗ L3,∞ ≤ C2 sup ∥⃗u1 (t, .) ⊗ w(t,
⃗ .)∥L3/2,∞
0<t<τ
0<t<τ
≤ C3 sup ∥⃗u1 (t, .)∥3 sup ∥w(t,
⃗ .)∥L3,∞
0<t<τ
0<t<τ
and
sup ∥B(w,
⃗ ⃗v1 )∥L3,∞ ≤ C3 sup ∥⃗v1 (t, .)∥3 sup ∥w(t,
⃗ .)∥L3,∞
0<t<τ
0<t<τ
0<t<τ
3. moreover, we have
∥B(Wνt ∗ ⃗u0 , w)(t,
⃗
.)∥L3,∞
Z t
≤ C4
0
1
3
(t − s) 4
∥(Wνs ∗ ⃗u0 ) ⊗ w(t,
⃗ .)∥L2,∞ ds
1
≤ C5 sup s 4 ∥Wνs ∗ ⃗u0 ∥6 ∥w(s,
⃗ .)∥L3,∞
0<s<t
4. thus, we find
sup ∥w(t,
⃗ .)∥L3,∞ ≤ C6 A(τ ) sup ∥w(t,
⃗ .)∥L3,∞
0<t<τ
0<t<τ
with
1
A(τ ) = sup ∥⃗u1 (t, .)∥3 + sup ∥⃗v1 (t, .)∥3 + sup t 4 ∥Wνt ∗ ⃗u0 ∥6 .
0<t<τ
0<t<τ
0<t<τ
As we have limτ →0+ A(τ ) = 0, the theorem is proved.
ˆ Lp Lq maximal regularity: Monniaux’s proof replaces the role of real interpolation
by the Lp LqRmaximal regularity for the heat kernel [313] and shows that it is easier
τ
⃗ .)∥3 . Monniaux’s
to estimate 0 ∥|w(t,
⃗ .)∥p3 dt (for 2 < p < +∞) than sup0<t<τ ∥w(t,
proof is then the following one:
1. Lp Lq maximal regularity for the heat kernel states that, for 1 < p < +∞ and
1 < q < +∞, there exists a constant C1 (which depends on p, q and ν, but not
on τ ) such that
Z
∥
t
Wν(t−s) ∗ ∆F (s, .) ds∥Lp ((0,τ ),Lq ) ≤ C1 ∥F |Lp ((0,τ ),Lq )
0
150
The Navier–Stokes Problem in the 21st Century (2nd edition)
2. we have
∥
1
P div(⃗u1 (s, .) ⊗ w(s,
⃗ .))∥3 ≤ C2 ∥I1 (⃗u1 (s, .) ⊗ w(s,
⃗ .))∥3
∆
≤ C3 ∥⃗u1 (s, .)∥3 ∥w(s,
⃗ .)∥3
3. from this inequality and maximal regularity, we get, for 1 < p < +∞,
∥B(⃗u1 , w)∥
⃗ Lp ((0,τ ),L3 ) ≤ C4 sup ∥⃗u1 (t, .)∥3 ∥w(t,
⃗ .)∥Lp ((0,τ ),L3 )
0<t<τ
4. moreover, we have
t
Z
1
∥B(Wνt ∗ ⃗u0 , w)(t,
⃗
.)∥3 ≤ C5
≤ C6
∥(Wνs ∗ ⃗u0 ) ⊗ w(t,
⃗ .)∥3 ds
1
0 (t − s) 2
Z
1
2
sup s ∥Wνs ∗ ⃗u0 ∥∞
0<s<t
t
0
1
1
1
2
1
s (t − s) 2
∥w(s,
⃗ .)∥3 ds
If 2 < p < +∞, we find
∥B(Wνt ∗ ⃗u0 , w)(t,
⃗
.)∥Lp ((0,τ ),L3 )
Z
1
t
≤ C7 sup s 2 ∥Wνs ∗ ⃗u0 ∥∞
0<s<t
0
1
1
1
s (t − s) 2
≤ C8 sup s 2 ∥Wνs ∗ ⃗u0 ∥∞
0<s<t
1
1
2
1
1
s2
∥w(s,
⃗ .)∥3 ds
)
Lp ((0,τ )
∥w(s,
⃗ .)∥3 ds
2p
L 2+p
,p
((0,τ ))
1
2
≤ C9 sup s ∥Wνs ∗ ⃗u0 ∥∞ ∥∥w(s,
⃗ .)∥3 ds∥Lp ((0,τ ))
0<s<t
5. thus, we find
∥w(t,
⃗ .)∥Lp ((0,τ ),L3 ) ≤ C10 A(τ )∥w(t,
⃗ .)∥Lp ((0,τ ),L3 )
with
1
A(τ ) = sup ∥⃗u1 (t, .)∥3 + sup ∥⃗v1 (t, .)∥3 + sup t 2 ∥Wνt ∗ ⃗u0 ∥∞ .
0<t<τ
0<t<τ
0<t<τ
As we have limτ →0+ A(τ ) = 0, the theorem is proved.
Chapter 8
Mild Solutions in Besov or Morrey Spaces
Soon after the release of Kato’s paper on strong solutions in Lebesgue spaces, there has
been a flourishing of papers on solutions in Morrey spaces: Giga and Miyakawa [212] in
1989, Kato [256] and Taylor [467] in 1992, Federbush [169] in 1993. Those examples have
been extended in the books of Cannone [81] in 1995 and Lemarié-Rieusset [313] in 2002,
and recently in 2013 in the book of Triebel [476].
8.1
Morrey Spaces
The simplest generalization one can introduce uses only basic properties of those spaces:
scaling, shift invariance and stability under bounded pointwise multiplication. We shall use
real interpolation spaces as auxiliary spaces where to look for solutions.
We begin with an easy lemma on shift invariance:
Shift invariant estimates
Lemma 8.1. Let E be a Banach space such that:
• E ⊂ S ′ (R3 ) (continuous embedding)
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
Then
• for every φ ∈ S and every x0 ∈ R3 , ∥φ(x − x0 )∥E ′ = ∥φ∥E ′ .
• for every φ ∈ S, the map f 7→ f ∗ φ is bounded from E to L∞ .
R
Proof. If θ ∈ D be an even function with θdx = 1, then φ(x − x0 ) is the limit (for ϵ → 0)
0
in S of φ ∗ θϵ,x0 , where θϵ,x0 (x) = ϵ13 θ( x−x
ϵ ). Thus, we have
⟨φ(x − x0 )|f ⟩E ′ ,E =⟨φ(x − x0 )|f ⟩S,S ′
= lim ⟨φ|θϵ,−x0 ∗ f ⟩S,S ′
ϵ→0
and thus
∥φ(x − x0 )∥E ′ ≤ ∥φ∥E ′ .
In particular, we have
|f ∗ φ(x)| = |⟨φ(x − y)|f (y)⟩S,S ′ | ≤ ∥f ∥E ∥φ(−x)∥E ′ .
DOI: 10.1201/9781003042594-8
151
152
The Navier–Stokes Problem in the 21st Century (2nd edition)
Navier–Stokes equations and local measures: the bilinear operator
Theorem 8.1.
Let E be a Banach space such that:
• E ⊂ S ′ (R3 ) (continuous embedding)
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
• E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
• E is stable under dilations with a (sub)critical scaling: for λ ≤ 1, ∥f (λx)∥E ≤
Cλ−α ∥f ∥E for two constants C > 0 and 0 ≤ α ≤ 1.
The bilinear operator B defined as
Z t
⃗
⃗
⃗ ds
B(F , G) =
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is continuous on the space YT of Lebesgue measurable vector fields on (0, T ) × R3 such
that t 7→ ⃗u(t, .) is continuous from (0, T ) to ([E, L∞ ]1/2,1 )3 and
∥⃗u∥YT = sup tα/4 ∥⃗u(t, .)∥[E,L∞ ]1/2,1 < +∞.
0<t<T
More precisely, we have:
α
⃗ [E,L∞ ]
sup t 4 ∥B(F⃗ , G)∥
≤ Cν T
1/2,1
1−α
2
⃗ Y
(1 + (νT )α/4 )∥F⃗ ∥YT ∥G∥
T
(8.1)
0<t<T
⃗
where the constant Cν does not depend on T , F⃗ nor G.
⃗
⃗
Moreover t 7→ B(F , G)(t, .) is continuous from [0, T ] to E 3 if α < 1. If α = 1, t 7→
⃗
⃗ belong
B(F⃗ , G)(t,
.) is continuous and bounded from (0, T ] to E 3 ; if moreover F⃗ or G
1/4 ⃗
⃗
⃗ ∈
∞
to the space YT,0 = {f ∈ YT / limt→0 t ∥f (t, .)∥[E,L ]1/2,1 = 0}, then B(F⃗ , G)
3
YT,0 ∩ C([0, T ], E ).
Proof. Pointwise product maps E × L∞ and L∞ × E to E, hence maps [E, L∞ ]1/2,1 ×
[E, L∞ ]1/2,1 to E. Moreover convolution with Wt maps E to E with a bounded operator
norm while it maps E to L∞ with an operator norm which is O(max(1, t−α/2 )). Thus, we
find, writing
⃗ = Wν(t−s)/2 ∗ P div(Wν(t−s)/2 ∗ (F⃗ ⊗ G)),
⃗
Wν(t−s) ∗ P div(F⃗ ⊗ G)
that (using the inequality max(1, (ν(t − s))−α/4 ) ≤ (ν(t − s))−α/4 max(1, (νT )α/4 )),
1
max(1, (νT )α/4 ) 1 ⃗
⃗ [E,L∞ ]
⃗
p
∥Wν(t−s) ∗ P div(F⃗ ⊗ G)∥
≤
C
α ∥F ∥YT ∥G∥YT
1/2,1
ν(t − s) (ν(t − s))α/4 s 2
so that
⃗
tα/4 ∥B(F⃗ , G)(t,
.)∥[E,L∞ ]1/2,1 ≤ C ′
1 + (νT )α/4
ν
2+α
4
t
1−α
2
⃗ Y .
∥F⃗ ∥YT ∥G∥
T
Mild Solutions in Besov or Morrey Spaces
153
Similarly, we have
1 ⃗
⃗ E ≤ Cp 1
⃗
∥Wν(t−s) ∗ P div(F⃗ ⊗ G)∥
α ∥F ∥YT ∥G∥YT
ν(t − s) s 2
and
⃗
∥B(F⃗ , G)(t,
.)∥E ≤ Ct
1−α
2
1
⃗ Y .
√ ∥F⃗ ∥YT ∥G∥
T
ν
The proof of continuity follows the same line as for Theorem 7.5. Given a time t > 0,
⃗ = B(F⃗ , G),
⃗ we write
we consider θ close to t: |t − θ| < 13 t; let η = |t − θ|. For H
Z θ Z t−2η
⃗ x) − H(θ,
⃗
⃗ ds dτ
H(t,
x) =
∆Wν(τ −s) ∗ P div(F⃗ ⊗ G)
t
Z
0
t
+
⃗ ds −
Wν(t−s) ∗P div(F⃗ ⊗ G)
Z
t−2η
θ
⃗ ds
Wν(θ−s) ∗ P div(F⃗ ⊗ G)
t−2η
so that
⃗ .) − H(θ,
⃗
∥H(t,
.)∥[E,L∞ ]1/2,1 ≤
Z
Z
Cν,T
[t,θ]
t/3
1
ds
3/2+α/4
(τ − s)
sα/2
0
Z
Z
t−2η
+ Cν,T
[t,θ]
Z
+Cν,T
t/3
t
− s)
⃗ Y
dτ ∥F⃗ ∥YT ∥G∥
T
1
ds
(τ − s)3/2+α/4 sα/2
1
t−2η (t
Z
!
1
α
2+ 4
!
⃗ Y
dτ ∥F⃗ ∥YT ∥G∥
T
ds ⃗
⃗ Y
∥F ∥YT ∥G∥
T
tα/2
θ
1
ds ⃗
⃗ Y
∥F ∥YT ∥G∥
1
T
α/2
+α
t
2
4
(θ
−
s)
t−2η
1
1
1
1
′
−α
⃗
⃗
2
4
≤ Cν,T ∥F ∥YT ∥G∥YT |t − θ| 1 + 3α + |t − θ| 1 + α
+η
tα/2
t2 4
η 2 4 tα/2
+Cν,T
1
′
⃗ Y
≤ 3 Cν,T
∥F⃗ ∥YT ∥G∥
T
α
|t − θ| 2 − 4
tα/2
and
⃗ .) − H(θ,
⃗
∥H(t,
.)∥E ≤ Cν,T
Z
t/3
Z
[t,θ]
Z
0
Z
1
ds
(τ − s)3/2 sα/2
t−2η
+ Cν,T
[t,θ]
Z
t
+ Cν,T
t−2η
Z
θ
+ Cν,T
t−2η
′
⃗ Y
≤ Cν,T
∥F⃗ ∥YT ∥G∥
T
t/3
1
ds
3/2
(τ − s) sα/2
1
(t − s)
ds
⃗ Y
dτ ∥F⃗ ∥YT ∥G∥
T
!
⃗ Y
dτ ∥F⃗ ∥YT ∥G∥
T
⃗ Y
∥F⃗ ∥YT ∥G∥
T
1
2
tα/2
1
2
ds ⃗
⃗ Y
∥F ∥YT ∥G∥
T
tα/2
1
(θ − s)
|t − θ|
1
1
α
t2+ 2
+ |t − θ|
1
≤
!
′
⃗ Y
3 Cν,T
∥F⃗ ∥YT ∥G∥
T
|t − θ| 2
.
tα/2
1
1
η 2 tα/2
+η
1
2
1
tα/2
154
The Navier–Stokes Problem in the 21st Century (2nd edition)
Navier–Stokes equations and local measures: mild solutions
Theorem 8.2.
Let E be a Banach space such that:
• E ⊂ S ′ (R3 )
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
• E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
• E is stable under dilations with a (sub)critical scaling: for λ ≤ 1, ∥f (λx)∥E ≤
Cλ−α ∥f ∥E for two constants C > 0 and 0 ≤ α ≤ 1.
Let YT be the space of Lebesgue measurable vector fields on (0, T ) × R3 such that t 7→
⃗u(t, .) is continuous from (0, T ) to ([E, L∞ ]1/2,1 )3 and
∥⃗u∥YT = sup tα/4 ∥⃗u(t, .)∥[E,L∞ ]1/2,1 < +∞.
0<t<T
Then
• If ⃗u0 ∈ E 3 , then Wνt ∗ ⃗u0 ∈ YT .
• If f⃗ is defined on (0, T ) × R3 and satisfies Pf⃗ ∈ L1 ((0, T ), E 3 ) and
Rt
sup0<t<T t∥Pf⃗(t, .)∥E < +∞, then 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C((0, T ], E 3 ) ∩ YT
and limt→0 tα/4 ∥Pf⃗(t, .)∥[E,L∞ ]1/2,1 = 0.
Rt
• Case α < 1: If ⃗u0 ∈ E and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ YT , then there exists a
T0 ∈ (0, T ) and a mild solution
Z
t
Wν(t−s) ∗ Pf⃗(s, .) ds − B(⃗u, ⃗u)
⃗u = Wνt ∗ ⃗u0 +
0
of Equation (7.4) on (0, T0 ) × R3 such that ⃗u ∈ C([0, T0 ], E 3 ) ∩ YT0 .
Rt
• Case α = 1: If ⃗u0 ∈ E and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ YT and if Wνt ∗ ⃗u0 and
Rt
Wν(t−s) ∗Pf⃗(s, .) ds are small enough in YT , then there exists a mild solution
0
Z
⃗u = Wνt ∗ ⃗u0 +
t
Wν(t−s) ∗ Pf⃗(s, .) ds − B(⃗u, ⃗u)
0
of Equation (7.4) on (0, T ) × R3 such that ⃗u ∈ C((0, T ], E 3 ) ∩ YT .
• Case α = 1: if S ⊂ E, if ⃗u0 belongs to the closure of S 3 in E 3 , if f⃗ is defined on
(0, T ) × R3 and satisfies Pf⃗ ∈ L1 ((0, T ), E 3 ) and sup0<t<T t∥Pf⃗(t, .)∥E < +∞,
then there exists a T0 ∈ (0, T ) and a mild solution ⃗u of Equation (7.4) in
C([0, T0 ], E 3 ) ∩ YT0 .
Remark: Under those assumptions
on E, in the case α = 1, we have limT →0 ∥Wνt ∗ ⃗u0 ∥YT
√
as soon as limt→0 sup0<t<T t∥Wνt ∗ ⃗u0 ∥∞ = 0. In particular, when ⃗u0 belongs to the
closure of S 3 in E 3 .
Mild Solutions in Besov or Morrey Spaces
155
Proof. If ⃗u0 ∈ E 3 , we know that ∥Wνt ∗ ⃗u0 ∥[E,L∞ ]1/2,1 ≤ C max(1, νt)−α/2 ∥⃗u0 ∥E ; moreover,
1
we have ∥∂t Wνt ∗ ⃗u0 ∥[E,L∞ ]1/2,1 ≤ C νt
max(1, νt)−α/2 ∥⃗u0 ∥E . Thus, Wνt ∗ ⃗u0 ∈ YT . Simi1
larly, we have ∥Wνt ∗ ⃗u0 ∥E ≤ ∥⃗u0 ∥E and ∥∂t Wνt ∗ ⃗u0 ∥E ≤ C νt
∥⃗u0 ∥E ; thus, Wνt ∗ ⃗u0 ∈
3
3
3
C((0, T ], E ). If ⃗u0 belongs to S ⊂ E , then we have ∥Wνt ∗ ⃗u0 − ⃗u0 ∥E ≤ t∥∆⃗u0 ∥E
and ∥Wνt ∗ ⃗u0 ∥∞ ≤ ∥⃗u0 ∥∞ ; thus, if ⃗u0 belongs to the closure of S 3 in E 3 , we have
limt→0 tα/4 ∥Wνt ∗ ⃗u0 ∥[E,L∞ ]1/2,1 = 0 and limt→0 ∥Wνt ∗ ⃗u0 − ⃗u0 ∥E = 0, so that Wνt ∗ ⃗u0 ∈
C([0, T ], E 3 ).
If Pf⃗ ∈ L1 ((0, T ), E 3 ) and sup0<t<T t∥Pf⃗(t, .)∥E < +∞, we find that
Z
∥
t
Wν(t−s) ∗ Pf⃗(s, .) ds∥[E,L∞ ]1/2,1 ≤ Cν,T
Z
0
Z
t/2
t−α/4 ∥Pf⃗∥E ds
0
t
(t − s)−α/4
+ Cν,T
t/2
≤
and
Z
∥
0
′
Cν,T
t−α/4 (∥Pf⃗∥L1t E
ds
sup s∥Pf⃗(s, .)∥E
t 0<s<t
+ sup s∥Pf⃗(s, .)∥E )
0<s<T
t
Wν(t−s) ∗ Pf⃗(s, .) ds∥E ≤ ∥Pf⃗∥L1t E .
Moreover, given a time t > 0, we consider θ close to t: |t − θ| < 13 t; let η = |t − θ|. For
Rt
F⃗ = 0 Wν(t−s) ∗ Pf⃗(s, .) ds, we write
F⃗ (t, x) − F⃗ (θ, x) =
Z
θ
Z
Z
t
t
t−2η
∆Wν(τ −s) ∗ Pf⃗(s, .) ds dτ
0
+
Wν(t−s) ∗ Pf⃗ ds −
t−2η
Z
θ
Wν(θ−s) ∗ Pf⃗ ds
t−2η
so that
!
1
⃗
∥F⃗ (t, .) − F⃗ (θ, .)∥[E,L∞ ]1/2,1 ≤ Cν,T
dτ
α ∥Pf (s, .)∥E ds
(τ − s)1+ 4
[t,θ]
0
!
Z
Z t−2η
1
ds
+Cν,T
dτ sup s∥Pf⃗(s, .)∥E
α
(τ − s)1+ 4 s
0<s<T
[t,θ]
t/3
Z t
1
ds
+ Cν,T
sup s∥Pf⃗(s, .)∥E
α
s 0<s<T
t−2η (t − s) 4
Z θ
1
ds
+ Cν,T
sup s∥Pf⃗(s, .)∥E
α
4
s 0<s<T
(θ
−
s)
t−2η
Z t/3
α
η
η 1− 4
′
′
⃗
≤ Cν,T
∥P
f
∥
ds
+
C
sup s∥Pf⃗(s, .)∥E
α
E
ν,T
t 0<s<T
t1+ 4 0
Z T
1− α
4
′′ η
≤ Cν,T
(
∥Pf⃗∥E ds + sup s∥Pf⃗(s, .)∥E )
t
0<s<T
0
Z
Z
t/3
156
The Navier–Stokes Problem in the 21st Century (2nd edition)
and similarly
∥F⃗ (t, .) − F⃗ (θ, .)∥E ≤ Cν,T
!
1
⃗
∥Pf (s, .)∥E ds dτ
(τ − s)
0
!
Z t−2η
1 ds
dτ sup s∥Pf⃗(s, .)∥E
(τ − s) s
0<s<T
t/3
Z
[t,θ]
Z
+ Cν,T
[t,θ]
Z
t
+ Cν,T
t−2η
Z θ
+ Cν,T
t−2η
η
t
Z
1
′
+ Cν,T
t
Z
≤
′
Cν,T
t/3
Z
′′ η
≤ Cν,T
(
t
ds
s
ds
s
sup s∥Pf⃗(s, .)∥E
0<s<T
sup s∥Pf⃗(s, .)∥E
0<s<T
t/3
′
∥Pf⃗∥E ds + Cν,T
0
t−2η
Z
|
t/3
Z
[t,θ]
η
sup s∥Pf⃗(s, .)∥E
t 0<s<T
dτ
| ds sup s∥Pf⃗(s, .)∥E
τ −s
0<s<T
T
∥Pf⃗∥E ds + sup s∥Pf⃗(s, .)∥E )
0<s<T
0
We now discuss the behavior near t = 0. We have, under the sole assumption that
Pf⃗ ∈ L1 ((0, T ), E 3 ) that limt→0 ∥F⃗ ∥E = 0: for every M > 0 and t ∈ (0, T ), we have
∥F⃗ (t, .)∥E ≤ M t + |{s ∈ (0, T ) / ∥Pf⃗(s, .)∥E > M }|.
Moreover, we have
∥F⃗ (t, .)∥[E,L∞ ]3/4,1 ≤ Cν,T t−3α/8 (∥Pf⃗∥L1t E + sup s∥Pf⃗(s, .)∥E ),
0<s<T
hence
1/3
tα/2 ∥F⃗ (t, .)∥[E,L∞ ]1/2,1 ≤ Cν,T (∥Pf⃗∥L1t E + sup s∥Pf⃗(s, .)∥E )2/3 ∥F⃗ (t, .)∥E
0<s<T
→t→0 0.
The rest of the proof (on existence of mild solutions) is then a direct application of
Theorem 8.1.
Examples: There are many examples of spaces satisfying the hypotheses of Theorem 8.2.
We may quote for instance
ˆ Lebesgue space Lp with p ≥ 3
ˆ uniform local Lebesgue space Lpuloc with p ≥ 3 : f ∈ Lpuloc if supx0 ∈R3 (
< +∞
R
B(x0 ,1)
|f (y)| dy)1/p
ˆ weak Lebesgue space Lp,∗ with p ≥ 3
ˆ more generally, Lorentz spaces Lp,q with p ≥ 3 and 1 ≤ q ≤ +∞ [remark: Lp,p = Lp
and Lp,∞ = Lp,∗ ]
Mild Solutions in Besov or Morrey Spaces
157
ˆ Morrey spaces M p,q with q ≥ 3 and 1 ≤ p ≤ q: if p > 1, this is the space of locally
p-integrable functions such that
Z
1/p
1
∥f ∥M p,q =
sup
|f (x)|p dx
< +∞.
(8.2)
p
3(1−
)
q
0<R≤1,x0 ∈R3 R
|x−x0 |<R
For p = 1, this is the space of locally finite Borel measures dµ (a larger space than
the spaces of locally integrable functions f , i.e., of absolutely continuous measures
f (x)dx) such that
Z
1/p
1
sup
d|µ|(x)
< +∞
1
3(1−
)
q
0<R≤1,x0 ∈R3 R
|x−x0 |<R
ˆ homogeneous Morrey spaces Ṁ p,q with q ≥ 3 and 1 ≤ p ≤ q, defined (with the usual
modification when p = 1) by
Z
1/p
1
∥f ∥Ṁ p,q =
sup
|f (x)|p dx
< +∞.
(8.3)
p
3(1−
)
q
0<R,x0 ∈R3 R
|x−x0 |<R
ˆ multiplier spaces M(H α → L2 ) and V α = M(Ḣ α → L2 ) with 0 ≤ α ≤ 1.
When there is no forcing term, another way to state the results in Theorem 8.2 could
be the following one:
Navier–Stokes equations and local measures: mild solutions
Theorem 8.3.
Let E be a Banach space such that:
• E ⊂ S ′ (R3 )
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
• E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
• E is stable under dilations with a (sub)critical scaling: for λ ≤ 1, ∥f (λx)∥E ≤
Cλ−α ∥f ∥E for two constants C > 0 and 0 ≤ α ≤ 1.
For T > 0, there exists a constant ϵν,T,E > 0 such that, if ⃗u0 ∈ E 3 with div ⃗u0 = 0 and
√
∥⃗u0 ∥E sup
t∥Wνt ∗ ⃗u0 ∥∞ < ϵν,T,E ,
0<t<T
then there exists a mild solution of Equation
⃗u = Wνt ∗ ⃗u0 − B(⃗u, ⃗u)
on (0, T ) × R3 such that ⃗u ∈ L∞ ((0, T ], E 3 ) with sup0<t<T
√
t∥⃗u(t, .)∥∞ < +∞.
Proof. First, we remark that Wνt ∗ ⃗u0 belongs to YT (the space introduced in Theorem 8.2),
with
√
∥Wνt ∗ ⃗u0 ∥YT ≤ C∥⃗u0 ∥E sup
t∥Wνt ∗ ⃗u0 ∥∞
0<t<T
158
The Navier–Stokes Problem in the 21st Century (2nd edition)
1/2
1/2
t
1
(since ∥v∥[E,L∞ ]1/2,1 ≤ ∥v∥E ∥v∥∞ ). We know that, if ∥Wνt ∗ ⃗u0 ∥YT is small enough, the
⃗ 0 = Wνt ∗ ⃗u0 and U
⃗ n+1 = U
⃗ 0 − B(U
⃗ n, U
⃗ n ) converge to a solution ⃗u in YT .
Picard iterates U
⃗n − U
⃗ n−1 ∥Y (with U
⃗ −1 = 0), we have P+∞ ϵn < +∞.
In particular, if ϵn = ∥U
T
n=0
√
⃗ n (t, .)∥∞ , we have β0 ≤ CT ∥⃗u0 ∥E and
If βn = sup0<t<T t∥U
⃗ n+1 − U
⃗ n ∥∞ ≤Cν,T
∥U
Z
1
0
α
(t − s) 2 + 4
⃗ n+1 − U
⃗ n ∥[E,L∞ ] (∥U
⃗ n ∥∞ + ∥U
⃗ n−1 ∥∞ ) ds
∥U
1 ,1
2
t
Z
≤Cν,T ϵn (βn + βn−1 )
(t − s)
0
′
≤Cν,T
Thus, if BN =
PN
n=0
1
ds
1
α
2+ 4
s
1
α
2+ 4
1
√ ϵn (βn + βn−1 ).
t
βn , we have BN +1 ≤ BN (1 + CϵN ) ≤ B0
sup
√
⃗ n+1 − U
⃗ n ∥ ∞ ≤ ϵN B 0
t∥U
0<t<T
Y
Q
n≥0 (1
+ Cϵn ) and
(1 + Cϵn ).
n≥0
⃗ n ∥E , we have
If γn = sup0<t<T ∥U
γ0 ≤ CT ∥⃗u0 ∥E
and
⃗ n+1 − U
⃗ n ∥E ≤Cν,T
∥U
Z
t
1
1
0
(t − s) 2
⃗ n+1 − U
⃗ n ∥∞ (∥U
⃗ n ∥E + ∥U
⃗ n−1 ∥E ) ds
∥U
Z
≤Cν,T βn (γn + γn−1 )
0
t
1
ds
1
2
1
(t − s) s 2
′
≤Cν,T
βn (γn + γn−1 ).
Thus, if CN =
PN
n=0
γn , we have CN +1 ≤ CN (1 + CβN ) ≤ B0
⃗ n+1 − U
⃗ n ∥ E ≤ βN B 0
sup ∥U
0<t<T
8.2
Y
Q
n≥0 (1
+ Cβn ) and
(1 + Cβn ).
n≥0
Morrey Spaces and Maximal Functions
An alternative proof for existence of mild solutions can be given as a generalization of
Theorem 7.6. More precisely, we will say that a Banach space of distributions E satisfy
hypothesis (Hα ) for some α ∈ (0, 1] when
ˆ E ⊂ S ′ (R3 )
ˆ E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
ˆ E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
ˆ The Hardy–Littlewood maximal function operator is bounded on E.
ˆ The operator (u, v) 7→
R Iα (uv) is continuous from E × E to E, where Iα is the Riesz
potential Iα (g) = cα R3 |x−y|1 3−α g(y) dy.
Mild Solutions in Besov or Morrey Spaces
159
−α
The last point implies that E ⊂ V α = M(Ḣ α → L2 ) ⊂ Ḃ∞,∞
. In particular, when f⃗ ∈ E 3 ,
Pf⃗ is well defined in (V α )3 . Another important point is that we have supt>0 tα/2 ∥Wt ∗u∥∞ ≤
C∥u∥E .
We shall be interested in functions f (t, x) such that f ∈ L1t E or f ∈ L∞
t E. We do
not need vector integrals or measurability, as we are dealing with Lebesgue measurable
functions; instead of using vector integration, we shall use the Fubini theorem combined
with norm estimates and duality to give a meaning to integrals in E 1 . More precisely,
1
3
as E ⊂ L1loc , the notation f ∈ L1t E or f ∈ L∞
t E will mean that f ∈ Lloc ((0, T ) × R so
that for almost every t ∈ (0, T ) the locally integrable f (t, x) is well defined and belongs
RT
to E, with 0 ∥f (t, .)∥E < +∞ or ess sup0<t<T ∥f (t, .)∥E < +∞. We shall as well assume
that E is a dual space and that S is dense in the pre-dual of E. In that case, we have
RT
RT
RT
f ∈ L1t E ⇒ 0 f (s, .) ds ∈ E and ∥ 0 f (s, .) ds∥E ≤ 0 ∥f (s, .)∥E ds.
Navier–Stokes equations, Morrey spaces and maximal functions
Theorem 8.4.
Let E be a Banach space that satisfies hypothesis (Hα ) for some α ∈ (0, 1]. We define
the following Banach space YT :
YT = {⃗u ∈ L∞ ((0, T ), E 3 ) / ess sup |⃗u(t, x)| ∈ E}.
0<t<T
We then have the following results:
• The bilinear operator B is continuous on YT .
• For ⃗u0 ∈ E 3 , we have Wνt ∗ ⃗u0 ∈ YT with
sup |Wνt ∗ ⃗u0 (x)| ≤ M⃗u0 (x)
0<t<T
(where M⃗u0 is the Hardy–Littlewood maximal function of ⃗u0 ).
Rt
• If Pf⃗ ∈ L1 ((0, T ), E 3 ), then 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ YT .
• Let ⃗u0 ∈ E 3 and Pf⃗ ∈ L1 ((0, T ), E 3 ). There exists a positive constant ϵ0
1−α
1+α
(independent of ν, T , ⃗u0 and f⃗), such that, if ∥⃗u0 ∥E < ϵ0 ν 2 T − 2 and
RT
1−α
1+α
∥Pf⃗(s, .)∥E ds < ϵ0 ν 2 T − 2 , then there exists a mild solution ⃗u of Equa0
tion (7.4) on (0, T ) × R3 such that ⃗u ∈ YT .
Proof. Let ⃗u and ⃗v in YT : |⃗u(t, x)| ≤ U (x) ∈ E, |⃗v (t, x)| ≤ V (x) ∈ E. We have
|B(⃗u, ⃗v )| ≤ C
Z Z
(
0
t
1
(νT )
ds) U (y)V (y) dy ≤ Cα
2
2
4
ν |t − s| + |x − y|
ν
1−α
2
Iα (U V )(x)
Thus, the bilinear operator B is continuous on YT .
The inequality sup0<t<T |Wνt ∗ ⃗u0 (x)| ≤ M⃗u0 (x)) is given by Lemma 7.4. Thus, for
⃗u0 ∈ E 3 , Wνt ∗ ⃗u0 ∈ YT .
1 See
the discussion in [313].
160
The Navier–Stokes Problem in the 21st Century (2nd edition)
Rt
Now, we consider F⃗ = 0 Wν(t−s) ∗ Pf⃗(s, .) ds with Pf⃗ ∈ L1 ((0, T ), E 3 ). We have
|F⃗ (t, x)| ≤
Z
T
MPf⃗(s, .)(x) ds
0
hence
∥ sup |F⃗ (t, x)|∥E ≤
T
Z
0<t<T
∥MPf⃗(s, .)(x)∥E ds ≤ C
0
Z
T
∥Pf⃗(s, .)∥E ds.
0
The end of the proof is now easy.
Examples: There are many examples of spaces E satisfying hypothesis (Hα ) for some
α ∈ (0, 1]. We may quote for instance
ˆ Lebesgue space Lp with 3 ≤ p < ∞
ˆ weak Lebesgue space Lp,∗ with p ≥ 3
ˆ more generally, Lorentz spaces Lp,q with p ≥ 3 and 1 ≤ q ≤ +∞
ˆ homogeneous Morrey spaces Ṁ p,q with 3 ≤ q < ∞ and 2 < p ≤ q
ˆ multiplier spaces V α = M(Ḣ α → L2 ) with 0 < α ≤ 1
In all those examples, the Leray projection operator P is bounded on E 3 .
The role played by the maximal function may be underlined in another way. We know
that the Navier–Stokes equations can be written as
Z t
⃗u = Wνt ∗ ⃗u0 − B(⃗u, ⃗u) −
Wν(t−s) Pf⃗ ds
0
where
Z tX
3
B(⃗u, ⃗v ) =
∂j O(ν(t − s)) :: uj ⃗v ds.
0 j=1
As we have
1
|∂j O(ν(t − s))(x − y)| ≤ C √
,
( t − s + |x − y|)4
we write
∂j O(ν(t − s)) :: uj ⃗v = Wν(t−s)/2 ∗ ∂j O(ν(t − s)/2) :: uj ⃗v
and get
Z
|B(⃗u, ⃗v )| ≤ C
t
√
0
1
Wν(t−s)/2 ∗ M|⃗u(s,.)||⃗v(s,.)| ds.
t−s
This gives us the following version of Theorem 8.3:
Theorem 8.5.
Let E be a Banach space such that:
• E ⊂ S ′ (R3 )
• E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
Mild Solutions in Besov or Morrey Spaces
161
• The Hardy–Littlewood maximal function operator is bounded on E.
−1
• E is embedded in B∞,∞
For T > 0, there exists a constant ϵν,T,E > 0 such that, if ⃗u0 ∈ E 3 with div ⃗u0 = 0 and
√
∥⃗u0 ∥E sup
t∥Wνt ∗ ⃗u0 ∥∞ < ϵν,T,E ,
0<t<T
then there exists a mild solution of Equation
⃗u = Wνt ∗ ⃗u0 − B(⃗u, ⃗u)
on (0, T ) × R3 such that ⃗u ∈ L∞ ((0, T ], E 3 ) with sup0<t<T
√
t∥⃗u(t, .)∥∞ < +∞.
Proof. Once again, the proof is given by a Picard iteration in the space YT defined as the
space of Lebesgue measurable vector fields on (0, T ) × R3 such that t 7→ ⃗u(t, .) is continuous
from (0, T ) to ([E, L∞ ]1/2,1 )3 and
∥⃗u∥YT = sup t1/4 ∥⃗u(t, .)∥[E,L∞ ]1/2,1 < +∞.
0<t<T
Pointwise multiplications maps [E, L∞ ]1/2,1 × [E, L∞ ]1/2,1 to E, so that
Wν(t−s)/2 ∗ M|⃗u(s,.)||⃗v(s,.)|
E
≤ C∥MM|⃗u||⃗v| ∥E ≤C ′ ∥|⃗u||⃗v |∥E
≤C ′′ ∥⃗u∥[E,L∞ ]1/2,1 ∥⃗v ∥[E,L∞ ]1/2,1
and
Wν(t−s)/2 ∗ M|⃗u(s,.)||⃗v(s,.)|
∞
≤C max(1, p
1
)∥M|⃗u||⃗v| ∥E
ν(t − s)
1
≤C ′ max(1, p
)∥⃗u∥[E,L∞ ]1/2,1 ∥⃗v ∥[E,L∞ ]1/2,1 ,
ν(t − s)
so that
Wν(t−s)/2 ∗ M|⃗u(s,.)||⃗v(s,.)|
≤ C ′ max(1,
[E,L∞ ]1/2,1
1
)∥⃗u∥[E,L∞ ]1/2,1 ∥⃗v ∥[E,L∞ ]1/2,1 .
(ν(t − s))1/4
Now, the solution ⃗u we get in YT satisfies
Z t
1
1
√
√ (s1/4 ∥⃗u(s, .)∥[E,L∞ ]1/2,1 )2 ds,
∥B(⃗u, ⃗u)∥E ≤ Cν
t−s s
0
∥B(⃗u, ⃗u)∥[E,L∞ ]3/4,1
Z t
1
1
1
√
≤ Cν
max(1,
) √ (s1/4 ∥⃗u(s, .)∥[E,L∞ ]1/2,1 )2 ds,
3/8
s
(t
−
s)
t
−
s
0
so that ⃗u ∈ L∞ ((0, T, E)) and t3/8 ⃗u ∈ L∞ ((0, T ), [E, L∞ ]3/4,1 ). Moreover, pointwise multiplication maps [E, L∞ ]1/2,1 ×[E, L∞ ]1/2,1 to E, and maps [E, L∞ ]1/2,1 ×L∞ to [E, L∞ ]1/2,1 ,
162
The Navier–Stokes Problem in the 21st Century (2nd edition)
−3/4
thus maps [E, L∞ ]1/2,1 × [E, L∞ ]3/4,1 to [E, L∞ ]1/4,1 ⊂ B∞,∞ ; thus, we find
∥B(⃗u, ⃗v )∥∞
Z t
1
1
C
√ ν max(1,
) 5/8 (s1/4 ∥⃗u∥[E,L∞ ]1/2,1 )(s3/8 ∥⃗u∥[E,L∞ ]3/4,1 ) ds
≤
3/8
(t − s)
s
t−s
0
and
√
t⃗u ∈ L∞ ((0, T ), L∞ ).
Example: An easy example of a space E that satisfies the assumptions of Theorem 8.5 is
the variable exponent Lebesgue space Lp() under some conditions on p(). Recall that, if p
is a continuous function on R3 such that 1 ≤ p− ≤ p(x) ≤ p+ < +∞, f ∈ Lp() means that
f is measurable and
Z
|f (x)|p(x) dx < +∞.
R3
The space Lp() is normed by
Z
∥f ∥Lp() = inf{λ > 0 /
|
R3
f (x) p(x)
|
dx ≤ 1}.
λ
Let us remark that, if p is not constant, then Lp() is stable neither under convolution with
L1 nor under dilations. Thus, two assumptions in Theorem 8.3 are not fulfilled. Let us
consider whether Lp() satisfies the assumptions of Theorem 8.5. Stability under bounded
pointwise multiplication is obvious. As we have the embedding Lp() ⊂ Lp− + Lp+ , we find
−1
if p− ≥ 3. Finally, Cruz-Uribe and Fiorenza [133] have shown that
that Lp() ⊂ B∞,∞
the Hardy–Littelwood maximal function defines a bounded operator on Lp() if p− > 1,
1
|p(x) − p(y)|≤ C ln( 11 ) when |x − y| < 1/2 and, for some constant p∞ , | p(x)
− p1∞ | ≤
|x−y|
1
.
C ln(e+|x|)
8.3
Uniqueness of Morrey Solutions
Analogous to the case of uniqueness for mild solutions in Lebesgue spaces, we shall
study uniqueness for mild solutions in C([0, T ), E 3 ), where E is a space of local measures.
We consider a space E that satisfies the hypotheses of Theorem 8.2:
ˆ E ⊂ S ′ (R3 )
ˆ E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
ˆ E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
ˆ E is stable under dilations with a (sub)critical scaling: for λ ≤ 1, ∥f (λx)∥E ≤
Cλ−α ∥f ∥E for two constants C > 0 and 0 ≤ α ≤ 1.
To give meaning to the Equation (7.4) with the sole assumption that ⃗u ∈ C([0, T ), E 3 ), we
need to define the term ⃗u ⊗ ⃗u, hence to assume that E ⊂ L2loc . In order to adapt the proof
of Theorem 7.7, we need that Wνt ∗ ⃗u0 is small enough when t is close to 0; this will be
granted if S is dense in E.
Mild Solutions in Besov or Morrey Spaces
163
3
From those assumptions, we get that E ⊂ M̃ 2, α , the closure of S in the Morrey space
M :
!1/2
Z
1
∥f ∥ 2, α3 =
.
sup
|f (y)|2 dy
M
R3−2α |y−x0 |<R
x0 ∈R3 ,0<R≤1
3
2, α
In the case α = 1, however, uniqueness in C([0, T ), (M̃ 2,3 )3 ) is still an open problem. Uniqueness has been proved for some subspaces close to M̃ 2,3 : M̃ p,3 with 2 < p ≤ 3 by Furioli,
Lemarié-Rieusset and Terraneo [187], M̃(H 1 → L2 ) (the closure of S in M(H 1 → L2 )) by
2,1
2,1
May [325] and M̃ L ,3 (the closure of S in M L ,3 ) by Lemarié-Rieusset [316], where L2,1
is a Lorentz space and
∥f ∥M L2,1 ,3 =
sup
x0 ∈R3 ,0<R≤1
1
∥1|y−x0 |<R f (y)∥L2,1
R
1/2
.
Uniqueness of Morrey solutions
Theorem 8.6.
2,1
3
Let E = M̃ 2, α with 0 < α < 1, M̃ p,3 with 2 < p ≤ 3, M̃(H 1 → L2 ) or M̃ L ,3 . Let
R
t
⃗u0 ∈ E 3 and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C([0, T ), E 3 ). If ⃗u and ⃗v are two solutions of the
Equation (7.4) on (0, T ) × R3 that belong to C([0, T ), E 3 ), then ⃗u = ⃗v .
Proof. As for the proof of Theorem 7.7, we reduce the problem to local uniqueness and we
look for a contractive estimate for the fixed-point problem
w
⃗ = B(⃗u1 , w)
⃗ + B(w,
⃗ ⃗v1 ) − B(Wνt ∗ ⃗u0 , w)
⃗ − B(w,
⃗ Wνt ∗ ⃗u0 ).
(8.4)
with w
⃗ = ⃗u − ⃗v , ⃗u1 = Wνt ∗ ⃗u0 − ⃗u and ⃗v1 = Wνt ∗ u0 − ⃗v .
⃗ belong to C([0, ϵ], (M̃ 2, α3 )3 ) (with 0 < ϵ < 1), then
The case α < 1 is easy: if F⃗ and G
3
⃗ .) ∈ (M̃ 1, 2α )9 and we have, for 0 < s < t < ϵ,
F⃗ (s, .) ⊗ G(s,
⃗ .)∥
∥Wν(t−s) ∗ P div(F⃗ (s, .) ⊗ G(s,
3
M 1, 2α
≤ Cν √
1
⃗ (G, .)∥ 2, 3
3 ∥F
∥F⃗ (s, .)∥ 2, 2α
M
M 2α
t−s
and
⃗ .)∥∞ ≤ Cν
∥Wν(t−s) ∗ P div(F⃗ (s, .) ⊗ G(s,
1
1
(t − s) 2 +α
∥F⃗ (s, .)∥
3
M 2, 2α
∥F⃗ (G, .)∥
3
M 2, 2α
so that
sup ∥w∥
⃗
0<t<ϵ
3
M 2, 2α
≤ Cϵ
1−α
2
sup ∥w∥
⃗
0<t<ϵ
3
( sup ∥⃗u∥
M 2, 2α 0<t<ϵ
3
M 2, 2α
+ sup ∥⃗v ∥
0<t<ϵ
3
M 2, 2α
).
(Remark: in the case α < 1, this proof (and the result) is still valid for ⃗u and ⃗v *-weakly
3
continuous from [0, T ) with values in the (non-separable) space (M 2, α )3 ).
2,1
The case α = 1 is more difficult. As M p,3 ⊂ M L ,3 for p > 2 we consider only the
2,1
cases E = M L ,3 and E = M(H 1 → L2 ). The proof for M p,3 given by Furioli, LemariéRieusset and Terraneo [187] used Morrey-Besov spaces, following their use of Besov spaces
2,1
for the proof of uniqueness in the case E = L3 . For the case M L ,3 , we shall adapt the
proof by Lemarié-Rieusset [316], which is a variation of the proof by Meyer [359] who used
the Lorentz space L3,∞ to prove uniqueness in the case E = L3 . Similarly, for the case
164
The Navier–Stokes Problem in the 21st Century (2nd edition)
M(H 1 → L2 ), we shall follow the proof by May [325], which is a variation of the proof by
Monniaux [368] who used the maximal regularity of the heat kernel to prove uniqueness in
the case E = L3 .
Case E = M L
2,1
2,∞
: We introduce the space M L ,3 , where
1/2
1
.
∥1|y−x0 |<R f (y)∥L2,∞
∥f ∥M L2,∞ ,3 =
sup
x0 ∈R3 ,0<R≤1 R
We have M L
2,1
,3
,3
2,∞
⊂ ML
,3
. We are going to estimate
1/2
1
sup ∥w(t,
⃗ .)∥M L2,∞ ,3 = sup
sup
.
∥1|y−x0 |<R w(t,
⃗ y)∥L2,∞
0<t<ϵ
0<t<ϵ x0 ∈R3 ,0<R≤1 R
We have
∥B(Wνt ∗ ⃗u0 , w)∥
⃗ M L2,∞ ,3 ≤ C sup
√
s∥Wνs ∗ ⃗u0 ∥∞ sup ∥w(s,
⃗ .)∥M L2,∞ ,3
∥B(w,
⃗ Wνt ∗ ⃗u0 )∥M L2,∞ ,3 ≤ C sup
√
s∥Wνs ∗ ⃗u0 ∥∞ sup ∥w(s,
⃗ .)∥M L2,∞ ,3 .
0<s<t
0<s<t
and
0<s<t
0<s<t
3
In order to estimate B(⃗u1 , w),
⃗ we write ⃗u1 (s, .) ⊗ w(s,
⃗ .) ∈ (M 1, 2 )9 . For 0 < t < ϵ < 1, we
have
∥Wν(t−s) ∗ P div ⃗u1 (s, .) ⊗ w(s,
⃗ .)∥
3
M 1, 2
1
∥⃗u1 (s, .)∥M L2,1 ,3 ∥w(s,
⃗ .)∥M L2,∞ ,3
≤ Cν √
t−s
and
∥Wν(t−s) ∗ P div ⃗u1 (s, .) ⊗ w(s,
⃗ .)∥∞ ≤ Cν
1
∥⃗u1 (s, .)∥M L2,1 ,3 ∥w(s,
⃗ .)∥M L2,∞ ,3
(t − s)3/2
⃗α+Z
⃗α
On the ball B(x0 , R) with 0 < R ≤ 1, for −∞ < A < t, we write B(⃗u1 , w)(y)
⃗
=W
with
Z max(0,A)
Wα (t, y) =
Wν(t−s) ∗ P div ⃗u1 (s, .) ⊗ w(s,
⃗ .) ds.
0
We have
max(0,A)
Z
∥Wα ∥∞ ≤ Cν
0
1
∥⃗u1 (s, .)∥M L2,1 ,3 ∥w(s,
⃗ .)∥M L2,∞ ,3 ds
(t − s)3/2
hence
1
sup ∥⃗u1 (s, .)∥M L2,1 ,3 sup ∥w(s,
⃗ .)∥M L2,∞ ,3 ds
∥1B(x0 ,R) Wα ∥∞ ≤ Cν′ √
t − A 0<s<t
0<s<t
Similarly, we have
Z
∥Zα ∥
3
M 1, 2
t
≤ Cν
max(0,A)
1
∥⃗u1 (s, .)∥M L2,1 ,3 ∥w(s,
⃗ .)∥M L2,∞ ,3 ds
(t − s)1/2
hence
√
∥1B(x0 ,R) Zα ∥1 ≤ Cν′ t − A sup ∥⃗u1 (s, .)∥M L2,1 ,3 sup ∥w(s,
⃗ .)∥M L2,∞ ,3 ds R
0<s<t
0<s<t
Mild Solutions in Besov or Morrey Spaces
165
From this, we find that
⃗ .)∥M L2,∞ ,3 ds
∥1B(x0 ,R) B(⃗u1 , w)∥
⃗ L2,∞ ≤ Cν sup ∥⃗u1 (s, .)∥M L2,1 ,3 sup ∥w(s,
√
R
0<s<t
0<s<t
hence we get the control of B(⃗u1 , w)(t,
⃗
.)∥M L2,∞ ,3 . Finally, we get
sup ∥w(t,
⃗ .)∥M L2,∞ ,3 ≤ Cν A(ϵ)∥w(t,
⃗ .)∥M L2,∞ ,3
0<t<ϵ
with
A(ϵ) = sup ∥⃗u1 (s, .)∥M L2,1 ,3 + sup ∥⃗v1 (s, .)∥M L2,1 ,3 + sup
0<s<ϵ
0<s<ϵ
√
s∥Wνs ∗ ⃗u0 ∥∞
0<s<ϵ
and
lim A(ϵ) = 0.
ϵ→0
Case E = M(H 1 → L2 ): We are going to estimate
Z
ϵ
!1/p
p/2
|w(t,
⃗ .)|2 dx)
dt
Z
Iϵ (w)
⃗ = sup sup
0<t<ϵ x0 ∈R3
0
B(x0 ,1)
for 2 < p < +∞ and 0 < ϵ < 1.
⃗ = 1B(x ,4) w
⃗ =w
⃗ . We write
Fix x0 ∈ R3 . Let W
⃗ and Z
⃗ −W
0
⃗ ) + B(W
⃗ , ⃗v1 ) − B(Wνt ∗ ⃗u0 , W
⃗ ) − B(W
⃗ , Wνt ∗ ⃗u0 ) − B(⃗u, Z)
⃗ − B(Z,
⃗ ⃗v ).
w
⃗ = B(⃗u1 , W
⃗ is easy:
We want to estimate ∥1B(x0 ,1) w∥
⃗ Lpt L2x ((0,ϵ)×R3 ) . Estimating the terms involving Z
we have, for |x − x0 | < 1,
Z tZ
1
⃗ y)||⃗u(s, y)| ds dy
⃗
|Z(s,
|B(⃗u, Z)(t,
x)| ≤ C
4
0
|x0 −y|>4 |x − y|
X Z tZ
1
⃗ y)||⃗u(s, y)| ds dy
≤ C′
|Z(s,
1
+
|k|4
3
0
x
+k+[0,1]
0
3
k∈Z
X
p−1
1
′′
Iϵ (w)t
⃗ p
≤ Cp sup ∥⃗u(s, .)∥M 2,3
4
1 + |k|
0<s<t
3
k∈Z
and we find finally
⃗ Lp L2 ((0,ϵ)×R3 ) ≤ Cp ϵIϵ (w)
∥1B(x0 ,1) B(⃗u, Z)∥
⃗ sup ∥⃗u(s, .)∥M(H 1 →L2 ) .
t x
0<s<ϵ
⃗ ⃗v ).
A similar estimate holds for B(Z,
For the terms involving Wνt ∗ ⃗u0 , we just write that the operator h 7→ H with H(t) =
Rt 1 1
√
√ H(s), ds is bounded on Lp ((0, +∞), dt) for p > 2 (as it can be checked with
0
t−s s
the estimates for convolution and products in Lorentz spaces [313]: 1t>0 √1t ∈ L2,∞ , the
2p
2p
pointwise product maps Lp × L2,∞ to L p+2 ,p and the convolution maps L p+2 ,p × L2,∞ to
Lp ). Writing
⃗ )(t, .)∥2
∥B(Wνs ∗ ⃗u0 , W
Z t
√
1
1 ⃗
p
√ ∥W
≤C
(s, .)∥2 ds sup s∥Wνs ∗ ⃗u0 ∥L∞ (dx)
s
0<s<t
ν(t − s)
0
166
The Navier–Stokes Problem in the 21st Century (2nd edition)
we get
√
⃗ )∥Lp L2 ((0,ϵ)×R3 ) ≤ Cp √1 Iϵ (w)
⃗ sup s∥Wνs ∗ ⃗u0 ∥L∞ (dx) .
∥1B(x0 ,1) B(Wνs ∗ ⃗u0 , W
t x
ν
0<s<ϵ
⃗ , Wνs ∗ ⃗u0 ).
A similar estimate holds for B(W
For the terms involving ⃗u1 and ⃗v1 , we use the maximal Lp L2 regularity for the heat
kernel [313]: for 1 < p < +∞
Z t
Wν(t−s) ∗ ∆f (s, .) ds∥Lp ((0,+∞),L2 (R3 ) ≤ Cν,p ∥f ∥Lp L2 .
∥
0
On the other hand, we have
Z t
Wν(t−s) ∗ f (s, .) ds∥Lp ((0,ϵ),L2 (R3 ) ≤ ϵ∥f ∥Lp L2 .
∥
0
Thus, for ϵ < 1, we have
Z t
Wν(t−s) ∗ f (s, .) ds∥Lp ((0,ϵ),L2 (R3 ) ≤ Cν,p ∥f ∥Lp H −2 .
∥
0
Moreover, M(H → L2 ) = M(L2 → H −1 ) (by duality), so that
1
⃗ )∥H −2 ≤ C∥⃗u1 (s, .)∥M(H 1 →L2 ) ∥W
⃗ (s, .)∥2 .
∥P div(⃗u1 ⊗ W
We thus get
⃗ )∥Lp L2 ((0,ϵ)×R3 ) ≤ Cν,p Iϵ (w)
∥1B(x0 ,1) B(⃗u1 , W
⃗ sup ∥⃗u1 (s, .)∥M(H 1 →L2 ) .
t x
0<s<ϵ
⃗ , ⃗v1 ).
A similar estimate holds for B(W
Finally, we get
Z
ϵ
!1/p
p/2
|w(t,
⃗ .)| dx)
dt
≤ Cν,p A(ϵ)Iϵ (w)
⃗
Z
2
Iϵ (w)
⃗ = sup sup
0<t<ϵ x0 ∈R3
0
B(x0 ,1)
with
A(ϵ) = sup ∥⃗u1 (s, .)∥M(H 1 →L2 ) + sup ∥⃗v1 (s, .)∥M(H 1 →L2 )
0<s<ϵ
0<s<ϵ
√
+ sup s∥Wνs ∗ ⃗u0 ∥∞ + ϵ sup ∥⃗u(s, .)∥M(H 1 →L2 )
0<s<ϵ
0<s<ϵ
and
lim A(ϵ) = 0.
ϵ→0
The theorem is proved.
8.4
Besov Spaces
Besov spaces play an important role in the analysis of the Navier–Stokes equations, as
the regularization properties of the heat kernel may often be expressed in terms of Besov
norms. We have seen in the previous sections estimates of the type:
sup tα/2 ∥Wνt ∗ u0 ∥∞ ≤ C∥u0 ∥E
0<t<1
−α
for some α > 0. Such an estimate is equivalent to the continuous embedding E ⊂ B∞,∞
.
Mild Solutions in Besov or Morrey Spaces
167
Similarly, in Theorem 7.4, global existence of a solution ⃗u ∈ Cb ([0, +∞), (H 1/2 )3 ) ∩
L2 ((0, T0 ), (H 3/2 )3 ) was granted under the hypothesis that ⃗u0 ∈ (H 1/2 (R3 ))3 , f⃗ ∈
Rt
L2 ((0, +∞), (Ḣ −1/2 (R3 )3 ) with ∥Wνt ∗ ⃗u0 ∥L4 Ḣ 1 and ∥ 0 Wν(t−s) ∗ Pf⃗(s, .) ds∥L4 Ḣ 1 small
enough. The assumption on ⃗u0 can, again, be expressed in terms of Besov spaces, as we
have the equivalence
∥Wνt ∗ u0 ∥L4 ((0,+∞),Ḣ 1 ) ≈ ∥u0 ∥Ḃ −1/2 ≈ ∥u0 ∥Ḃ 1/2
Ḣ 1 ,4
1/2
(8.5)
2,4
1/2
(together with the embedding H 1/2 ⊂ Ḣ 1/2 = Ḃ2,2 ⊂ Ḃ2,4 ).
Similarly, in Theorem 7.5, global existence of a solution ⃗u ∈ Cb ([0, +∞), (L3 )3 ) was
proved under the assumptions that ⃗u0 and f⃗ were small enough, the assumptions on ⃗u0
being that ⃗u0 ∈ (L3 )3 and sup0<t tβ ∥Wνt ∗ ⃗u0 ∥q small enough for some q ∈ (3, +∞) and
3
. This is again an assumption on a Besov norm, as underlined by Cannone [81]:
β = 21 − 2q
sup tβ ∥Wνt ∗ ⃗u0 ∥q ≈ ∥⃗u0 ∥Ḃq,∞
−2β .
(8.6)
0<t
In the same way, the control norm on ⃗u0 we used in Theorem 8.1 was ∥Wνt ∗ ⃗u0 ∥YT =
.
sup0<t<T tα/4 ∥⃗u(t, .)∥[E,L∞ ]1/2,1 , which is equivalent to the norm ∥⃗u0 ∥B −α/2
[E,L∞ ]1/2,1 ,∞
All those examples suggest to investigate the Navier–Stokes equations with an initial
−α
value ⃗u0 in a Besov space BE,q
with α > 0 and with solutions in the space Yα,E,q =
dt
α/2
q
−1+α
{⃗u / t ∥⃗u(t, .)∥E ∈ L ((0, T ), t )}. For scaling arguments, we restrict to E ⊂ B∞,∞
. If
∞
B is not bounded on Yα,E , we may consider as well the space Yα,E,q ∩ Y1,L ,∞ and use the
following lemma:
Lemma 8.2.
Let 0 ≤ α < 1. The operator f 7→ F defined by
t
Z
F (t) =
0
1
√
√
t−s s
α/2
t
f (s) ds
s
is bounded on Lp ((0, +∞), dt
t ) for 1 ≤ p ≤ +∞.
Proof. The case p = +∞ is obvious, since
Z
0
t
1
√
√
t−s s
α/2
α/2
Z 1
t
1
1
√
ds =
ds < +∞.
√
s
1−s s s
0
For p = 1, we write
Z
+∞
0
dt
|F (t)| ≤
t
Z
+∞
s
0
and
Z
s
s
+∞
Z
1
√
√
t−s s
s
+∞
1
√
√
t−s s
α/2 !
t
dt
ds
|f (s)|
s
t
s
α/2 ! Z +∞
t
dt
1
dt
√
=
tα/2 < +∞.
s
t
t
t
−
1
1
By interpolation, we get the result for all values of p ∈ [1, +∞].
168
The Navier–Stokes Problem in the 21st Century (2nd edition)
The Navier–Stokes bilinear operator and Besov spaces
Theorem 8.7. Let E ⊂ S ′ (R3 ) be a Banach space such that:
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
• E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
−1+α
• E ⊂ B∞,∞
for some α ∈ (0, 1).
The bilinear operator B defined as
Z t
⃗
⃗
⃗ ds
B(F , G) =
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is continuous on the space YT of Lebesgue measurable vector fields on (0, T ) × R3 such
that
∥⃗u∥YT = ∥tα/2 ∥⃗u(t, .)∥E ∥Lq ((0,T ), dt ) + ∥t1/2 ∥⃗u(t, .)∥∞ ∥L∞ ((0,T ), dt ) < +∞
t
t
where 1 ≤ q ≤ +∞.
−1+α
Moreover, if E ⊂ Ḃ∞,∞
, the same result holds with T = +∞.
Proof. The proof is based on the inequality, for 0 < t < T ,
∥Wνt ∗ f ∥∞ ≤ Cν,T t(α−1)/2 ∥f ∥E
−1+α
(which is valid as well for T = +∞ when E ⊂ Ḃ∞,∞
).
⃗ = B(F⃗ , G).
⃗ We have
Let H
⃗ .))∥E ≤ C p 1
⃗ .)∥∞
∥Wν(t−s) ∗ P div(F⃗ (s, .) ⊗ G(s,
∥F⃗ (s, .)∥E ∥G(s,
ν(t − s)
so that, by Lemma 8.2, we have
⃗ .)∥E ∥ q
∥tα/2 ∥H(t,
L ((0,T ), dt )
t
≤ Cν ∥t
α/2
⃗ .)∥∞ ∥ ∞
∥F⃗ (t, .)∥E ∥Lq ((0,T ), dt ) ∥t1/2 ∥G(t,
L ((0,T ), dt ) .
t
t
Moreover, we have
⃗ .))∥∞ ≤ Cν,T
∥Wν(t−s) ∗ P div(F⃗ (s, .) ⊗ G(s,
1
⃗ .)∥∞
∥F⃗ (s, .)∥∞ ∥G(s,
(t − s)1/2
⃗ .))∥∞ ≤ Cν,T
∥Wν(t−s) ∗ P div(F⃗ (s, .) ⊗ G(s,
1
⃗
⃗
α ∥F (s, .)∥E ∥G(s, .)∥∞
(t − s)1− 2
and
Mild Solutions in Besov or Morrey Spaces
169
so that
⃗ .)∥∞ ≤ Cν,T ∥F⃗ ∥Y ∥G∥
⃗ Y ×
∥H(t,
T
T

Z t/2 "

×
0
1−α
2
s
α
(t − s)1− 2
q
# q−1
1− q1
ds 
s

Z
t
+
t/2
√
1 ds 

t−s s
′
⃗ Y √1 .
≤ Cν,T
∥F⃗ ∥YT ∥G∥
T
t
The theorem is proved.
We shall be interested in a smaller class of Besov spaces for which we do not need the
estimates on the size of ⃗u in L∞ :
The Navier–Stokes bilinear operator and Besov spaces II
Theorem 8.8.
Let E, F ⊂ S ′ (R3 ) be two Banach spaces such that:
• E ⊂ L2loc and F ⊂ L1loc
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
• pointwise product is bounded from E × E to F
• For every finite positive T , we have sup0<t<T t
some α ∈ (0, 1).
1−α
2
∥Wνt ∗ f ∥E ≤ CT ∥f ∥F for
The bilinear operator B defined as
Z t
⃗ =
⃗ ds
B(F⃗ , G)
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is continuous on the space YT of Lebesgue measurable vector fields on (0, T ) × R3 such
that
∥⃗u∥YT = ∥tα/2 ∥⃗u(t, .)∥E ∥Lq ((0,T ), dt ) < +∞
t
where
2
α
≤ q ≤ +∞.
⃗ = B(F⃗ , G).
⃗ Writing
Proof. Let H
⃗ = Wν(t−s)/2 ∗ P div(Wν(t−s)/2 ∗ [F⃗ ⊗ G]),
⃗
Wν(t−s) ∗ P div(F⃗ ⊗ G)
we get
⃗ .))∥E ≤ Cν,T
∥Wν(t−s) ∗ P div(F⃗ (s, .) ⊗ G(s,
1
⃗
⃗
α ∥F (s, .)∥E ∥G(s, .)∥E
(t − s)1− 2
Hence, we need to estimate the Lq ( dt
t ) norm of
H(t) = tα/2
Z
0
t
1
−α
F (s) ds
α s
(t − s)1− 2
170
The Navier–Stokes Problem in the 21st Century (2nd edition)
with F ∈ Lq/2 ( dt
t ). When q = +∞, we just write:
t
α/2
Z
0
When q =
2
α,
t
1
−α
ds =
α s
(t − s)1− 2
1
Z
0
1
−α
ds < +∞.
α s
(1 − s)1− 2
we need to estimate the Lq (dt) norm of
Z
K(t) =
t
1
1
0
(t − s)1− q
G(s) ds
with G ∈ Lq/2 (dt) and q > 2. This is easy with the Hardy–Littlewood–Sobolev inequality:
Z t
1
∥
f (s) ds∥q ≤ Cr,p ∥f ∥p
r
0 (t − s)
for 1 < p < +∞, 1 − p1 < r < 1 and 1q = r +
interpolation. Thus the theorem is proved.
1
p
− 1. For the other values of q, we use
From Theorems 8.7 and 8.8, it is easy to deduce some conditions to ensure existence of
solutions to the Navier–Stokes equations with initial values in Besov spaces or to deduce
some regularity estimates for solutions in more regular spaces. For example, Giga [208]
described the Lpt Lqx properties of the solutions associated to an initial value in L3 ; the case
of L3 has been later fully commented by Cannone and Planchon [85].
Solutions in Lp Lq were first described in 1972 by Fabes, Jones and Rivière [168]. The
corresponding initial values belong to a homogeneous Besov space:
−2
Wνt ∗ u0 ∈ Lpt Lqx ⇔ u0 ∈ Ḃq,pp
Solutions such that
1
sup |t| p ∥⃗u(t, x)∥Lq (dx) < +∞
t∈R
2
3
p+q
with
= 1 and 3 < q < +∞ were described in 1995 by Cannone [81]. The corresponding
initial values belong to a homogeneous Besov space:
1
−2
p
sup t p ∥Wνt ∗ u0 ∥q < +∞ ⇔ u0 ∈ Ḃq,∞
0<t
Similarly, Besov spaces based on Morrey–Campanato spaces were defined by Kozono and Yamazaki in 1994 [279] and led to the existence of solutions such that
√
1
supt>0 t p ∥⃗u(t, x)∥Ṁ r,q (dx) < +∞ and sup0<t t∥⃗u(t, x)∥L∞ (dx) < +∞ (with 1 ≤ r ≤ q
and p2 + 3q = 1).
Here, we give an example of such an existence theorem:
Navier–Stokes equations and Besov spaces: mild solutions
Theorem 8.9.
Let E, F, G ⊂ S ′ (R3 ) be three Banach spaces such that:
• E ⊂ L2loc
Mild Solutions in Besov or Morrey Spaces
171
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
• pointwise product is bounded from E × E to F
• For every finite positive T , we have sup0<t<T t
some α ∈ (0, 1)
• For every finite positive T , we have sup0<t<T t
some β ∈ (0, 1), with α + β < 1.
1−α
2
1−β
2
∥Wνt ∗ f ∥E ≤ CT ∥f ∥F for
∥Wνt ∗ f ∥E ≤ CT ∥f ∥G for
Then
−α 3
• If ⃗u0 ∈ (BE,
u0 ∈ Lpt Ex on (0, T ) × R3 for all (finite) positive
2 ) , then Wνt ∗ ⃗
α
T , with p =
2
α.
• If f⃗ is defined on (0, T )×R3 and satisfies Pf⃗ ∈ Lq ((0, T ), G3 ) with 1q = 1+α+β
,
2
Rt
p
3
⃗
then
Wν(t−s) ∗ Pf (s, .) ds ∈ L Ex on (0, T ) × R for all (finite) positive T ,
t
0
2
α.
with p =
−α 3
• If ⃗u0 ∈ (BE,
and Pf⃗ ∈ Lq ((0, T ), G3 ), then there exists a T0 ∈ (0, T ) and a
2 )
α
mild solution ⃗u of Equation (7.4) on (0, T0 ) × R3 such that ⃗u ∈ Lp ((0, T0 ), E 3 ).
Assume moreover that we have the global inequalities
sup t
1−α
2
∥Wνt ∗ v∥E ≤ C∞ ∥v∥F and sup t
0<t
1−β
2
∥Wνt ∗ v∥E ≤ C∞ ∥v∥G .
0<t
−α 3
Then, if ⃗u0 is small enough in (ḂE,
and Pf⃗ is small enough in Lq ((0, +∞), G3 ),
2 )
α
then there exists a mild solution ⃗u of Equation (7.4) on (0, +∞) × R3 such that ⃗u ∈
Lp ((0, +∞), E 3 ).
Proof.
This is a direct consequence of Theorem 8.8; the only thing we need to check is that
Rt
p
⃗
W
ν(t−s) ∗ Pf (s, .) ds ∈ Lt Ex . We have (for 0 < t < T )
0
Z
∥
t
Wν(t−s) ∗ Pf⃗(s, .) ds∥E ≤ CT
0
Z
0
t
1
(t − s)
1−β
2
∥Pf⃗(s, .)∥G ds
thus we have only to use the Hardy–Littlewood–Sobolev inequality, since
α
2.
1
q
+
1−β
2
= 1+
Examples:
ˆ Lebesgue spaces: E = Lq , 3 < q < +∞: in that case, we have global solutions in Lpt Lqx
−1+ 3
with 2 + 3 = 1 when ⃗u0 is small enough in (Ḃq,p q )3 and when f⃗ is small enough
p
q
in (Lrt Lsx )3 with
2
r
+
3
s
= 3 and
3q
3+q
3( q1 − 1s )
< s < 3. The latter condition may be relaxed to
3
r
(Lrt Ls,∗
)3 .
x ) or even to (Lt Ḃq,∞
For the case q = 3, see Theorem 15.12.
172
The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ Morrey spaces: E = Ṁ ρ,q with max(2, 3q ) < ρ ≤ q and 3 < q < +∞: in that case, we
−1+ 3
have global solutions in Lpt Ṁ ρ,q with p2 + 3q = 1 when ⃗u0 is small enough in (ḂṀ ρ,q q,p )3
s
and when f⃗ is small enough in (Lr Ṁ ρ q ,s )3 with 2 + 3 = 3 and max( q , 3q ) < s < 3.
t
r
s
3( 1 − 1 )
ρ 3+q
q
s
The latter condition may be relaxed to (Lrt ḂṀ ρ,q
)3 .
,∞
8.5
Regular Besov Spaces
In the preceding section, we considered Besov spaces with negative regularity indexes.
The case of Besov spaces with positive regularity indexes is easier to deal with, in a complete
analogy to the case of Morrey spaces and Theorem 8.1.
Navier–Stokes equations and regular Besov spaces
Theorem 8.10.
Let E ⊂ S ′ (R3 ) be a Banach space such that:
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E
• E is stable under bounded pointwise multiplication: ∥f g∥E ≤ ∥f ∥∞ ∥g∥E
−α
• E ⊂ B∞,∞
for some α > 0.
Let β > 0 such that α − β ≤ 1. Let YT be the space of Lebesgue measurable vector fields
on (0, T ) × R3 such that
t(α−β)/4 ∥⃗u(t, .)∥
β
2
B[E,L
∞]
∈ L∞ ((0, T ))
1/2,1
normed with
∥⃗u∥YT = sup t(α−β)/4 ∥⃗u(t, .)∥
0<t<T
< +∞.
β
2
B[E,L
∞]
1/2,1
Then
• The bilinear operator B is continuous on the space YT :
⃗ [E,L∞ ]
sup t(α−β)/4 ∥B(F⃗ , G)∥
≤ Cν T
1/2,1
1−α
2
⃗ Y (8.7)
(1 + (νT )α/4 )∥F⃗ ∥YT ∥G∥
T
0<t<T
⃗
where the constant Cν does not depend on T , F⃗ nor G.
β
⃗
⃗
Moreover t 7→ B(F , G)(t, .) is bounded from (0, T ] to (BE,∞
)3 .
β
• If ⃗u0 ∈ (BE,∞
)3 , then Wνt ∗ ⃗u0 ∈ YT .
Mild Solutions in Besov or Morrey Spaces
173
• If f⃗ is defined on (0, T ) × R3 and satisfies Pf⃗ ∈ L1 ((0, T ), E 3 ) and
Rt
sup0<t<T t∥Pf⃗(t, .)∥E < +∞, then 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C((0, T ], E 3 ) ∩ YT
and limt→0 tα/4 ∥Pf⃗(t, .)∥[E,L∞ ]1/2,1 = 0.
Rt
• Case α < 1: If ⃗u0 ∈ E 3 and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ YT , then there exists a
T0 ∈ (0, T ) and a mild solution ⃗u of Equation (7.4) on (0, T0 ) × R3 such that
Rt
⃗u − Wνt∗u0 − 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C([0, T0 ], E 3 ) ∩ YT0 .
Rt
• Case α = 1: If ⃗u0 ∈ E and 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ YT are small enough,
then there exists a mild solution ⃗u of Equation (7.4) on (0, T ) × R3 such that
Rt
⃗u − Wνt∗u0 − 0 Wν(t−s) ∗ Pf⃗(s, .) ds ∈ C((0, T ], E 3 ) ∩ YT .
• Case α = 1: if S ⊂ E 3 , if ⃗u0 belongs to the closure of S 3 in E 3 , if f⃗ is defined on
(0, T ) × R3 and satisfies Pf⃗ ∈ L1 ((0, T ), E 3 ) and sup0<t<T t∥Pf⃗(t, .)∥E < +∞,
then there exists a T0 ∈ (0, T ) and a mild solution ⃗u of Equation (7.4) in
C([0, T0 ], E 3 ) ∩ YT0 .
8.6
Triebel–Lizorkin Spaces
Our study of the Cauchy problem for the Navier–Stokes problem with initial value in a
−α
Besov space BE,q
(with α > 0) relied on the thermic characterization of the Besov space:
−α
f ∈ Ḃq,E
⇔ tα/2 Wνt ∗ f ∈ Lqt Ex .
Similar results hold when we change the order of integration in t and in x:
The Navier–Stokes bilinear operator and Triebel–Lizorkin spaces
Theorem 8.11.
Let E, F ⊂ S ′ (R3 ) be two Banach spaces such that:
• E ⊂ L2loc and F ⊂ L1loc .
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E .
• If h ∈ E and if g ∈ L2loc with |g| ≤ |h|, then g ∈ E and ∥g∥E ≤ ∥h∥E .
• Pointwise product is bounded from E × E to F .
• The Riesz potential I1−α is bounded from F to E for some α ∈ (0, 1).
The bilinear operator B defined as
Z t
⃗
⃗
⃗ ds
B(F , G) =
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
174
The Navier–Stokes Problem in the 21st Century (2nd edition)
is continuous on the space YT of measurable vector fields on (0, T ) × R3 such that
∥⃗u∥YT = ∥ ∥tα/2 |⃗u(t, x)| ∥Lq ((0,T ), dt ) ∥E < +∞
t
where
2
α
≤ q ≤ +∞.
⃗ = B(F⃗ , G).
⃗ Writing
Proof. Let H
⃗ = Wν(t−s)/2 ∗ P div(Wν(t−s)/2 ∗ [F⃗ ⊗ G]),
⃗
Wν(t−s) ∗ P div(F⃗ ⊗ G)
we get
⃗ .))| ≤
|Wν(t−s) ∗ P div(F⃗ (s, .) ⊗ G(s,
Z
Cν,α
α
(t − s)1− 2
1
|F⃗ (s, y)||G(s, y)| dy
|x − y|2+α
Thus,
⃗ x)| ∥ q dt ≤
∥tα/2 |H(t,
L ( t )
Z
Z t
Cν,α
1
α/2
⃗ y)| ds∥ q dt dy
∥t
s−α sα/2 |F⃗ (s, y)|sα/2 |G(s,
L ( t )
1− α
2+α
2
|x − y|
0 (t − s)
We alreaby proved (for Theorem 8.8) that
Z t
1
α/2
∥t
s−α F (s) ds∥Lq ( dt ) ≤ C∥F ∥Lq/2 ( dt )
1− α
t
t
2
0 (t − s)
for
2
α
≤ q ≤ +∞. Thus, we find
⃗ x)| ∥ q dt ∥E ≤
∥ ∥tα/2 |H(t,
L ( )
t
⃗ .)|∥ q dt )∥E
C∥I1−α (∥tα/2 |F⃗ (s, .)|∥Lq ( dt ) ∥tα/2 |G(s,
L ( )
t
t
Thus the theorem is proved.
This gives readily the following theorem:
The Navier–Stokes bilinear operator and Triebel–Lizorkin spaces: mild
solutions
Theorem 8.12.
Let E, F, G ⊂ S ′ (R3 ) be three Banach spaces such that:
• E ⊂ L2loc .
• E is stable under convolution with L1 : ∥f ∗ g∥E ≤ ∥f ∥1 ∥g∥E .
• If h ∈ E and if g ∈ L2loc with |g| ≤ |h|, then g ∈ E and ∥g∥E ≤ ∥h∥E .
• Pointwise product is bounded from E × E to F .
• The Riesz potential I1−α is bounded from F to E for some α ∈ (0, 1).
Mild Solutions in Besov or Morrey Spaces
175
• The Riesz potential I1−β is bounded from G to E for some β ∈ (0, 1) with
α + β < 1.
−α
Then let ḞE,p
be defined by
−α
h ∈ ḞE,p
⇔ ∥tα/2 |Wνt ∗ h| ∥Lp ( dt ) ∥E < +∞.
t
−α 3
If ⃗u0 is small enough in (ḞE,p
) with α2 = p and ∥Pf⃗∥Lqt is small enough in G with
1+α+β
1
, then there exists a mild solution ⃗u of Equation (7.4) on (0, +∞) × R3 such
q =
2
R +∞
that ( 0 |⃗u|p dt)1/p ∈ E.
Proof. RThis is a direct consequence of Theorem 8.11, the only thing we need to check is
t
that ∥ 0 Wν(t−s) ∗ Pf⃗(s, .) ds∥Lpt ∈ E. We have
Z
Cν,β
1
|Wν(t−s) ∗ Pf⃗(s, .)| ≤
|Pf⃗(s, y)| dy
β
1
−
|x − y|2+β
(t − s) 2 2
Thus,
t
Z
∥|
0
Thus, as
1
p
=
α
2
Wν(t−s) ∗ Pf⃗(s, .) ds ∥Lpt ≤
Z
Z t
Cν,β
1
∥
|Pf⃗(s, y)| ds∥Lpt dy
|x − y|2+β 0 (t − s) 12 − β2
=
1
q
+ 1−β
2 − 1, we find (by the Hardy–Littlewood–Sobolev inequality) that
Z
∥|
0
t
Wν(t−s) ∗ Pf⃗(s, .) ds ∥Lpt ≤
Z
Cν,β
∥|Pf⃗(t, y)|∥Lpt dy
|x − y|2+β
Thus the theorem is proved.
In particular, we may easily find a solution in Lqx Lpt (with p2 + 3q = 1) when the initial
value belongs to a (classical) homogeneous Triebel–Lizorkin space [36, 475]:
−2
Wνt ∗ u0 ∈ Lqx Lpt ⇔ u0 ∈ Ḟq,pp .
Another interesting example is the case of homogeneous Triebel–Lizorkin–Morrey spaces
(studied by Sickel, Yang and Yuan in [436]): if E = Ṁ p,q with 2 < p ≤ q and F = Ṁ p/2,q/2
then pointwise product maps E × E to F and the Riesz potential I1−α maps F to E,
−1+ 3
3
q
q
(and 3 < q < +∞, to ensure that 0 < α < 1). If ⃗u0 ∈ ḞṀ p,q ,r
with
R +∞ r
2
3
1/r
p,q
|⃗u| dt)
∈ Ṁ . In the case p = r and
r + q = 1, we may find a solution such that ( 0
2 < p ≤ 5, we have seen on page 99 that
provided that 1 − α =
−2
p
1t>0 Wνt ∗ u0 ∈ Ṁ p,5 (R × R3 ) ⇔ u0 ∈ ḞṀ p,q
.
,p
176
The Navier–Stokes Problem in the 21st Century (2nd edition)
−1+ 3
q
An interesting subspace of ḞṀ p,q ,r
with 2 < p ≤ q and
2
r
+
3
q
−1+ 3
q
= 1 is ḞṀ p,q ,2
=
√
1− 3
−∆ q Ṁ p,q . The limit case p = 2 has been considered by Xiao in [507] within the
theory of Q-spaces, as indeed we have
−1+ 3
3
q
ḞṀ 2,q ,2
= Q−1, q .
8.7
Fourier Transform and Navier–Stokes Equations
The Navier–Stokes equations have a simple structure: they have constant coefficients
and the non-linearity is quadratic. Thus, the Fourier transform turns out to be an efficient
tool to describe some classes of solutions. This is well-known for periodic solutions [471],
especially in the setting of Sobolev spaces.
But the Fourier transform is useful as well for the problem in the whole space. We shall
give examples in the general setting of Fourier-Lebesgue spaces or Fourier–Herz spaces:
existence of mild solutions (as in the results of Le Jan and Sznitman [305], of Lei and Lin
[306] or of Cannone and Wu [86]); analyticity of the solutions (when f⃗ = 0), following the
formalism of Foias and Temam [181] or of Lemarié-Rieusset [312].2
Let us consider a mild solution of the Navier–Stokes equations
Z t
⃗u = Wνt ∗ ⃗u0 +
Wν(t−s) ∗ P(f⃗ − div(⃗u ⊗ ⃗u)) ds.
0
⃗ be the (spatial) Fourier transform of ⃗u and F⃗ be the (spatial) Fourier transform of
Let U
f⃗. We find
Z t
2
ξ⊗ξ ⃗
−νt|ξ|2 ⃗
⃗
U0 (ξ) +
U =e
e−ν(t−s)|ξ| (Id −
)F (s, ξ) ds
|ξ|2
0
(8.8)
Z t
Z
dη
ξ⊗ξ
−ν(t−s)|ξ|2
⃗
⃗
−
)(iξ) · ( U (s, ξ − η)U (s, η)
) ds.
e
(Id −
|ξ|2
(2π)3
0
Thus, we may look for Fourier transforms of mild solutions in some space Y, with Fourier
transform of the initial value in some space X and Fourier transform of the force in some
space Z, where X is a lattice Banach space of measurable functions on R3 and Y and Z are
lattice Banach spaces of measurable functions on (0, +∞) × R3 , provided we have
2
∥e−νt|ξ| U0 ∥Y ≤ C0 ∥U0 ∥X
t
Z
2
e−ν(t−s)|ξ| F (s, ξ) ds∥Y ≤ C1 ∥F ∥Z
∥
0
and
Z tZ
∥
2
e−ν(t−s)|ξ| |ξ||V (s, η)||W (s, ξ − η)| ds dη∥Y ≤ C2 ∥V ∥Y ∥W ∥Y
0
for some constants that depend on ν. Then we know that we have a mild solution on
(0, +∞) × R3 provided that
⃗ 0 ∥X + C1 ∥F⃗ ∥Z ≤
C0 ∥U
2A
(2π)3
.
4C2
more general result on analyticity will be proved in Section 9.9.
Mild Solutions in Besov or Morrey Spaces
First example:
Let α > 0 and 1 < q < +∞ be such that moreover 2 <
Y = {U / |ξ|α U ∈ Lpt Lqξ }, where
3
q
177
+ α < 3. We choose
2
3
= + α − 2.
p
q
We then have 2 < p < +∞. We write
2
2
e−(t−s)|ξ| |ξ| ≤ C|ξ| p −1
1
1
(t − s)1− p
.
Thus, if
Z tZ
Z=
2
e−ν(t−s)|ξ| |ξ||V (s, η)||W (s, ξ − η)| ds dη,
0
we have
|ξ|α Z ≤ C
Z
0
t
1
(ν(t − s))
2
1
1− p
|ξ|α+ p −1
Z
|V (s, η)||W (s, ξ − η)| dη ds
and thus
t
Z
∥Z∥Y ≤ C∥
1
3
ν((t − s))
0
1
1− p
∥|ξ|2α+ q −3
Z
|V (s, η)||W (s, ξ − η)| dη∥Lq (dξ) ds∥Lp(dt)
We now write, for 0 < δ < α, |ξ|δ ≤ C(|ξ − η|δ + |η|δ ) and thus
Z
|ξ|δ |V (s, η)||W (s, ξ − η)| dη ≤ C(W ∗ (|η|δ V ) + V ∗ (|η|δ W )).
We have |η|δ = |η|δ−α |η|α V ∈ Lr,q and W = |η|−α |η|α W ∈ Lρ,q with
1
1
α
ρ = q + 3 . Thus, by convolution inequalities, we find that
|ξ|δ
Z
1
r
=
1
q
+
α−δ
3
and
|V (s, η)||W (s, ξ − η)| dη ∈ Lτ,q
δ
3
3
3
where τ1 = 2q + 2α
3 − 3 −1 (provided that 0 < δ < 2( q +α− 2 )). As we have 2α+ q −3−δ < 0,
we find that
Z
3
|ξ|2α+ q −3−δ |ξ|δ |V (s, η)||W (s, ξ − η)| dη ∈ Lv,q
with
1
1
3
2 2α δ
1
= − (2α + − 3 − δ) + +
− −1= .
v
3
q
q
3
3
q
Thus, we find
Z
∥Z∥Y ≤ C∥
0
t
1
(ν(t − s))
1
1− p
∥|ξ|α V (s, ; )∥q ∥|ξ|α W (s, .)∥q ds∥p ≤
C2
1
ν 1− p
∥V ∥Y ∥W ∥Y .
Of course, we have now to identify X. We could define X just as the space of U0 such
that
2
∥|ξ|α e−νt|ξ| U0 ∥Lp Lq < +∞.
178
The Navier–Stokes Problem in the 21st Century (2nd edition)
If we perform a dyadic partition of unity on R3 :
1=
X
ψ(
j∈Z
for some smooth ψ supported in {ξ
on (0, +∞)
1
2
ξ
)
2j
≤ ξ ≤ 4} and similarly we have a partition of unity
1=
X
k∈Z
ω(
t
),
4k
we find
2
∥|ξ|α e−νt|ξ| U0 ∥qq ≈
X
2
2qαj ∥e−νt|ξ| ψ(
j∈Z
ξ
)U0 (ξ)∥qq
2j
and thus

1/q 1/p
X
2
t
ξ
X

∥ω( k ) 
∥U0 ∥X ≈ 
2qαj ∥e−νt|ξ| ψ( j )U0 (ξ)∥qq  ∥pp 
4
2

j∈Z
k∈Z
Thus, for two positive constants A and B, we have
p/q 1/p
ξ
X k X qαj −256 νq4k+j

A
4
2 e
∥ψ( j )U0 (ξ)∥qq  
2


k∈Z
j∈Z
≤∥U0 ∥X


p/q 1/p
1
ξ
X k X qαj − 16

νq4k+j
4
2 e
∥ψ( j )U0 (ξ)∥qq  
≤B 
2
j∈Z
k∈Z
Restricting in the first term the sum over j to the sole value j = −k, we find
1/p

Ae−256ν 
X
2−2j 2jαp ∥ψ(
j∈Z
ξ
)U0 (ξ)∥pq 
2j
≤ ∥U0 ∥X
(8.9)
s
The Herz spaces Bq,p
are defined by
s
∥U0 ∥Bq,p

1/p
X
ξ
=
2sjp ∥ψ( j )U0 (ξ)∥pq 
< +∞.
2
j∈Z
s
They have been introduced by Herz [234], and the Fourier-Herz spaces FBq,p
(i.e., the image
of Herz spaces through the Fourier transform) have been recently used in the context of
parabolic equations, first by Iwabuchi [242] for the Keller–Segel equation then by Cannone
and Wu [86] for the Navier–Stokes equations. Let 1 < r < +∞ and 1q + 1r = 1. Then, if
−1+ r3
r ≤ 2, we have Ḃr,p
2− 3
2− 3
−1+ r3
⊂ FBq,p q while, if r ≥ 2, we have FBq,p q ⊂ Ḃr,p
.
Mild Solutions in Besov or Morrey Spaces
α− 2
179
2− 3
From (8.9), we find that X ⊂ Bq,p p = Bq,p q . On the other hand, we have, by the
Minkowski and Young inequalities,

p/q 1/p
X
X
k+j
1
ξ


4k 
2qαj e− 16 νq4 ∥ψ( j )U0 (ξ)∥qq  

2

j∈Z
k∈Z

p 1/p
X
X
k+j
1
ξ
≤
4k 
2αj e− 16 ν4 ∥ψ( j )U0 (ξ)∥q  
2

j∈Z
k∈Z

=

X
X

X
2
e
1
− 16
k+j
ν4
2
2
(α− p
)j
j∈Z
k∈Z
≤(
2
(k+j) p
2
k
1
2k p e− 16 ν4 )(
X
2
2j(α− p )p ∥ψ(
j∈Z
k∈Z
so that
∥U0 ∥X ≤ B(
X
2
p 1/p
ξ
∥ψ( j )U0 (ξ)∥q  
2
ξ
)U0 (ξ)∥pq )1/p
2j
k
1
2k p e− 16 ν4 )∥U0 ∥
α− 2
Bq,p p
k∈Z
.
α− 2
Thus, X = Bq,p p .
For the choice of the space Z, we may take Z = {F / |ξ|α−δ F ∈ Lr Lq }, where
and 1r = 1 + p1 − 2δ (so that 1 < r < p): indeed, we have
|ξ|α
t
Z
2
e−ν(t−s)|ξ| |F (s, ξ)| ds ≤ C
t
Z
0
0
1
|ξ|α−δ |F (s, ξ)| ds
(ν(t − s))δ/2
Z
t
2
p
<δ<2
so that
∥|ξ|α
Z
t
2
e−ν(t−s)|ξ| |F (s, ξ)| ds∥q ≤ C
0
∥|ξ|α−δ F (s, ξ)∥q
0
ds
.
(ν(t − s))δ/2
As 0 < δ/2 < 1, we use the Hardy–Littlewood–Sobolev inequality and find that
Rt
2
∥|ξ|α 0 e−ν(t−s)|ξ| |F (s, ξ)| ds∥q belongs to Lσ with σ1 = 1r + 2δ − 1 = p1 .
Recollecting all those results, we find the theorem:
Theorem 8.13.
Let α > 0 and 1 < q < +∞ be such that moreover 2 < 3q + α < 3. Let δ be such that
3
2
3
1
1
δ
q + α − 2 < δ < 2. Let p = q + α − 2 and r = 1 + p − 2 . Then there exists a positive
2− 3
constant ϵ0 (depending on ν, α, q and δ) such that, if ⃗u0 ∈ FBq,p q with div ⃗u0 = 0 and
(−∆)(α−δ)/2 f⃗ ∈ Lr FLq , and if moreover
∥F⃗u0 ∥
2− 3
Bq,p q
+ ∥|ξ|α−δ F f⃗∥Lr Lq < ϵ0 ,
then the Cauchy problem for the Navier–Stokes equations with initial value ⃗u0 and forcing
term f⃗ has a global mild solution ⃗u such that (−∆)α/2 ⃗u ∈ Lp FLq .
180
The Navier–Stokes Problem in the 21st Century (2nd edition)
Second example:
We now consider the case p = +∞. Here, we interchange the order of integration between
t and ξ. Thus, we choose, for 0 ≤ α < 2,
Y = {U / |ξ|α sup |U (t, ξ)| ∈ Lqξ }, where 0 =
t>0
3
+ α − 2.
q
Thus, if
Z tZ
Z=
2
e−ν(t−s)|ξ| |ξ||V (s, η)||W (s, ξ − η)| ds dη,
0
with |V (s, ξ)| ≤ V (ξ)|ξ|−α and |W (s, ξ)| ≤ W (ξ)|ξ|−α , we have
Z
1
V (ξ − η) W (η)
|ξ|α Z ≤ |ξ|α−1
dη
ν
|ξ − η|α |η|α
Z
1
W (η) V (ξ − η)
≤C
V (ξ − η)
+
W (η) dη.
ν|ξ|
|η|α
|ξ − η|α
W
r,q
V belongs to Lq and |η|
with 1r = 1q + α3 . Thus, supt>0 |ξ|α Z(t, ξ) belongs
α belongs to L
to Lρ,q with ρ1 = 2q + α3 − 1 + 13 = 1q and we find ∥Z∥Y ≤ C2 ∥V ∥Y ∥W ∥Y .
The associated space X is easily identified, as
−νt|ξ|2
∥e0
∥Y = ∥|ξ|α U0 ∥q .
The associated space Z can be chosen as L1 ((0, +∞), X). Indeed, we have
Z t
Z +∞
2
|
e−ν(t−s)|ξ| F (s, ξ) ds| ≤
|F (s, ξ)| ds
0
so that
∥|ξ|α
Z
0
t
2
e−ν(t−s)|ξ| F (s, ξ) ds∥q ≤
0
Z
+∞
∥|ξ|α F (s, ξ)∥q ds.
0
We thus find the theorem:
Theorem 8.14.
Let α ≥ 0 and 32 ≤ q < +∞ with 2 =
3
q
+ α. Then there exists a positive constant ϵ0
(depending on ν, and q) such that, if (−∆) ⃗u0 ∈ FLq with div ⃗u0 = 0 and (−∆)α/2 f⃗ ∈
L1 FLq , and if moreover
α/2
∥∥|ξ|α F⃗u0 ∥q + ∥|ξ|α F f⃗∥L1 Lq < ϵ0 ,
then the Cauchy problem for the Navier–Stokes equations with initial value ⃗u0 and forcing
term f⃗ has a global mild solution ⃗u such that supt>0 |ξ|α |F⃗u(t, ξ)| ∈ Lq .
In particular, we recover the results of Theorem 7.4 on global existence of solutions in
the Sobolev space Ḣ 1/2 (except that we replaced the condition f⃗ small in L2 Ḣ −1/2 by f⃗
small in L1 Ḣ 1/2 ).
Third example:
We consider the case q = +∞. This is a simple case. Let α > 0 be such that moreover
2 ≤ α < 3. We choose
Y = {U / |ξ|α U ∈ Lpt L∞
ξ }, where
2
= α − 2.
p
Mild Solutions in Besov or Morrey Spaces
We then have 2 < p ≤ +∞.
If
Z Z
t
Z=
181
2
e−ν(t−s)|ξ| |ξ||V (s, η)||W (s, ξ − η)| ds dη,
0
we write |V (s, η)| ≤
V (s)
|η|α
with V (s) ∈ Lp and similarly |W (s, ξ − η)| ≤
Z
Z
|V (s, η)||W (s, ξ − η)| dη ≤ V (s)W (s)
and
|ξ|α Z(t, ξ) ≤ C
Z
t
W (s)
|ξ−η|α ,
so that
dη
≤ C|ξ|3−2α V (s)W (s)
|η|α |ξ − η|α
2
e−ν(t−s)|ξ| |ξ|4−α V (s)W (s) ds.
0
If α = 2, we obtain
|ξ|α Z(t, ξ) ≤C∥V (s)∥∞ ∥W (s)∥∞
Z
t
2
e−ν(t−s)|ξ| |ξ|2 ds
0
1
≤C ∥V (s)∥∞ ∥W (s)∥∞ .
ν
2−α/2
2
1
If α > 2, we write e−ν(t−s)|ξ| ≤ C ν(t−s)ξ|
and obtain
2
|ξ|α Z(t, ξ) ≤ C
1
Z
1
1
ν 2−α/2
(t − s)1− p
V (s)W (s) ds.
In every case, we have ∥Z∥Y ≤ C2 ∥V ∥Y ∥W ∥Y .
2
. The associated Fourier-Herz space
The space X will then again be a Herz space B∞,p
3
is then a Besov space based on pseudo-measures , as studied by Cannone and Karch [82]:
2
2
FB∞,p
= ḂPM,p
. The case p = +∞ and α = 2 is the case considered by Le Jan and
Sznitman [305].
For the choice of the space Z, we may again take, if p < +∞, Z = {F / |ξ|α−δ F ∈ Lr Lq },
where p2 < δ < 2 and 1r = 1 + p1 − 2δ (so that 1 < r < p). For p = +∞, we want to get
|ξ|
2
Z
t
2
∞
e−ν(t−s)|ξ| |F (s, ξ)| ds ∈ L∞
t Lξ .
0
2
2
r
As ∥e−νt|ξ| ∥Lr ((0,+∞) = Cr (ν|ξ|)−2/r , we may take Z = {F / |ξ|2− r F ∈ Ltr−1 L∞
ξ }. We
thus get (in the case p = +∞) the theorem:
Theorem 8.15.
Let r ∈ [1, +∞]. Then there exists a positive constant ϵ0 (depending on ν, and r) such that,
1
if ∆⃗u0 ∈ FL∞ with div ⃗u0 = 0 and (−∆) r f⃗ ∈ Lr FL∞ , and if moreover
2
∥|ξ|2 F⃗u0 ∥∞ + ∥|ξ| r F f⃗∥Lr L∞ < ϵ0 ,
then the Cauchy problem for the Navier–Stokes equations with initial value ⃗u0 and forcing
term f⃗ has a global mild solution ⃗u such that supt>0 |ξ|2 |F⃗u(t, ξ)| ∈ L∞ .
3A
pseudo-measure is a tempered distribution whose Fourier transform belongs to L∞ .
182
The Navier–Stokes Problem in the 21st Century (2nd edition)
Let us remark that the case f⃗ = βδ⃗e3 (where ⃗e3 is the unit vector in the x3 axis, β is
a positive constant and δ is the Dirac mass at x = 0) has been discussed by Cannone and
Karch [82], in relation with the Landau self-similar solutions [301, 439, 447] (see Section
10.8).
Fourth example:
We consider the case q = 1. Of course, the conditions α > 0 and 3q + α < 3 become
incompatible, and we shall deal with the limit case α = 0. Thus, we choose
Y = L2t L1ξ .
If
Z tZ
Z=
2
e−ν(t−s)|ξ| |ξ||V (s, η)||W (s, ξ − η)| ds dη,
0
we write
Z
Z(t, ξ)
dξ
≤
|ξ|
Z t ZZ
|V (s, η)||W (s, ξ − η)| dη dξ ds
0
Z
t
∥V (s, .)∥1 ∥W (s, .)∥1 ds
=
0
≤∥V ∥L2 L1 ∥W ∥L2 L1
1 dξ
so that Z ∈ L∞
t (L ( |ξ| ). Moreover, we have
Z +∞ Z
Z(t, ξ) |ξ| dξ dt
0
Z +∞Z Z Z +∞
2
=
(
e−ν(t−s)|ξ| |ξ|2 dt)|V (s, η)||W (s, ξ − η)| dη dξ ds
0
s
Z
ZZ
1 +∞
|V (s, η)||W (s, ξ − η)| dη dξ ds
=
ν 0
Z
1 +∞
=
∥V (s, .)∥1 ∥W (s, .)∥1 ds
ν 0
1
≤ ∥V ∥L2 L1 ∥W ∥L2 L1
ν
so that Z ∈ L1 (L1 (|ξ| dξ)). Thus, we find
Z +∞ Z
Z +∞ Z
Z
dξ
1
( Z(t, ξ) dξ)2 dt ≤
( Z )( Z|ξ| dξ) dt ≤ ∥V ∥2L2 L1 ∥W ∥2L2 L1 .
|ξ|
ν
0
0
dξ
)), then
A similar proof gives that, if F ∈ Z = L1t (L1 ( |ξ|
Z Z t
2
dξ
(
e−ν(t−s)|ξ| |F (s, ξ) ds)
≤ ∥F ∥L1 (L1 ( dξ ))
t
|ξ|
|ξ|
0
and
Z
+∞
0
Z Z t
2
1
(
e−ν(t−s)|ξ| |F (s, ξ) ds) |ξ| dξ dt ≤ ∥F ∥L1 (L1 ( dξ ))
t
|ξ|
ν
0
so that
Z
t
2
1
e−ν(t−s)|ξ| |F (s, ξ) ds∥L2 L1 ≤ √ ∥F ∥L1 (L1 ( dξ ))
(8.10)
t
|ξ|
ν
0
Moreover, we see that the associated space X for the initial value U0 is the Herz space
−1
B1,2
. We thus find the theorem:
∥
Mild Solutions in Besov or Morrey Spaces
183
Theorem 8.16.
−1
There exists a positive constant ϵ0 (depending on ν) such that, if ⃗u0 ∈ FB1,2
with div ⃗u0 = 0
(−1/2 ⃗
1
1
and (−∆)
f ∈ L FL , and if moreover
∥F⃗u0 ∥B−1 + ∥
1,2
1 ⃗
F f ∥L1 L1 < ϵ0 ,
|ξ|
then the Cauchy problem for the Navier–Stokes equations with initial value ⃗u0 and forcing
term f⃗ has a global mild solution ⃗u such that ⃗u ∈ L2 FL1 .
−1
Cannone and Wu [86] studied the more general case of ⃗u0 ∈ FB1,q
with 1 ≤ q ≤ 2. The
dξ
case q = 1 corresponds to F⃗u0 ∈ L1 ( |ξ ), i.e., to the theorem of Lei and Lin [306]:
Corollary 8.1.
There exists a positive constant ϵ0 (depending on ν) such that, if
Z
Z +∞ Z
dξ
dξ
|F⃗u0 (ξ)|
+
|F f⃗(t, ξ)| dt < ϵ0 ,
|ξ|
|ξ|
0
and if div ⃗u0 = 0 then the Cauchy problem for the Navier–Stokes equations with initial
1 dξ
value ⃗u0 and forcing term f⃗ has a global mild solution ⃗u such that F⃗u ∈ L∞
t (L ( |ξ| )) ∩
L1 (L1 (|ξ| dξ)).
2
1 dξ
1
1
Proof. It is enough to check that e−νt|ξ| F⃗u0 (ξ) ∈ L∞
t (L ( |ξ| ) ∩ L (L (|ξ| dξ)). Then, we
−1
have that ⃗u0 belongs to FB1,2
and we may apply Theorem 8.16 to get the existence of
2
1
the mild solution ⃗u ∈ L FL . But the proof of Theorem 8.16 shows that F(⃗u − Wνt ∗ ⃗u0 )
1 dξ
1
1
belongs to L∞
t (L ( |ξ| ) ∩ L (L (|ξ| dξ)).
It is very easy to slightly modify the proofs of Theorems 8.13 to 8.16 to get Gevrey-type
estimates for our solutions. Indeed, we have
2
1
√
νt|ξ|
2
e−tν|ξ| ≤ Ce− 2 tν|ξ| e−
.
2
α− p
Thus, if U0 ∈ Bq,p , then
|ξ|α e
√
νt|ξ| −tν|ξ|2
e
U0 ∈ Lp Lq .
Moreover, if
Z tZ
Z=
2
√
νs|η|
e−ν(t−s)|ξ| |ξ|e−
√
νs|ξ−η|
|V (s, η)|e−
|W (s, ξ − η)| ds dη,
0
we use the inequalities
|ξ − η| + |η| ≥ |ξ|
and
√
√
√
s+ t−s≥ t
to get that
√
− νt|ξ|
Z tZ
Z ≤ Ce
0
Thus, we find
1
2
e− 2 ν(t−s)|ξ| |ξ||V (s, η)||W (s, ξ − η)| ds dη.
184
The Navier–Stokes Problem in the 21st Century (2nd edition)
Theorem 8.17.
For the following spaces X, Y and Z, there exists a positive
constant ϵ1 (depending on ν, X,
√
−νt∆ ⃗
Y and Z) such that, if ⃗u0 ∈ FX with div ⃗u0 = 0 and e
f ∈ FZ, and if moreover
√
νt |ξ|
∥F⃗u0 ∥X + ∥e
F f⃗∥Z < ϵ1 ,
then the Cauchy problem for the Navier–Stokes√equations with initial value ⃗u0 and forcing
term f⃗ has a global mild solution ⃗u such that e −νt∆ ⃗u ∈ FY:
2− 3
1
1
p q
r q
|ξ|α L L , Z = |ξ|α−δ L L ,
< 2, p2 = 3q + α − 2 and 1r =
• X = Bq,p q , Y =
3
q
+α−2<δ
• X=
1
q
|ξ|α L ,
Y=
q ∞
1
|ξ|α Lξ Lt ,
• X=
1
q
|ξ|2 L ,
Y=
1
∞
|ξ|2 Lt,ξ ,
Z=
Z=
1
1 q
|ξ|α Lt Lξ ,
1
Lr L∞ ,
|ξ|2/r t ξ
with α > 0, 1 < q < +∞, 2 <
1+
1
p
−
3
q
+ α < 3,
δ
2.
with α ≥ 0 and
3
2
≤ q < +∞.
with r ∈ [1, +∞].
dξ
−1
• X = B1,2
, Y = L2 L1 , Z = L1 (L1 ( |ξ|
)).
dξ
1 dξ
1
1
1
1 dξ
• X = L1 ( |ξ|
), Y = L∞
t (L ( |ξ| ) ∩ L (L (|ξ| dξ)), Z = L (L ( |ξ| )).
This result can be extended to the case of more general spaces, where the absolute
value does not operate on the Fourier transforms, so that the proofs given here do not
apply and must be replaced by more delicate estimates on singular integrals. For instance,
Lemarié-Rieusset [312, 313, 314] proved the following theorem:
Theorem 8.18.
There exists a positive constant ϵ1 (depending on ν) such that, if ⃗u0 ∈ L3 with div ⃗u0 = 0
and ∥⃗u0 ∥3 < ϵ1 , then the Cauchy problem for the Navier–Stokes equations with initial value
⃗u0 (and forcing term f⃗ = 0) has a global mild solution ⃗u such that
√
t1/8 F −1 e νt(|ξ1 |+|ξ2 |+|ξ3 |) F⃗u ∈ L∞ L4 .
√
−νt∆)β
Theorem 8.17 may easily be adapted to Gevrey regularity of the form e(
where 0 < β ≤ 1:
⃗u ∈ Y,
Theorem 8.19.
Let 0 < β ≤ 1 and let X, Y and Z be the same spaces as in Theorem 8.17. There exists a
positive
constant ϵ1 (depending on ν, X, Y and Z) such that, if ⃗u0 ∈ FX with div ⃗u0 = 0
√
( −νt∆)β ⃗
and e
f ∈ FZ, and if moreover
∥F⃗u0 ∥X + ∥e(
√
νt |ξ|)β
F f⃗∥Z < ϵ1 ,
then the Cauchy problem for the Navier–Stokes √equations with initial value ⃗u0 and forcing
β
term f⃗ has a global mild solution ⃗u such that e( −νt∆) ⃗u ∈ FY.
Proof. Same proof as for Theorem 8.17, replacing the inequality
√
√
√
√
e− ν(t−s)|ξ| e− νs|η| e− νs|ξ−η| ≤ e− νt|ξ|
with
√
√
√
√
β
β
β
β
e−( ν(t−s)|ξ|) e−( νs|η|) e−( νs|ξ−η|) ≤ e−( νt|ξ|) .
Mild Solutions in Besov or Morrey Spaces
8.8
185
The Cheap Navier–Stokes Equation
In chapter 5, we replaced the integral vector equation
Z tX
3
⃗u = Wνt ∗ ⃗u0 −
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G + uj ⃗u ds
0 j=1
with the study of the integral scalar equation
Z tZ
1
Ω(s, y)2 dy ds.
Ω(t, x) = Ω0 (t, x) + C0
2
2
ν
(t
−
s)
+ |x − y|4
0
R3
Of course, we can do the same for the Fourier transform of the Navier–Stokes equations
(8.8)
Z t
2
ξ⊗ξ ⃗
⃗ =e−νt|ξ|2 U
⃗ 0 (ξ) +
e−ν(t−s)|ξ| (Id −
)F (s, ξ) ds
U
|ξ|2
0
Z
Z t
2
ξ⊗ξ
⃗ (s, ξ − η) ⊗ U
⃗ (s, η) dη ) ds
U
)(iξ)
·
(
−
e−ν(t−s)|ξ| (Id −
2
|ξ|
(2π)3
0
and study the scalar equation
Z t
2
2
W (t, ξ) =e−νt|ξ| W0 (ξ) +
e−ν(t−s)|ξ| F (s, ξ) ds
0
Z t
Z
1
−ν(t−s)|ξ|2
e
|ξ|
W (s, η)W (s, ξ − η) dη ds.
+
(2π)3 0
R3
(8.11)
Taking the inverse Fourier transforms w = F −1 W of W and f = F −1 F of F , equation
(8.11 becomes the equation
(
√
∂t w = ∆w + f + −∆(w2 )
(8.12)
w(0, .) = w0 .
Equation (8.12) is known as the cheap Navier–Stokes equation. It has been introduced in
2001 by S. Montgomery-Smith [369] as a toy model for the Navier–Stokes equations. He gave
an example of an initial value w0 in the Schwartz class (w0 ∈ S(R3 )) with a non-negative
Fourier transform W0 such that the solution w for the equation with a null force (f⃗ = 0)
blows up in finite time (see section 11.2).
This cheap equation allows very simple computations for the search of solutions. Indeed,
⃗ 0 is controlled by a function W0 and F⃗ by a function F :
let us assume that U
1
1
W0 (ξ) and |F⃗ (t, ξ)| ≤
F (t, ξ)
18
18
and that W (t, ξ) is measurable, almost everywhere finite and is a non-negative solution of
the integral inequation for every t ∈ [0, T ] and every ξ ∈ R3
Z t
2
2
e−νt|ξ| W0 (ξ) +
e−ν(t−s)|ξ| F (s, ξ) dξ + B0 (W, W )(t, ξ) ≤ W (t, ξ)
⃗ 0 (ξ)| ≤
|U
0
with
B0 (W, V )(t, ξ) =
Define
1
(2π)3
Z
0
t
2
e−ν(t−s)|ξ| |ξ|
Z
W (s, η)V (s, ξ − η) dη ds.
R3
186
The Navier–Stokes Problem in the 21st Century (2nd edition)
Rt
2
2
• W [0] (t, ξ) := e−νt|ξ| W0 (ξ) + 0 e−ν(t−s)|ξ| F (s, ξ) dξ
• W [n+1] (t, ξ) := W [0] (t, ξ) + B0 (W [n] , W [n] )(t, ξ)
R
⃗ [0] := e−νt|ξ|2 U
⃗ 0 (ξ) + t e−ν(t−s)|ξ|2 (Id − ξ⊗ξ2 )F⃗ (s, ξ) ds
• U
|ξ|
0
⃗ [n+1] (t, ξ) := U
⃗ [0] (t, ξ) − B(U
⃗ [n] , U
⃗ [n] )(t, ξ)
• U
where
⃗,V
⃗)=
B(U
t
Z
2
e−ν(t−s)|ξ| (Id −
0
Z
ξ⊗ξ
⃗ (s, ξ − η) ⊗ V
⃗ (s, η) dη ) ds.
)(iξ)
·
(
U
|ξ|2
(2π)3
By induction on n, we find that we have the pointwise inequalities
• 0 ≤ W [n] (t, ξ) ≤ W [n+1] (t, ξ) ≤ W (t, ξ)
⃗ [n] (t, ξ)| ≤
• |U
1
[n]
(t, ξ)
18 W
⃗ [n+1] (t, ξ) − U
⃗ [n] (t, ξ)| ≤
• |U
1
[n+1]
(t, ξ)
18 (W
− W [n] (t, ξ)).
We find that W [n] is pointwisely convergent to a function W [∞] ≤ W . By monotonous
convergence, we have
W [∞] = W [0] + B0 (W [∞] , W [∞] ).
⃗ [n] converges to a limit U
⃗ [∞] such that
Then, by dominated convergence, we find that U
⃗ [∞] = U
⃗ [0] − B(U
⃗ [∞] , U
⃗ [∞] ).
U
⃗ [∞] is then the Fourier transform of a solution to the Navier–Stokes problem with initial
U
value ⃗u0 and forcing term f⃗.
The same formalism allows one to get easily Gevrey-type analyticity estimates. Let us
⃗ 0 is controlled by a function X0 and F⃗ by a function G:
assume more precisely that U
⃗ 0 (ξ)| ≤
|U
1
1 −√νt|ξ|
X0 (ξ) and |F⃗ (t, ξ)| ≤
e
G(t, ξ)
18e
18e
and that X(t, ξ) is measurable, almost everywhere finite and is a non-negative solution of
the integral inequation for every t ∈ [0, T ] and every ξ ∈ R3
ν
2
e− 2 t|ξ| X0 (ξ) +
Z
t
2
ν
e− 2 (t−s)|ξ| G(s, ξ) dξ + B1 (X, X)(t, ξ) ≤ X(t, ξ)
0
with
B1 (X, Y )(t, ξ) =
1
(2π)3
Z
t
ν
2
e− 2 (t−s)|ξ| |ξ|
Z
W (s, η)V (s, ξ − η) dη ds.
R3
0
Define
ν
2
• Z [0] (t, ξ) := e− 2 t|ξ| X0 (ξ) +
Rt
0
ν
2
e− 2 (t−s)|ξ| G(s, ξ) dξ
• Z [n+1] (t, ξ) := Z [0] (t, ξ) + B0 (Z [n] , Z [n] )(t, ξ)
R
⃗ [0] := e−νt|ξ|2 U
⃗ 0 (ξ) + t e−ν(t−s)|ξ|2 (Id − ξ⊗ξ2 )F⃗ (s, ξ) ds
• U
|ξ|
0
⃗ [n+1] (t, ξ) := U
⃗ [0] (t, ξ) − B(U
⃗ [n] , U
⃗ [n] )(t, ξ) .
• U
Mild Solutions in Besov or Morrey Spaces
187
Again, we find that 0 ≤ Z [n] (t, ξ) ≤ Z [n+1] (t, ξ) ≤ X(t, ξ), so that Z [n] is pointwisely
convergent to a function Z [∞] ≤ X. By monotonous convergence, we have
Z [∞] = Z [0] + B0 (Z [∞] , Z [∞] ).
We have
1
2
sup ez− 2 z =
√
e
z≥0
and, for 0 ≤ s ≤ t
√
e
√
√
νt|ξ| − νs|ξ−η| − νs|η|
e
e
√
≤ e(
√
νt− νs)|ξ|
√
≤e
ν(t−s)|ξ|
.
By induction on n, we then find that we have the pointwise inequalities
⃗ [n] (t, ξ)| ≤
• |U
√
1√ − νt|ξ| [n]
e
Z (t, ξ)
18 e
⃗ [n+1] (t, ξ) − U
⃗ [n] (t, ξ)| ≤
• |U
√
1√ − νt|ξ|
e
(Z [n+1] (t, ξ)
18 e
− Z [n] (t, ξ)).
This gives the Gevrey estimate
⃗ (t, ξ)| ≤
|U
1 −√νt|ξ| [∞]
√ e
Z (t, ξ).
18 e
(8.13)
The study of the cheap equations
W = W [0] + B0 (W, W )
(8.14)
Z = Z [0] + B1 (Z, Z)
(8.15)
or
thus provides simple classes of solutions to the Navier–Stokes equations.
For instance, if M0 (ξ) and M1 (ξ) are non-negative measurable functions that satisfy the
following inequation
Z
1
M0 (ξ) +
M1 (ξ − η)M1 (η) dη ≤ M1 (ξ),
(2π)3 |ξ| R3
and if W [0] (t, ξ) ≤ M0 (ξ) for 0 ≤ t ≤ T , we get, by induction on n, that W [n] (t, ξ) ≤ M1 (ξ).
This means that, if M0 belongs to a lattice Banach space of functions E such that the
1
operator (Z, V ) 7→ |ξ|
(Z ∗ V ) is bounded on E and if ∥M0 ∥E is small enough, then the
Navier–Stokes equations with initial value ⃗u0 and force f⃗ such that their Fourier transforms
satisfy
Z T
1
⃗
|U0 | +
|F⃗ (s, ξ)| dt ≤
M0 (ξ)
18
0
have a global solution ⃗u with sup0<t<+T |F⃗u| ∈ E. Two simple instances can be found in
the litterature:
• the case where E = L2 (|ξ| dξ): if Z ∈ E, it means that Z =
1
Z
|ξ|1/2 0
with Z0 ∈ L2 ;
3/2,2
thus Z belongs to the Lorentz space L
(as a product of a function in L6,∞ by
2
3,1
a function in L ), thus Z ∗ Z belongs to L ⊂ L3,2 and |ξ|11/2 Z ∗ Z ∈ L2,2 = L2 .
1
As E = F(Ḣ 2 ), we find the theorem of Fujita and Kato for the Sobolev space Ḣ 1/2
(Theorem 7.4) .
188
The Navier–Stokes Problem in the 21st Century (2nd edition)
• the equality
Z
1
1
1
dη = C0
|ξ − η|2 |η|2
|ξ|
−2
allowed Le Jan and Sznitman [305] to consider the space E = F(ḂPM,∞
) defined by
Z ∈ E ⇔ Z ∈ L1loc and |ξ|2 Z ∈ L∞ .
If we look for local-in-time solutions, we must include the time variable in our estimations. For instance, since
e
−ν(t−s)|ξ|2
3/2
1
1
3
≤e
,
3/4
4
(ν(t − s)) |ξ|3/2
3
4
then, if M0 (ξ), M1 (ξ) and α(t) are non-negative measurable functions that satisfy on (0, T0 )
the following inequation
32
Z t
Z
3
e4
α(s)2
1
3
ds
M1 (ξ − η)M1 (η) dη ≤ α(t) M1 (ξ),
α(t)M0 (ξ) +
1
4
(2π)3 0 (ν(t − s)) 43
|ξ| 2 R3
and if W [0] (t, ξ) ≤ α(t)M0 (t), we get, by induction on n, that W [n] (t, ξ) ≤
α(t) M1 (ξ). Thus, if F is a lattice Banach space of functions such that the opera2
⃗ 0 (ξ)| ∈ F and
tor (Z, V ) 7→ |ξ|11/2 (Z ∗ V ) is bounded on F , if sup0<t<T t1/4 e−νt|ξ| |U
R
2
t
sup0<t<T t1/4 0 e−ν(t−s)|ξ| |F⃗ (s, ξ) ds ∈ F and if
2
⃗ 0 (ξ)|∥F + ∥ sup t1/4
lim ∥ sup t1/4 e−νt|ξ| |U
T0 →0
0<t<T0
0<t<T0
Z
t
2
e−ν(t−s)|ξ| |F⃗ (s, ξ) ds∥F = 0
0
2
∥ sup0<t<T t1/4 e−t|ξ| W 0 (ξ)∥F is small enough, then the Navier–Stokes equations with
initial value ⃗u0 and force f⃗ have a solution ⃗u on (0, T0 ) for T0 small enough, with
⃗ (t, ξ)| ∈ F . Let us look at our two simple instances:
sup0<t<T0 t1/4 |U
Cheap Navier–Stokes equation.
Let us look at our two simple instances:
• the case where E = L2 (|ξ| dξ) and F = L2 (|ξ|2 dξ): if Z ∈ F , it means that Z =
2
1
|ξ| Z0
6/5,2
with Z0 ∈ L ; thus Z belongs to the Lorentz space L
(as a product of a function
in L3,∞ by a function in L2 ) and V = |ξ|1/2 Z ∈ L3/2,2 , thus, writing
2
1
|Z ∗ Z| ≤
(|Z| ∗ |V |),
1/2
|ξ|
|ξ|
we get that |Z| ∗ |V | belongs to L6/5,2 ∗ L3/2,2 ⊂ L2,1 ⊂ L2,2 = L2 , so that we have
1
(Z ∗ Z) ∈ F . Moreover, if A > 0 and W0 ∈ E, we find that, for t > 0,
|ξ|1/2
2
1
|(νt)1/4 e−νt|ξ| W0 (ξ)| ≤ 1|ξ|≤A (νt) 4 W0 (ξ) + C1|ξ|>A
1
W0 (ξ)
|ξ|1/2
so that
2
∥ sup (νt)1/4 e−νt|ξ| |W0 (ξ)|∥F ≤ (νT )1/4 A1/2 ∥W0 ∥E + C∥1|ξ|>A W0 ∥E
0<t<T
Mild Solutions in Besov or Morrey Spaces
and
189
2
lim+ ∥ sup t1/4 e−νt|ξ| |W0 (ξ)|∥F = 0.
T →0
0<t<T
1
Similarly, if F0 ∈ L ((0, T ), E), t1/4 F0 ∈ L1 ((0, T ), F ), 0 < ϵ < 1 and A > 0, we have
Z t
2
(νt)1/4
e−ν(t−s)|ξ| |F0 (s, ξ)| ds
0
≤
1/4
ν
1−ϵ
Z
14 Z
t
1
s 4 |F0 (s, ξ)| ds
(1−ϵ)t
(1−ϵ)t
1|ξ|≤A |F0 (s, ξ|) dξ +
+(νt)
0
C
Z
t
1|ξ|>A
ϵ1/4
0
1
|F0 (s, ξ)| ds.
|ξ|1/2
We find that
sup (νt)1/4 ∥
t
Z
0<t<T0
2
e−ν(t−s)|ξ| |F0 (s, ξ)| ds∥F
0
41
Z
1
ν
sup
s 4 ∥F0 (s, ξ)∥F ds
1−ϵ
|I|≤ϵT I
Z T
Z T
C
+(νT0 )1/4 A1/2
∥F0 (s, ξ)∥E ds + 1/4
∥1|ξ|>A F0 (s, ξ)∥E ds.
ϵ
0
0
≤
and thus
lim
sup (νt)1/4 ∥
T0 →0 0<t<T0
Z
t
2
e−ν(t−s)|ξ| |F0 (s, ξ)| ds∥F = 0.
0
Thus, we find that if the initial value ⃗u0 belongs to the
√ homogeneous Sobolev space
Ḣ 1/2 and the force f⃗ belongs to L1 ((0, T ), Ḣ 1/2 and tf⃗ belongs to L1 ((0, T ), Ḣ 1 ),
then the Navier–Stokes problem has a local-in-time solution such that t1/4 ⃗u ∈
L∞ ((0, T0 ), Ḣ 1 ). This is the result of Fujita and Kato [185]. A similar proof willl
give us Gevrey regularity estimates for a data in the Sobolev space, a result first given
by Foias and Temam [181]:
Theorem 8.20.
If the initial value ⃗u√0 belongs to the homogeneous Sobolev space
Ḣ 1/2 and the
√
−νt∆ ⃗
1
1/2
1/4
−νt∆ ⃗
⃗
force f is such that e
f belongs to L ((0, T ), Ḣ
and t e
f belongs to
1
L1 ((0,
T
),
Ḣ
),
then
the
Navier–Stokes
problem
has
a
local-in-time
solution
such that
√
t1/4 e −νt∆ ⃗u ∈ L∞ ((0, T0 ), Ḣ 1 ).
• the case where E =
1
∞
|ξ|2 L (dξ)
Z
shows that
for t > 0,
1
(F
|ξ|1/2
and F =
1
L∞ (dξ):
|ξ|5/2
the equality
1
1
1
dη = C0 2
|ξ|
|ξ − η|5/2 |η|5/2
∗ F ) ⊂ F . Moreover, if A > 0 and W0 ∈ E, we write again that,
2
1
|(νt)1/4 e−t|ξ| W0 (ξ)| ≤ 1|ξ|≤A (νt) 4 W0 (ξ) + 1|ξ|>A
1
W0 (ξ)
|ξ|1/2
so that
2
∥ sup (νt)1/4 e−t|ξ| W0 (ξ)∥F ≤ (νT )1/4 A1/2 ∥W 0 ∥E + ∥1|ξ|>A W 0 ∥E
0<t<T
190
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
2
lim sup ∥ sup (νt)1/4 e−t|ξ| W 0 (ξ)∥F ≤ lim sup sup |ξ|2 |W0 (ξ)|.
T →0+
0<t<T
A→+∞ |ξ|>A
Similarly, if F0 ∈ L1 ((0, T ), E) and t1/4 F0 ∈ L1 ((0, T ), F ), we have, for every 0 < ϵ < 1
and every A > 0,
Z t
2
e−ν(t−s)|ξ| |F0 (s, ξ)| ds∥F
lim sup sup (νt)1/4 ∥
T →0+ 0<t<T0
≤
ν
1−ϵ
14
0
Z
|I|≤ϵT
1
s 4 ∥F0 (s, ξ)∥F ds +
sup
I
C
ϵ1/4
Z
T
∥1|ξ|>A F0 (s, ξ)∥E ds.
0
Hence, we get again the result of Le Jan and Sznitman [305]:
Theorem 8.21.
2
2
2
Let B̃PM,∞
be the closure of S in ḂPM,∞
. If the initial value ⃗u0 belongs to B̃PM,∞
1
2
1/4
and the force f⃗ is such that f⃗ belongs to L ((0, T ), B̃PM,∞ ) and t f⃗ belongs to
5/2
L1 ((0, T ), ḂPM,∞ ), then the Navier–Stokes problem has a local-in-time solution such
5/2
that t1/4 ⃗u belongs to L1 ((0, T0 ), ḂPM,∞ ).
8.9
Plane Waves
When dealing with Fourier transforms, we stated some results in terms of the L1 norm of
the Fourier transform. The results can be easily extended to the case of Fourier transforms
that are finite Borel measures. We shall write µ(dξ) for a locally finite measure and |µ|(dξ)
3
for
R its total variation. Let M(R ) be the space of finite Borel measure normed with ∥µ∥ =
|µ|(dξ). For µ1 , µ2 two finite Borel measures and σ a continuous bounded function on R3 ,
we have the easy estimates
∥µ1 ∗ µ2 ∥M ≤ ∥µ1 ∥M ∥µ2 ∥M ,
∥σµ1 ∥M ≤ ∥µ1 ∥M .
Thus, if ⃗u and ⃗v have their Fourier transform in (M)3 , which we write ⃗u, ⃗v ∈ FM, we find
that
1
∥eν(t−s)∆ P div(⃗u ⊗ ⃗v )∥F M ≤ C p
∥⃗u∥F M ∥⃗v ∥F M .
ν(t − s)
With these estimates one easily deals with the Navier–Stokes problem
(
∂t ⃗u = ∆⃗u − P div(⃗u ⊗ ⃗u)
⃗u(0, .) = ⃗u0 with div ⃗u0 = 0
(8.16)
when ⃗u0 ∈ FM and find a solution ⃗u ∈ L∞ ((0, T ), FM) with T = O( ∥⃗u0 ∥ν2
∞
1
) (where, to
FM
avoid measurability issues, L FM is defined as the dual space of L FC0 ).
When looking for global solutions, one may extend Lei-Lin’s theorem [306] (see Corollary
8.1):
Proposition 8.1.
There exists a positive constant ϵ0 (depending on ν) such that, if
Z
1
|F⃗u0 |(dξ) < ϵ0 ,
|ξ|
Mild Solutions in Besov or Morrey Spaces
191
and if div ⃗u0 = 0 then the Cauchy problem for the Navier–Stokes equations (8.16) with
1
initial value ⃗u0 has a global mild solution ⃗u such that |ξ|
F⃗u ∈ L∞
u ∈ L1 M.
t M and |ξ|F⃗
In 2008, Dinaburg and Sinai [153] discussed the special case of an initial value given by
a finite combination of plane waves:
⃗u0 =
N
X
eiωj ·x⃗aj with ωj · ⃗aj = 0 and ωj ̸= 0.
j=1
One easily checks that the solution of (8.16) can be written as a sum
X
⃗u(t, x) =
eiξ·x⃗aξ (t)
ξ∈Ξ
where Ξ is the set of finite sums ξ =
equations on ⃗aξ (t) can be written as
⃗aξ (t) =
N
X
2
δξ,ωj e−νt|ωj | ⃗aξj − i
j=1
j=1
kj ωj with N ∈ N∗ , kj ∈ N and ξ =
̸ 0. The
Z
X
X Z
η∈Ξ,ξ−η∈Ξ 1≤p,q≤3
0
t
t
2
e−ν(t−s)|ξ| ξ · ⃗aη (s) ⃗aξ−η (s) ds
0
η∈Ξ,ξ−η∈Ξ
X
+i
PN
2
e−ν(t−s)|ξ|
ξp ξq ξ
aη,p (s)aξ−η,q (s) ds.
|ξ|2
Existence of the solutions can thus be proved directly by a fixed-point estimate
P on the
set of coefficients (⃗aξ )ξ∈Ξ . Local existence is proved using the norm sup0<t<T ξ∈Ξ |⃗aξ (t)|
P
1
and global existence is proved for small data using the norm sup0<t ξ∈Ξ |ξ|
|⃗aξ (t)| +
R +∞ P
aξ (t)|. This can be done by Picard’s iteration and Banach contraction prinξ∈Ξ |ξ||⃗
0
ciple. Dinaburg and Sinai use the series method of Oseen (as discussed in Section 5.2), a
method emphasized by Sinai in his approach of the Navier–Stokes equations in the frequency
variable [438].
A very interesting feature of the series method is that it gives directly Gevrey regularity
estimates on Dinaburg and Sinai’s solutions. Indeed, let us use again the notations
P+∞ P of Section
5.2: the solution ⃗u is written as a sum of basic words w
⃗ ∈ W as ⃗u = n=1 w∈W
w
⃗ in
⃗
n
P+∞ P
the series method and as ⃗u = w
⃗ 1 + n=1 w∈V
w
⃗ in the Picard method. In both
⃗
n \Vn−1
methods, we have a control of norm of the n-th term in the sum (the norm is the norm of
the Banach space X where we solve the quadratic equation):
X
X
∥
w∥
⃗ X ≤ C0 ϵn and ∥
w∥
⃗ X ≤ C0 ϵn
w∈W
⃗
n
w∈V
⃗
n \Vn−1
where ϵ < 1. Moreover, when considering the problem with plane waves as initial data, we
have that Fu0 is supported in the ball B(0, R) with R = max1≤j≤N (|ωj |). If w
⃗ ∈ Wn , then
F(w)
⃗ is supported in B(0, nR); if w
⃗ ∈ Vn , then F(w)
⃗ is supported in B(0, 2n R). It means
that the Picard estimates seem to give at best polyniomally decaying spectral estimates (in
− ln(1/ϵ)
ln 2
)) while the Oseen estimates give exponentially decaying spectral estimates
O( |ξ|
R
|ξ|
(in O(e− ln(1/ϵ) R )4 .
4 One may, of course, recover Gevrey estimates through the Picard method by dealing with the corresponding Gevrey norm when studying the fixed-point problem.
Chapter 9
The Space BMO−1 and the Koch and Tataru
Theorem
9.1
The Koch and Tataru Theorem
The Koch and Tataru theorem [266] deals with the largest space where to search for
mild solutions, which is well fitted to the symmetries of the Navier–Stokes equations.
We recall that we have rewritten the Navier–Stokes equations

⃗ u + f⃗ − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
(9.1)
div ⃗u = 0

⃗u|t=0 = ⃗u0
into
⃗u = Wνt ∗ ⃗u0 −
Z tX
3
∂j O(ν(t − s)) :: f⃗ ∗ ∂j G + uj ⃗u ds
(9.2)
0 j=1
In order to give some meaning to the integral in Equation (9.2), we shall suppose that ⃗u
is locally square integrable on [0, T ) × R3 . Moreover, we shall suppose that the estimates are
uniform in x (in order to use translation invariance of the equations) and invariant under
the scaling ⃗u → λ⃗u(λ2 t, λx) (in order to use the scaling invariance of the equations). This
gives:
ZZ
1
sup
u(s, y)|2 ds dy < +∞.
√ |⃗
3/2
x∈R3 ,0<t<T t
(0,t)×B(x, t)
Koch and Tataru characterized then the associated space of initial values for the Navier–
Stokes equations [266, 313] as derivatives of functions in the BM O space of John and
Nirenberg [247] or in the local bmo space of Goldberg [214]:
Proposition 9.1. For a measurable function F on (0, T ) × R3 (with T ∈ (0, +∞]), define
s
ZZ
1
2
∥F ∥XT =
sup
√ |F (s, y)| ds dy.
t3/2
x∈R3 , 0<t<T
(0,t)×B(x, t)
Then, for T < +∞ and u0 ∈ S ′ (R3 ), we have
∥Wνt ∗ u0 ∥XT < +∞ ⇔ ∃(f0 , . . . , f3 ) ∈ (bmo(R3 ))4 u0 = f0 +
3
X
∂j fj
j=1
and the norm ∥Wνt ∗ u0 ∥XT is equivalent to the infimum of
P3
decompositions u0 = f0 + j=1 ∂j fj .
DOI: 10.1201/9781003042594-9
P3
j=0
∥fj ∥bmo over all possible
192
The Space BMO−1 and the Koch and Tataru Theorem
193
Simarly, we have
∥Wνt ∗ u0 ∥X∞ < +∞ ⇔ ∃(f1 , . . . , f3 ) ∈ (BM O(R3 ))3 u0 =
3
X
∂j fj
j=1
and the norm ∥Wνt ∗ u0 ∥X∞ is equivalent to the infimum of
P3
decompositions u0 = j=1 ∂j fj .
P3
j=1
∥fj ∥BM O over all possible
Koch and Tataru coined bmo−1 the space of distributions u0 such that ∥Wνt ∗ u0 ∥XT <
+∞ for finite T , and BM O−1 the space of distributions u0 such that ∥Wνt ∗ u0 ∥X∞ < +∞.
We define the norm of u0 in BM O−1 as the infimum of
P3
decompositions u0 = j=1 ∂j fj . In particular, we have
s
∥Wνt ∗ u0 ∥X∞ ≤ C
P3
j=1
∥fj ∥BM O over all possible
ln(e + ν1 )
∥u0 ∥BM O−1 .
ν
Koch and Tataru’s theorem is then the following one:
Theorem 9.1.
The bilinear operator
⃗ =
B(F⃗ , G)
Z tX
3
⃗ ds =
∂j O(ν(t − s)) :: Fj G
0 j=1
Z
t
⃗ ds
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is bounded on the space
ET = {F⃗ / ∥F⃗ ∥XT < +∞ and
sup
√
t∥F⃗ (t, .)∥∞ < +∞}
0<t<T
for every T ∈ (0, +∞].
In order to prove Theorem 9.1, we follow the strategy of Auscher and Frey [11]. We
begin with the following lemma:
Lemma 9.1.
R
⃗ belong to E∞ , then A(F⃗ , G)
⃗ = +∞ Wνs ∗ (F⃗ ⊗ G)
⃗ ds belongs to (BM O)9 and
If F⃗ and G
0
⃗ BM O ≤ Cν ∥F⃗ ∥E ∥G∥
⃗ E .
∥A(F⃗ , G)∥
∞
∞
R +∞
Proof. We want to estimate the BM O norm of H(x) = 0 Wνs ∗ h(s, .) ds, where h
satisfies
ZZ
1
∥h∥(1) = sup
√ |h(s, y)| ds dy < +∞
3/2
x∈R3 , 0<t t
(0,t)×B(x, t)
and
sup t∥h(t, .)∥∞ < +∞.
t>0
194
The Navier–Stokes Problem in the 21st Century (2nd edition)
Using the fact that BM O is the dual space of the Hardy space H1 [313, 448], we must prove
that, if A∞ is the set of atoms for H1 (i.e.,Ra ∈ A∞ if for some r > 0 and x0 ∈ R3 , a is
supported in B(x, r), ∥a∥∞ ≤ |B(x10 ,r)| , and a dx = 0), then
Z
sup | a(x)H(x) dx| ≤ Cν (∥h∥(1) + sup t∥h(t, .)∥∞ ).
t>0
a∈A∞
P3
If a is an atom (associated to a ball B(x0 , r)), then it can be written as a = i=1 ∂i αi ,
with αi supported in x0 + [−r, r]3 and ∥αj ∥∞ ≤ Cr−2 (and thus ∥αj ∥1 ≤ 8Cr). This gives
Z
Z +∞
Z +∞
⃗ νs ∗ h(s, .)∥∞ ds
| a(x)
Wνs ∗ h(s, .) ds dx| ≤24C
r∥∇W
r2
r2
Z +∞
dt
√
≤C ′ r(sup t∥h(t, .)∥∞ )
2
t
νt
t>0
r
′
C
= √ sup t∥h(t, .)∥∞
ν t>0
On the other hand, we write h = h1 + h2 , where h1 (s, y) = 1B(x0 ,3r) (y)h(s, y); we have
Z
|
Z
r2
Z
r2
Wνs ∗ h1 (s, .) ds dx| ≤∥a∥∞
a(x)
0
∥h1 (s, .)∥1 ds
0
≤C∥a∥∞ r3 ∥h∥(1)
≤C ′ ∥h∥(1)
while
Z
|
Z
r2
Wνs ∗ h2 (s, .) ds dx|
a(x)
0
Z
Z
r2
Z
≤C
νs
|h(s, y)| dy ds dx
|x − y|5
|a(x)|
B(x0 ,r)
|x0 −y|>3r
0
≤C ′ ν∥a∥1 r2
Z
r2
Z
Z
r2
|x0 −y|>3r
0
≤ C ′′ νr−3
X
k∈Z3 ,k̸=0
1
|k|5
1
|h(s, y)| ds dy
|x0 − y|5
Z
|h(s, y)| ds dy
0
x0 +kr+[−r/2,r/2]3
≤C ′′′ ν∥h∥(1) .
The lemma is proved.
Proof of Theorem 9.1:
A first remark is that when F belongs to ET and when G is defined on (0, +∞) × R3
by G(t, x) = 1(0,T ) (t)F (t, x), then G belongs to E∞ . Indeed, for t < T , we have
√
RR
√ |G(s, y)|2 ds dy ≤ ∥F ∥2 t3/2 ; for t ≥ T , we cover the ball B(x, t) by a
XT
(0,t)×B(x, t)
√
3/2
finite number Nt of balls B(xi , T ) with Nt = O( Tt
) and we write
ZZ
Z
Z
X
2
|F (s, y)|2 ds dy
√
√ |G(s, y)| ds dy ≤
(0,t)×B(x, t)
1≤i≤Nt
(0,T )×B(xi , T )
≤Nt ∥F ∥2XT T 3/2
≤C∥F ∥2XT t3/2 .
The Space BMO−1 and the Koch and Tataru Theorem
As
⃗ =
B(F⃗ , G)
195
t
Z
⃗ ds
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
⃗ for s < T , we find that it is enough to prove
only involves, for t < T , the values of F⃗ and G
the theorem for T = +∞.
⃗ in E∞ and we fix T > 0 and x0 ∈ R3 and we want to prove that,
Now, we take F⃗ and G
for a constant C which does not depend on T nor on x0 , we have
⃗
⃗ E T −1/2
|B(F⃗ , G)(T,
x0 )| ≤ C∥F⃗ ∥E∞ ∥G∥
∞
and
T
Z
0
Z
⃗
⃗ E T 3/2 .
|B(F⃗ , G)(t,
x)| dt dx ≤ ∥F⃗ ∥E∞ ∥G∥
∞
√
B(x0 , T )
Let χx0 ,T = 1(0,T ) (t)1B(x0 ,5√T ) (x) and ψx0 ,T = 1(0,T ) (t)(1 − 1B(x0 ,5√T ) (x)) . For 0 < t ≤ T ,
we have
⃗
⃗
⃗
B(F⃗ , G)(t,
x) = B(χx0 ,T F⃗ , G)(t,
x) + B(ψx0 ,T F⃗ , G)(t,
x).
√
If moreover x ∈ B(x0 , T ), we have
Z tZ
⃗
|B(ψx0 ,T F⃗ , G)(t,
x)| ≤C
0
≤C ′
T
Z
0
≤C
′
X
k∈Z3 ,k̸=0
1
|k|4 T 2
Z
T
ds dy
⃗ y)| p
ψx0 ,T (s, y)|F⃗ (s, y)| |G(s,
( ν(t − s) + |x − y|)4
Z
1
⃗ y)| ds dy
|F⃗ (s, y)| |G(s,
√
4
|y−x0 |>4 T |x − y|
Z
√
√
√
y∈x0 +k T +[− T /2, T /2]3
0
X
≤C ′′ T −1//2
k∈Z3 ,k̸=0
⃗ y)| ds dy
|F⃗ (s, y)| |G(s,
1 ⃗
⃗ X
∥F ∥X∞ ∥G∥
∞
|k|4
and thus
⃗
⃗ E T −1/2
|B(ψx0 ,T F⃗ , G)(T,
x0 )| ≤ C∥F⃗ ∥E∞ ∥G∥
∞
and
Z
0
T
Z
√
B(x0 , T )
⃗
⃗ E T 3/2 .
|B(ψx0 ,T F⃗ , G)(t,
x)|2 dt dx ≤ ∥F⃗ ∥E∞ ∥G∥
∞
⃗ we follow the strategy of Auscher and Frey [11] and
In order to estimate B(χx0 ,T F⃗ , G),
⃗
⃗
decompose B(χx0 ,T F , G) into
⃗ = A1 (χx ,T F⃗ , G)
⃗ + A2 (χx ,T F⃗ , G)
⃗ + A3 (χx ,T F⃗ , G)
⃗
B(χx0 ,T F⃗ , G)
0
0
0
with
⃗ =
A1 (χx0 ,T F⃗ , G)
Z
t
⃗ ds
(Wν(t−s) − Wν(t+s) ) ∗ P div(χx0 ,T F⃗ ⊗ G)
0
⃗ =
A2 (χx0 ,T F⃗ , G)
+∞
Z
⃗ ds
Wν(t+s) ∗ P div(χx0 ,T F⃗ ⊗ G)
0
⃗ =−
A3 (χx0 ,T F⃗ , G)
Z
t
+∞
⃗ ds.
Wν(t+s) ∗ P div(χx0 ,T F⃗ ⊗ G)
196
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ as
We further rewrite A1 (χx0 ,T F⃗ , G)
Z 1Z t
⃗
⃗
⃗
⃗
A1 (χx0 ,T F , G) = −2ν
∆Wν(t−s) ∗ W2νθs ∗ P div(χx0 ,T F ⊗ G) s ds dθ
0
0
or as
⃗ = −2ν
A1 (χx0 ,T F⃗ , G)
Z
1
⃗ dθ
A4 (A5,θ (χx0 ,T F⃗ , G))
0
with
t
Z
⃗ =
A4 (H)
⃗ ds
∆Wν(t−s) ∗ H
0
and
⃗
⃗
A5,θ (χx0 ,T F⃗ , G)(s,
.) = sW2νθs ∗ P div(χx0 ,T F⃗ ⊗ G).
⃗ as
We rewrite A2 (χx0 ,T F⃗ , G)
⃗ = Wνt ∗ P div A6 (χx ,T F⃗ , G)
⃗
A2 (χx0 ,T F⃗ , G)
0
with
⃗ =
A6 (χx0 ,T F⃗ , G)
+∞
Z
⃗ ds.
Wνs ∗ (χx0 ,T F⃗ ⊗ G)
0
We thus get the decomposition
Z 1
⃗
⃗
⃗ dθ + Wνt ∗ P div A6 (χx ,T F⃗ , G)
⃗
B(χx0 ,T F , G) = − 2ν
A4 (A5,θ (χx0 ,T F⃗ , G))
0
0
⃗
+ A3 (χx0 ,T F⃗ , G).
The last two terms are easily estimated:
⃗
ˆ A3 (χx ,T F⃗ , G)(T,
x0 ) = 0
0
ˆ we have
⃗ ∞ ≤CT −1/2 ∥A6 (χx ,T F⃗ , G)∥
⃗ BM O
∥WνT ∗ P div A6 (χx0 ,T F⃗ , G)∥
0
⃗ E .
≤Cν T −1/2 ∥F⃗ ∥E ∥G∥
∞
ˆ we have similarly
Z TZ
0
√
B(x0 , T )
∞
⃗ 2 dt dx
|WνT ∗ P divA6 (χx0 ,T F⃗ , G)|
⃗ 2X
≤ CT 3/2 ∥WνT ∗ P div A6 (χx0 ,T F⃗ , G)∥
∞
⃗ 2BM O
≤ C ′ T 3/2 ∥A6 (χx ,T F⃗ , G)∥
0
⃗ 2E .
≤ C ∥F⃗ ∥2E∞ ∥G∥
∞
√
⃗ .)∥∞ . We have
ˆ Let α(t) = ∥χx0 ,T (t, .)F⃗ (t, .)∥2 and β = supt>0 t∥G(t,
Z TZ
⃗
|A3 (χx0 ,T F⃗ , G)(t,
x)|2 dt dx
√
′′
0
B(x0 , T )
Z
≤
+∞Z
⃗
|A3 (χx0 ,T F⃗ , G)(t,
x)|2 dt dx
0
Z +∞
⃗
≤
(
∥Wν(t+s) ∗ P div(χx0 ,T F⃗ ⊗ G)(s,
.)∥2 ds)2 dt
0
t
Z +∞ Z +∞
1
⃗ .)∥2 ds)2 dt.
p
≤C
(
∥χx0 ,T F⃗ ⊗ G(s,
ν(t + s)
0
t
Z
+∞
The Space BMO−1 and the Koch and Tataru Theorem
Z
Z
1
C +∞ +∞
√
(
≤
√ βα(s) ds)2 dt
ν 0
t+s s
t
Z
Z
Z +∞
C +∞ +∞ 1 2 2
1
√ β α (s) ds)(
√ dτ ) dt
≤
(
ν 0
s
(t + τ ) τ
t
t
Z
Z +∞
1
C ′ +∞ 1
√ (
√ β 2 α2 (s) ds) dt
=
ν 0
s
t t
′ Z +∞
C
β 2 α2 (s) ds
=2
ν 0
√
C ′′ 3/2 ⃗ 2
⃗ ∞ )2 .
≤
T ∥F ∥X∞ (sup t∥G∥
ν
t>0
197
R
⃗ = t (Wν(t−s) −
Thus, we are left with the estimation of the first term, i.e., A1 (χx0 ,T F⃗ , G)
0
R
⃗ ds = −2ν 1 A4 (A5,θ (χx ,T F⃗ , G))
⃗ dθ.
Wν(t+s) ) ∗ P div(χx0 ,T F⃗ ⊗ G)
0
0
√
⃗ .)∥∞ . We use the maximal
Let again α(t) = ∥χx0 ,T (t, .)F⃗ (t, .)∥2 and β = supt>0 t∥G(t,
regularity of the heat kernel [313] to get
⃗ L2 L2 ((0,+∞)×R3 ) ≤ C∥H∥
⃗ L2 L2 ((0,+∞)×R3 )
∥A4 (H)∥
and thus
Z
∥
1
⃗ dθ∥L2 L2 ≤
A4 (A5,θ (χx0 ,T F⃗ , G))
1
Z
0
⃗ L2 L2 dθ
∥A4 (A5,θ (χx0 ,T F⃗ , G))||
0
1
Z
⃗ L2 L2 dθ
∥A5,θ (χx0 ,T F⃗ , G)||
≤C
0
≤C
′
Z
0
1
Z
0
+∞
1/2
s 2 β2
α (s) ds
dθ
νθ
s
=2C ∥χx0 ,T F⃗ ∥L2 L2 β
√
⃗ ∞.
≤C ′′ T 3/4 ∥F⃗ ∥X sup t∥G∥
′
∞
t>0
Finally, we write
⃗
|A1 (χx0 ,T F⃗ , G)(T,
x0 )| ≤C
Z
0
′
≤
C
T2
Z
T /2
Z
T
Z
χx0 ,T (s, y)
⃗ y)| ds dy
p
|F⃗ (s, y)||G(s,
( ν(T − s) + |x0 − y|)4
⃗ y)| ds dy
χx0 ,T (s, y)|F⃗ (s, y)||G(s,
0
Z
T
Z
ds dy
⃗ .)∥∞ p
∥F⃗ (s, .)∥∞ ∥G(s,
(
ν(T
−
s) + |x0 − y|)4
T /2
√
√
⃗ .)∥∞ ))
⃗ X + √1 (sup t∥F⃗ (t, .)∥∞ )(sup t∥G(t,
≤ C ′′ T −1/2 (∥F⃗ ∥X∞ ∥G∥
∞
ν t>0
t>0
+C
The theorem is proved.
□
198
The Navier–Stokes Problem in the 21st Century (2nd edition)
Theorem 9.1 gives the following theorem on Navier–Stokes equations:
Koch and Tataru theorem
Theorem 9.2.
Let, for T ∈ (0, +∞],
∥h∥ET = sup
√
t∥h(t, .)∥∞ + sup
sup
1
sZ Z
t
3/4
0<t<T x0 ∈R3 t
0<t<T
√ |h(s, y)|
B(x0 , t)
0
2
dy ds
and
∥h∥FT = sup t∥h(t, .)∥∞ + sup
sup
0<t<T x0 ∈R3 t
0<t<T
1
3/2
Z tZ
0
√ |h(s, y)|, dy ds.
B(x0 , t)
There exist two constants ϵ0 and C0 which do not depend on T (but depend on ν) such
that if
• f⃗ = div F with F ∈ FT and ∥F ∥FT < ϵ0
• ⃗u0 ∈ bmo−1 and div ⃗u0 = 0
• ∥10<t<T Wνt ∗ ⃗u0 ∥ET < ϵ0
then the Navier–Stokes equations
⃗ u − ∇p
⃗ + f⃗
∂t ⃗u = ν∆⃗u − ⃗u.∇⃗
with div ⃗u = 0 and ⃗u(0, .) = ⃗u0 have a unique solution ⃗u on (0, T ) such that ⃗u ∈ ET
and ∥⃗u∥ET ≤ C0 ϵ0 . This solution satisfies
∥⃗u∥ET ≤ C0 (∥10<t<T Wνt ∗ ⃗u0 ∥ET + ∥F ∥FT ).
Proof. As usual, by Picard’s iterative scheme, using the estimate given by Theorem 9.1.
The Koch and Tataru theorem gives criteria for local or global existence:
ˆ if ⃗u0 ∈ bmo−1 and Wνt ∗ ⃗u0 ∈ E∞ , then ⃗u0 belongs to the smaller space BM O−1 and
we have ∥Wνt ∗ ⃗u0 ∥E∞ ≈ ∥⃗u0 ∥BM O−1 .
ˆ Hence, if ∥F ∥F∞ < ϵ0 , then we have global existence of the solution of the Navier–
Stokes equations provided that ∥⃗u0 ∥BM O−1 is small enough.
ˆ If ⃗u0 belongs to the closure of test functions (or more generally of bounded functions)
in bmo−1 , then we have
lim ∥10<t<T Wνt ∗ ⃗u0 ∥ET = 0.
T →0
ˆ Hence, if ∥F ∥FT < ϵ0 , we have local existence (for some T0 ∈ (0, T ]) of a solution of
the Navier–Stokes equations provided that ⃗u0 is regular enough (i.e., belongs to the
closure of bounded functions in bmo−1 ).
The Space BMO−1 and the Koch and Tataru Theorem
9.2
199
A Variation on the Koch and Tataru Theorem
From Theorem 9.2, we find existence of a global mild solution to the Cauchy problem
for the Navier–Stokes equations if the initial value ⃗u0 is small enough in BM O−1 and if,
for the forcing term f⃗ = div F , F is small enough in F∞ . However, one easily sees that we
may allow more general forces. For instance, we have:
Theorem 9.3.
Let 2 < p ≤ 5.
A) The bilinear operator
⃗ =
B(F⃗ , G)
Z tX
3
⃗ ds =
∂j O(ν(t − s)) :: Fj G
Z
0 j=1
t
⃗ ds
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is bounded on the space E∞ + Mp,5
2 , where
s
ZZ
√
1
|F⃗ (s, y)|2 ds dy + sup t∥F⃗ (t, .)∥∞
∥F⃗ ∥E∞ = sup
√
3/2
t
0<t
x∈R3 , 0<t
(0,t)×B(x, t)
and
∥F⃗ ∥Mp,5 =
2
sup
r
x∈R3 , t∈R,r>0
5
1− p
!1/p
ZZ
|F⃗ (s, y)|p ds dy
s>0, (s,y)∈(t−r 2 ,t+r 2 )×B(x,r)
Rt
B) If ⃗u0 is small enough in BM O(−1) (and is divergence free) and if 0 e(t−s)∆ P div F ds
is small enough in E∞ + Mp,5
(in particular, if F = F1 + F2 , where F1 is small in
2
p/2,5/2
F∞ and F2 is small in the Morrey space M2
), then the Navier–Stokes equations
⃗ u − ∇p
⃗ + div F
∂t ⃗u = ν∆⃗u − ⃗u.∇⃗
with div ⃗u = 0 and ⃗u(0, .) = ⃗u0 have a global mild solution in E∞ + Mp,5
2 .
Proof. A) We already know that B is bounded from E∞ × E∞ to E∞ (Theorem 9.1).
Moreover, we know that, by Hedberg’s inequality (Lemma 5.3), that the parabolic Riesz
σ,s
potentialf 7→ K2,1 ∗t,x f is bounded from Mρ,r
for 1 < ρ ≤ r < 5, 1s = 1r − 15 and
2 to M2
p/2,5/2
s
σ = r ρ. In particular, it is bounded from M2
to Mp,5
2 ; as
⃗ ≤ CK2,1 ∗t,x (|F⃗ | |G|),
⃗
|B(F⃗ , G)|
p,5
B is bounded from Mp,5
to Mp,5
2 × M2
2 . Finally, we check that (F, ) 7→ K2,1 ∗t,x (F G)
p,5
2,∞ ∞
3
is bounded from L L × M2 to Mp,5
2 . Let (t0 , x0 ) ∈ R × R and r < 0. We want to
p p
2
2
estimate the L L norm of K2,1 ∗t,x (F G) on (t0 −r , t0 +r )×B(x0 , r); we write G = G1 +G2
with G1 = 1(t0 −16r2 ,t0 +16r2 )×B(x0 ,4r) G. We have
Z
∥K2,1 ∗t,x (F G1 )∥Lp (dx) ≤ ∥K2,1 (t − s, .)∥L1 (dx) ∥F (s, .)∥L∞ (dx) ∥G1 (s, .)∥Lp (dx) ds.
As ∥A ∗ (BC)∥p ≤ Cp ∥A∥L2,∞ ∥B∥L2,∞ ∥C∥p , we find that
5
∥K2,1 ∗t,x (F G1 )∥Lp Lp ≤ C∥F ∥L2,∞ L∞ ∥G1 ∥Lp Lp ≤ C ′ r p −1 ∥F ∥L2,∞ L∞ ∥G∥Mp,5 .
2
200
The Navier–Stokes Problem in the 21st Century (2nd edition)
p
On the other hand, L2,∞ L∞ ⊂ M2p−1
2
,5
so that ∥F G∥M1,5/2 ≤ C∥F ∥L2,∞ L∞ ∥G∥Mp,5 . We
2
2
2
then write, for (t, x) ∈ (t0 − r , t0 + r ) × B(x0 , r),
|K2,1 ∗t,x (F G2 )(t, x)|
+∞ Z Z
X
≤C
√
j=0
≤C ′
+∞
X
j=0
4j r≤
|t−0−s|+|x0
−y|<4j+1
1
p
|F G(s, y)| ds dy
( |t − s| + |x − y|)4
1
∥F G∥M1,5/2 (4j r)3
2
(4j r)4
1
≤C ′′ ∥F ∥L2,∞ L∞ ∥G∥Mp,5 ;
2
r
we et that
1
∥1(t0 −r2 ,t0 +r2 )×B(x0 ,r) K2,1 ∗t,x (F G2 )∥LpLp ≤ C∥F ∥L2,∞ L∞ ∥G∥Mp,5 r5/p .
2
r
p,5
p,5
p,5
Thus, B is bounded from E∞ × Mp,5
2 to M2 and from M2 × E∞ to M2 .
B) is then a direct consequence of A).
Remark: the assumptions of Theorem 9.3 allows one to consider singular
tensors F for
√
the Navier–Stokes problem; for instance, if F belongs to Lp Ẇ −1,q = ( −∆)(Lp Lq ) with
Rt
min(p,q),5
2 < p < +∞ and p2 + 3q = 1, then 0 e(t−s)∆ P div F ds ∈ Lp Lq ⊂ M2
.
2,5
2,5
We remark that, for 2 < p ≤ 5, E∞ + Mp,5
2 ⊂ M2 . However, M2 does not play the
role of maximal space where to find mild solutions for the Navier–tokes equations, as B is
2,5
2,5
not bounded from M2,5
2 ×M2 to M2 [323]. On the other hand, some mild solutions don’t
satisfy the assumptions of Theorem 9.3 and the proof of the theorem cannot be applied to
them. Consider for instance the space L2 FL1 ; we discussed mild solutions in this space in
Theorem 8.16 and the Corolllary 8.1, corresponding to theorems of Cannone and Wu [86]
and of Lei and Lin [306]. The bilinear operator B is not bounded from E∞ × L2 FL1 to
L2 FL1 nor to E∞ . Thus, we need a new space where to work.
We have a similar problem with the space V 2,1 (R × R3 ) we considered in Corollary 5.3.
2,1
We have, for 2 < p, Mp,5
(R × R3 ) ⊂ M2,5
2 ⊂V
2 . The idea developed by Lemarié-Rieusset
in [323] is to consider a broader space than E∞ , namely the space E∞,q , for 5 < q < +∞,
defined as the space of vector fields ⃗u on (0, +∞) × R3 such that
sup
T >0,x∈R3
and
sup
T >0,x∈R3
T −3/4 ∥1(0,T )×B(x,√T ) ⃗u∥L2t,x < +∞
5
1
T − 2q + 2 ∥1(T /2,T )×B(x,√T ) ⃗u∥
2q
Ṁ25
,q
< +∞.
We have E∞ ⊂ E∞,q ⊂ M2,5
2 . Lemarié-Rieusset’s theorem is then:
Theorem 9.4. Let 5 < q < +∞.
A) The bilinear operator
⃗ =
B(F⃗ , G)
Z tX
3
0 j=1
⃗ ds =
∂j O(ν(t − s)) :: Fj G
Z
t
⃗ ds
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
2,5
is bounded from E∞,q × M2,5
2 to E∞,q and from M2 × E∞,q to E∞,q .
The Space BMO−1 and the Koch and Tataru Theorem
201
B) If ⃗u0 is small enough in BM O(−1) (and is divergence free) and if
R t (t−s)∆
e
P div F ds is small enough in E∞,q + V 2,1 (R × R3 ), then the Navier–
0
Stokes equations
⃗ u − ∇p
⃗ + div F,
∂t ⃗u = ν∆⃗u − ⃗u.∇⃗
div ⃗u = 0,
, ⃗u(0, .) = ⃗u0
have a global mild solution in E∞,q + V 2,1 (R × R3 ).
C) Let 2 < p ≤ 5. If ⃗u0 is small enough in BM O(−1) (and is divergence free)
Rt
2
1
and if 0 e(t−s)∆ P div F ds is small enough in E∞,q + Mp,5
2 + L FL , then the
Navier–Stokes equations
⃗ u − ∇p
⃗ + div F,
∂t ⃗u = ν∆⃗u − ⃗u.∇⃗
div ⃗u = 0,
, ⃗u(0, .) = ⃗u0
2
1
have a global mild solution in E∞,q + Mp,5
2 + L FL .
9.3
Q-spaces
The origin of Q-spaces lies in the study of certain classes of holomorphic functions on the
disk that are invariant under Möbius transforms, i.e., under bi-holomorphic automorphisms
of the disk (see Xiao [506] for a survey). Those classes are connected to the notion of
Carleson measures and, as such, appear as generalizations of the class BM OA.
Then the notion was exported from the setting of holomorphic functions in complex
analysis to the setting of real variable harmonic analysis on Rd . For 0 < α < 1, the space
Qα (R3 ) is defined as the space of measurable functions such that
∥f ∥Qα =
1
sup
r>0,x∈R3
r3−2α
ZZ
B(x,r)×B(x,r)
|f (y) − f (z)|2
dy dz
|y − z|3+2α
!1/2
< +∞.
This definition is reminiscent both of the characterization of homogeneous Sobolev spaces1
ZZ
|f (y) − f (z)|2
α
f ∈ Ḣ ⇔
dy dz < +∞
|y − z|3+2α
and of Morrey–Campanato spaces
2,−α
f ∈L
⇔
sup
x0 ∈R3 , r>0
1
r3−2α
Z
|f (y) − mB(x0 ,r) f |2 dy < +∞.
B(x0 ,r)
2,−α
In the limit case α = 0, we have L2,0 = BM O; for 0 < α < 1, we find
if
R that f ∈ L
1
and only if f = g + C, where C is a constant (C = limr→+∞ |B(0,r)| B(0,r) f (x) dx) and g
belongs to the Morrey space Ṁ 2,3/α .
As a matter of fact, it turns out that, for 0 < α < 1,
f ∈ Qα ⇔ (−∆)α/2 f ∈ Ṁ 2,3/α .
1 When
the definition of Ḣ α is performed modulo constant functions.
202
The Navier–Stokes Problem in the 21st Century (2nd edition)
This was proved by Xiao [507] (a similar result was proved earlier by May [353], following
an idea of Meyer). In particular, we have Qα ⊂ BM O. In order to adapt Koch and Tataru’s
theorem to the setting of Q-spaces, one considers the derivatives of functions in Qα :
Q−1,α = (−∆)1/2 Qα = (−∆)
1−α
2
Ṁ 2,3/α .
Proposition 9.1 then becomes:
Proposition 9.2.
For a measurable function F on (0, +∞) × R3 ), define
s
α
ZZ
t
1
2
|F
(s,
y)|
ds dy.
∥F ∥Xα = sup
√
3/2
s
t
x∈R3 , 0<t
(0,t)×B(x, t)
Then, we have the equivalence:
∥Wνt ∗ u0 ∥Xα < +∞ ⇔ ∃(f1 , . . . , f3 ) ∈ (Qα (R3 ))3 u0 =
3
X
∂j fj
j=1
and the norm ∥Wνt ∗ u0 ∥Xα is equivalent to the infimum of
P3
decompositions u0 = j=1 ∂j fj .
P3
j=1
∥fj ∥Qα over all possible
Theorem 9.1 was then adapted by May [353] and Xiao [507] to the setting of Q−1,α
spaces:
Theorem 9.5.
The bilinear operator
⃗ =
B(F⃗ , G)
Z tX
3
⃗ ds =
∂j O(ν(t − s)) :: Fj G
0 j=1
Z
t
⃗ ds
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is bounded on the space
√
Eα = {F⃗ / ∥F⃗ ∥Xα < +∞ and sup t∥F⃗ (t, .)∥∞ < +∞}.
0<t
Proof. As the space Eα is clearly embedded in the space E∞ of Theorem 9.1, we may use
⃗
the results of the √
proof of Theorem 9.1. For T > 0, x0 ∈ R3 , we decompose again B(F⃗ , G)
on (0, T ) × B(x0 , T ) into
⃗
⃗
⃗
B(F⃗ , G)(t,
x) = B(χx0 ,T F⃗ , G)(t,
x) + B(ψx0 ,T F⃗ , G)(t,
x)
√
and we already know that, on (0, T ) × B(x0 , T ),
⃗
⃗ X ≤ CT −1/2 ∥F⃗ ∥X ∥G∥
⃗ X
|B(ψx0 ,T F⃗ , G)(t,
x)| ≤ CT −1/2 ∥F⃗ ∥X∞ ∥G∥
∞
α
α
so that
Z
0
T
Z
√
B(x0 , T )
⃗
|B(ψx0 ,T F⃗ , G)(t,
x)(t, x)|2
α
T
⃗ X .
dt dx ≤ CT 3/2 ∥F⃗ ∥Xα ∥G∥
α
t
⃗
Thus, we are left with the estimation of I(t, x) = |B(χx0 ,T F⃗ , G)(t,
x)(t, x)|. We write
Z tZ
1
⃗ y)| ds dy.
p
|I(t, x)| ≤ C
χx0 ,T (s, y)|F⃗ (s, y)| |G(s,
(9.3)
( ν(t − s) + |x − y|)4
0
The Space BMO−1 and the Koch and Tataru Theorem
203
In particular,
α
T
ds dy
√
t
0
B(x0 ,5 T )
Z T Z
√
√
1
ds dy
⃗ .)∥∞ )
p
+C (sup t∥F⃗ (t, .)∥∞ )(sup t∥G(t,
T t>0
t>0
(
ν(T
−
s) + |x0 − y|)4
T /2
√
√
−1/2
⃗ X + (sup t∥F⃗ (t, .)∥∞ )(sup t∥G(t,
⃗ .)∥∞ ))
≤Cν T
(∥F⃗ ∥Xα ∥G∥
α
1
I(T, x0 ) ≤C
(νT )2
Z
T /2
Z
⃗ y)|
|F⃗ (s, y)| |G(s,
t>0
t>0
Moreover,
H(t, x) = χx0 ,T (t, x)
belongs to L2 L2 and
ZZ
|F⃗ (t, x)| √ ⃗
t|G(t, x)|
α
t2
3
H(t, x)2 dt dx ≤ CT 2 −α ∥F⃗ ∥2Xα (sup
√
⃗ .)∥∞ )2 .
t∥G(t,
t>0
We then write, defining h(s) = ∥H(s, .)∥2 ,
Z α
T
I(t, x)2 dt dx
t
0
!
Z +∞ α Z Z t Z
α−1
T
1
2
p
≤C
(
s 2 H(s, y) ds dy) dx dt
t
( ν(t − s) + |x − y|)4
0
0
!1/2 2
Z +∞ α Z t Z Z
α−1
T
s 2 dy
( H(s, y) p
≤C
)2 dx
ds dt
t
( ν(t − s) + |x − y|)4
0
0
Z +∞ α Z t
α−1
T
1
′
p
≤C
(
s 2 h(s) ds)2 dt
t
ν(t − s)
0
0
Z
T
We then get, fixing β such that 1 − α < β < 1,
Z T Z α
T
I(t, x)2 dt dx
t
0
≤ C′
1
ν
α Z t
Z t
T
1
1 dτ
√
√
(
sβ+α−1 h(s)2 ds)(
) dt
t
t
−
s
t
− τ τβ
0
0
0
Z +∞ Z t
1
1
α
α+β−1
√
=Cβ,ν T
h(s)2 ds dt
1 s
α+β−
2
t−st
0
0
Z +∞
α
=Cα,β,ν T
h(s)2 ds
0
√
3
′
⃗ .)∥∞ )2 .
≤Cα,β,ν
T 2 ∥F⃗ ∥2Xα (sup t∥G(t,
Z
∞
t>0
The theorem is proved.
Remark: As we did not use the oscillations of the kernel but directly estimated integrals
involving absolute values of the integrands (see inequality (9.3)), we may suspect that the
204
The Navier–Stokes Problem in the 21st Century (2nd edition)
space Q−1,α is such that ⃗u0 ∈ Q−1,α ⇒ 1t>0 Wνt ∗ ⃗u0 belongs to the space V 1,2 (R × R3 ) we
discussed in Chapter 5. This is indeed the case: as a matter of fact we have more precisely
∥1t>0 Wνt ∗ ⃗u0 ∥M2(1+α),5 ≤ C∥⃗u0 ∥Q−1,α
2
2(1+α),5
where M2
is a parabolic Morrey space on R × R3 . The Fefferman–Phong inequality
gives the required embedding.
9.4
A Special Subclass of BM O−1
In this section, we offer a few words to 2D space–periodical problems. It is well known
that, in the 2D-case, the L2 theory works very well (see for existence the classical theory
in Ladyzhenskaya’s book [293]). In particular, the Cauchy problem for ⃗u = (⃗u1 , ⃗u2 ) in
C([0, +∞), L2 (R2 /2πZ2 )) ∩ L2 (Ḣ 1 (R2 /2πZ2 )) is well posed.
Here, we shall consider three-dimensional vector fields that depend only on the first two
variables:
⃗u = (u1 (t, x1 , x2 ), u2 (t, x1 , x2 ), u3 (t, x1 , x2 )).
Bertozzi and Majda labeled those vector fields as “two-and-a-half dimensional flows” [40].
We consider the Navier–Stokes equations with a null force:

⃗ u − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
(9.4)
div ⃗u = 0

⃗u|t=0 = ⃗u0
RR
with ⃗u0 ∈ L2 (R2 /2πZ2 ), div ⃗u0 = 0 and [0,2π]2 ⃗u0 (t, x1 , x2 ) dx1 dx2 = 0. As a matter of
fact, such an initial value belongs to BM O−1 (R3 ):
Proposition 9.3.
Let E2 be the space of measurable functions u : R3 7→ R such that
• ∂3 u = 0: u(x1 , x2 , x3 ) does not depend on x3 (we shall write u(x) = u(x1 , x2 ) or
u(x) = u(x1 , x2 [, x3 ]))
• u is 2πZ2 periodical:
u(x1 , x2 [, x3 ]) = u(x1 + 2π, x2 [, x3 ]) = u(x1 , x2 + 2π[, x3 ])
•
RR
•
RR
[0,2π]2
[0,2π]2
|u(x1 , x2 [, x3 ])|2 dx1 dx2 < +∞
u(x1 , x2 [, x3 ]) dx1 dx2 = 0
RR
endowed with the norm ∥u∥E2 = ( [0,2π]2 |u(x1 , x2 [, x3 ])|2 dx1 dx2 )1/2 .
Then we have the embeddings:
−1
E2 ⊂ Ḃ∞,2
(R3 ) ⊂ BM O−1 (R3 ).
−1
−1
Proof. For the proof of E2 ⊂ Ḃ∞,2
, the simplest way is to use the characterization of Ḃ∞,2
P
D
through the Littlewood–Paley decomposition u = j∈Z ∆j u with ∆j u = ψ( 2j )u, where
The Space BMO−1 and the Koch and Tataru Theorem
205
3
the Fourier multiplier is an even
P smooth−jfunction ψ supported in the annulus {ξ ∈ R / 1 ≤
|ξ| ≤ 4} such that, for ξ ̸= 0, j∈Z ψ(2 ξ) = 1.
For u ∈ E2 , we have a Fourier decomposition
X
u=
ak cos(k1 x1 + k2 x2 ) + bk sin(k1 x1 + k2 x2 )
k∈Z2 ,k̸=0
with
X
∥u∥2E2 = 2π 2
a2k + b2k
k∈Z2 ,k̸=0
We have
X
∆j u =
ψ(2−j (k, 0))(ak cos(k1 x1 + k2 x2 ) + bk sin(k1 x1 + k2 x2 ))
2j <|k|<4 2j
so that ∆j u = 0 for j ≤ −2, while, for j ≥ −2,
X
∥∆j u∥∞ ≤∥ψ∥∞ (
a2k + b2k )1/2 (
2j <|k|<4
≤C2j (
X
2j
X
2)1/2
2j <|k|<4 2j
a2k + b2k )1/2 .
2j <|k|<4 2j
P
−1
This gives the embedding E2 ⊂ Ḃ∞,2
(R3 ): u = j∈Z ∆j u (convergence in S ′ ) and
X
(2−j ∥∆j u∥∞ )2 ≤ C∥u∥E2 .
j∈Z
−1
For the proof of Ḃ∞,2
⊂ BM O−1 , the simplest way is to use the thermic characterization
−1
−1
of Ḃ∞,2 : if u ∈ Ḃ∞,2 , then Wt ∗ u ∈ L2 ((0, +∞), L∞ ). We then write
Z t
Z tZ
2
3/2
|W
∗
u(s,
y)|
ds
dy
≤
Ct
∥Ws ∗ u∥2∞ ds ≤ Ct3/2 ∥Ws ∗ u∥2L2 L∞ .
s
√
0
B(x0 , t)
0
The proposition is proved.
As L∞ ∩ E2 is dense in E2 (proof: just truncate the Fourier series), we see that E2 is
more precisely embedded into the closure of L∞ ∩ BM O−1 in BM O−1 ; thus, Theorem 9.2
ensures the local existence of a solution to Equations (9.4) when ⃗u0 belongs to E2 . It turns
out that this solution has global existence:
Two-and-a-half dimensional Navier–Stokes equations
Theorem 9.6.
If ⃗u0 ∈ E2 with div ⃗u0 = 0, then there exists a unique global mild solution ⃗u of equations

⃗ u = ν∆⃗u − ∇p
⃗
∂t ⃗u + (⃗u.∇)⃗
(9.5)
⃗u(0, .) = ⃗u0

div ⃗u = 0
⃗ u ∈ L2 E2 .
such that ⃗u ∈ L∞
t E2 and ∇ ⊗ ⃗
206
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proof. We know that there is a small time T for which the Picard iterates converge to a
solution in the ET norm (Theorem 9.2). Here, we shall use another norm that ensures the
convergence.
First, we remark that the solution we construct does not depend on x3 and is 2πZ2
periodical (by the translation invariance of the Navier–Stokes equations), and the same
holds for the associated pressure. More precisely, writing u1 (t, x) = v1 (t, x1 , x2 ), u2 (t, x) =
v2 (t, x1 , x2 ), u3 (t, x) = w(t, w1 , x2 ), and p(t, x) = q(t, x1 , x2 ), we find that (⃗v , w) solve the
following equations
• ⃗v satisfies a 2D Navier–Stokes equation:

⃗ v = ν∆⃗v − ∇p
⃗
 ∂t⃗v + (⃗v .∇)⃗
⃗v (0, x1 , x2 ) = (u0,1 (x), u0,2 (x))

div ⃗v = 0
• w satisfies a linear advection-diffusion scalar equation:
⃗
∂t w + (⃗v .∇)w
= ν∆w
w(0, x1 , x2 ) = u0,3 (x)
(9.6)
(9.7)
Step 1: Local existence for the 2D Navier–Stokes equations.
The study of 2D Navier–Stokes equations with initial value ⃗v0 ∈ L2 (R2 /2πZ2 ) was
initiated by the works of Leray [327, 328, 329], and fully developed by Ladyzhenskaya,
Lions and Prodi [293, 339].
We rewrite (9.6) into
⃗v =
(2)
Wνt
Z
∗ ⃗v0 −
0
t
(2)
Wν(t−s) ∗ P(2) div(⃗v ⊗ ⃗v ) ds
(9.8)
(2)
where Wt (x1 , x2 ) is the 2D heat kernel and P(2) is the 2D Leray projection operator.
For a periodical distribution vector field
X
⃗ (x1 , x2 ) =
⃗ k + sin(k · x)B
⃗ k,
V
cos(k · x)A
k∈Z2
we have
(2)
⃗ =
Wνt ∗ V
X
2
⃗ k + sin(k · x)B
⃗ k)
e−νt|k| (cos(k · x)A
k∈Z2
and for a periodical distribution tensor
X
T(x1 , x2 ) =
cos(k · x)Ak + sin(k · x)Bk ,
k∈Z2
we have
P(2) div T =
X
k∈Z2 ,k̸=0
⃗ ⃗
⃗k ⊗ ⃗k · Bk
⃗k) − sin(k · x)(⃗k · Ak − k ⊗ k.Ak ⃗k)
cos(k · x)(⃗k.Bk −
2
|k|
|k|2
We are going to look for a solution ⃗v ∈ L4 ((0, T0 ), L4 (R2 /2πZ2 )) for T0 small enough.
Indeed, we have the following estimates:
(2)
∥Wνt ∗ ⃗v0 ∥L∞ L2 (R2 /2πZ2 ) = ∥⃗v0 ∥L2 (R2 /2πZ2 )
The Space BMO−1 and the Koch and Tataru Theorem
207
1
(2)
∥Wνt ∗ ⃗v0 ∥L2 Ḣ 1 (R2 /2πZ2 ) = √ ∥⃗v0 ∥L2 (R2 /2πZ2 )
2ν
Z t
1
(2)
∥
Wν(t−s) ∗ P(2) div T ds∥L∞ ((0,T0 ),L2 (R2 /2πZ2 )) ≤ √ ∥T∥L2 L2 (R2 /2πZ2 )
2ν
0
Z t
1
(2)
∥
Wν(t−s) ∗ P(2) div T ds∥L2 ((0,T0 ),Ḣ 1 (R2 /2πZ2 )) ≤ ∥T∥L2 L2 (R2 /2πZ2 )
ν
0
Moreover, we have the Sobolev embedding
ZZ
f ∈ Ḣ 1/2 (R2 /2πZ2 ) and
f dx = 0 ⇒ f ∈ L4 (R2 /2πZ2 ).
R2 /2πZ2
Thus, we find that:
(2)
⃗0 = Wνt
⃗0 ∥L4 ((0,T ),L4 ) = 0.
ˆ V
∗ ⃗v0 belongs to L4 L4 . Moreover, limT0 →0 ∥V
0
4 4
2
2
ˆ If ⃗u and ⃗v belong to L L (R /2πZ ), we have that
Z t
(2)
B(⃗u, ⃗v ) =
Wν(t−s) ∗ P div(⃗u ⊗ ⃗v ) ds
0
∞
2
2
1
4
4
belongs to L L ∩ L Ḣ ∩ L L with
∥B(⃗u, ⃗v )∥L4 ((0,T0 ),L4 ) ≤ Cν ∥⃗u∥L4 ((0,T0 ),L4 ∥⃗v ∥L4 ((0,T0 ),L4 ) .
⃗0 ∥L4 ((0,T ),L4 ) < 1 , we shall find a solution ⃗v
ˆ If T0 is small enough, so that ∥V
0
4Cν
through Picard’s iterative process. The process will converge in L∞ L2 ∩ L2 Ḣ 1 ∩
L4 L4 .
Step 2: Global existence.
If the solution ⃗v belongs to L∞ ((0, T0 ), L2 ) ∩ L2 ((0, T0 ), Ḣ 1 ) then we find an estimate
slightly better than just ⃗v ∈ L∞ L2 . As a matter of fact, we have
Z t
(2)
(2)
(2)
⃗v = Wνt ∗ ⃗v0 −
Wν(t−s) ∗ P(2) div(⃗v ⊗ ⃗v )) ds = Wνt ∗ ⃗v0 + L(V)
0
where V = ⃗v ⊗ ⃗v . The operator L maps L2 L2 to L∞ L2 and L2 H 2 to Lip L2 ; as L2 H 2
is dense in L2 L2 , we find that L actually maps L2 L2 to C([0, T0 ], L2 ).
If T ∗ is the maximal time of existence of the solution ⃗v , so that ⃗v belongs to
L∞ ((0, T0 ), L2 ) ∩ L2 ((0, T0 ), Ḣ 1 ) for every T0 < T ∗ , we find that T ∗ = +∞ unless that T ∗ < +∞ and ⃗v does not belong to L∞ ((0, T ∗ ), L2 ) ∩ L2 ((0, T ∗ ), Ḣ 1 ): if ⃗v
belonged to L∞ ((0, T ∗ ), L2 ) ∩ L2 ((0, T ∗ ), Ḣ 1 ) with T ∗ < T , then it would belong to
C([0, T ∗ ], L2 ) and we could solve the Cauchy problem for the Navier–Stokes equations
on some interval [T ∗ , T ∗ + T0 ] with initial value ⃗v (T ∗ , .).
Thus, in order to show that we have a global solution, we only need to control the
⃗ ⊗ ⃗v on [0, T ∗ ). As ⃗v belongs to L2 H 1 and ∂t⃗v belongs to L2 H −1 on
sizes of ⃗v and ∇
every compact interval of [0, T ∗ ), we may write
d
⃗ ⊗ ⃗v (t, .)∥22
∥⃗v (t, .)∥2L2 (R2 /2πZ2 ) = 2⟨⃗v (t, .)|∂t⃗v (t, .)⟩H 1 ,H −1 = −2ν∥∇
dt
so that
∥⃗v (t, .)∥22 + 2ν
Z
t
⃗ ⊗ ⃗v (s, .)∥22 ds = ∥⃗v0 ∥22 .
∥|∇
0
Hence, T ∗ = +∞: we have a global solution.
208
The Navier–Stokes Problem in the 21st Century (2nd edition)
Step 3: Global existence of w.
For the existence of w, we write w as a fixed point of the transform
Z t
(2)
(2)
(2)
Wν(t−s) ∗ (div(ω⃗v )) ds = Wνt ∗ w0 + L(w).
ω 7→ Wνt ∗ w0 −
0
L is a bounded linear operator on L∞ ((0, T0 ), L2 ) ∩ L2 ((0, T0 ), Ḣ 1 ) and satisfies (uniformly in T0 )
∥L(w)∥L∞ ((0,T0 ),L2 )∩L2 ((0,T0 ),Ḣ 1 ) ≤ C0 ∥⃗v ∥L4 ((0,T0 ),L4 ) ∥w∥L∞ ((0,T0 ),L2 )∩L2 ((0,T0 ),Ḣ 1 ) .
Thus, L is a contraction as soon as T0 is small enough to grant that
C0 ∥⃗v ∥L4 ((0,T0 ),L4 ) < 1.
Global existence of w is then proved by splitting any given interval [0, T ] into a finite
union of intervals [Tj , Tj+1 ] with C0 ∥⃗v ∥L4 ((Tj ,Tj+1 ),L4 ) < 1: once w is constructed on
[0, Tj+1 ], one constructs w on [Tj+1 , Tj+2 ] by considering the Cauchy problem with
initial value w(Tj+1 , .) at t = Tj+1 . Thus, w exists up to the given arbitrary time T .
9.5
Ill-posedness
Thus far, the largest space of initial values that are well fitted for the Cauchy problem for
the Navier–Stokes equations is the space bmo−1 and its homogeneous counterpart BM O−1 .
For scaling properties of the equations, any such space that respects the shift invariance
−1
of the equations and their scaling properties should be embedded into B∞,∞
(for local
−1
existence results) or Ḃ∞,∞ (for global existence results).
−1
. More preBourgain and Pavlović [52] proved that the problem was ill-posed in Ḃ∞,∞
cisely, they proved a phenomenon of norm inflation:
Theorem 9.7.
−1
For every δ > 0, there exists a smooth divergence-free ⃗u0 ∈ E2 with a small norm in Ḃ∞,∞
−1
< δ) which generates a solution ⃗u of the Navier–Stokes equations
(i.e., ∥⃗u0 ∥Ḃ∞,∞

⃗ u − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u.∇)⃗
div ⃗u = 0

⃗u|t=0 = ⃗u0
(9.9)
which becomes very large in a very small time: for some τ ∈ (0, δ),
−1
∥⃗u(τ, .)∥Ḃ∞,∞
≥
1
.
δ
Of course, the norm of ⃗u0 must be large in BM O−1 : if ∥⃗u0 ∥BM O−1 is small enough, then
the Koch and Tataru theorem (Theorem 9.2) implies that there exists a global solution ⃗u
to (9.9) with
−1
∥⃗u(t, .)∥Ḃ∞,∞
≤ C∥⃗u∥BM O−1 ≤ C ′ ∥⃗u0 ∥BM O−1 .
The Space BMO−1 and the Koch and Tataru Theorem
209
Proof. The mild solution satisfies:
⃗u = Wνt ∗ ⃗u0 − B(⃗u, ⃗u)
with
Z
t
Wν(t−s) ∗ P div(⃗u ⊗ ⃗v ) ds.
B(⃗u, ⃗v ) =
0
The discussion will focus on the decomposition
⃗0 − U
⃗1 + U
⃗2
⃗u = U
with
⃗ 0 = Wνt ∗ ⃗u0 and U
⃗ 1 = B(U
⃗ 0, U
⃗ 0 ).
U
Bourgain and Pavlović’s choice for ⃗u0 (discussed as well by Sawada [422]) is given by a
lacunary sums of cosines
Q X
⃗u0 = √
w
⃗j
N j∈Λ
 −j

 
−2
0
X
Q
2j cos(2j x1 ) 1 + cos(2j x1 + x2 )  1 
=√
N j∈Λ
1
1
(9.10)
with Q a (large) integer and Λ a lacunary finite subset of N of the type {j0 < j1 < · · · <
jN −1 } with j0 ≥ 5 and jq+1 > 4jq . We assume that Q3 < N . (Thus √QN < Q−1/2 is small.)
In the following computations, C0 , C1 , . . . will denote positive constants that may depend
on ν but depend neither on Q, N , nor Λ.
Estimates on ⃗u0 :
−1
)
We have obviously div ⃗u0 = 0 and (using the Littlewood–Paley characterization of Ḃ∞,p
Q X −2j
∥⃗u0 ∥Ḃ −1 ≈ √ (
2 ∥w
⃗ j ∥2∞ )1/2 ≤ C0 Q
∞,2
N j∈Λ
and
Of course, we shall take
Q
Q
−1
≈ √ sup 2−j ∥w
∥⃗u0 ∥Ḃ∞,∞
⃗ j ∥∞ ≤ C0 √ .
N j∈Λ
N
√Q
N
small enough to ensure that
Q
−1
∥⃗u0 ∥Ḃ∞,∞
≤ C0 √ ≤ δ
N
while Q will be large with respect to δ −1 . (We shall see that ∥⃗u0 ∥BM O−1 ≈ Q).
⃗ 0:
Estimate on U
−1
−1
The initial assumption is that ⃗u0 ∈ B∞,∞
with ∥⃗u0 ∥Ḃ∞,∞
≤ δ. This gives, in particular,
⃗
the following estimate on U0 = Wνt ∗ ⃗u0 :
⃗ 0 (t, .)∥ −1 ≤ δ.
∥U
Ḃ∞,∞
−1
⃗ 0 remains small in Ḃ∞,∞
Thus, U
.
210
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ 1:
Estimates on U
⃗ 1 = B(Wνt ∗ ⃗u0 , Wνt ∗ ⃗u0 ). For j ∈ Λ, let kj = (2j , 0, 0), lj = (2j , 1, 0),
Now, 
we 
compute U
 −j 
0
−2
⃗j =  1  so that
α
⃗ j = 1 and β
1
1
Q X j
2 cos(kj · x)⃗
αj + cos(lj · x)β⃗j
⃗u0 = √
(9.11)
N j∈Λ
and
X
Q X j −νt22j Wνt ∗ ⃗u0 = √
2 e
cos(kj · x)⃗
αj + e−νt cos(lj · x)β⃗j =
⃗γj (t, x).
N j∈Λ
j∈Λ
⃗0 ⊗ U
⃗ 0 with the decomposition in paraproducts [313]: if j < l, then
We may compute U
the frequency localization of ⃗γj ⊗ ⃗γl and of ⃗γl ⊗ ⃗γj will be for frequencies of order 2l so that
∥
XX
−2
⃗γj ⊗ ⃗γl ∥L∞ Ḃ∞,∞
≤ C sup
t
l
j̸=l
X Q2
j<l
N
2j 2l 2−2l ∼ C
Q2
N
and we find finally
sup ∥
t>0
XX
−1
B(⃗γj , ⃗γl )(t, .)∥Ḃ∞,∞
≤ C1 δ 2 .
j̸=l
(We used the maximal regularity of the heat kernel for Besov spaces
Z t
1
−3
−1
∥
Wν(t−s)∆ ∗ f (s, .) ds∥L∞ Ḃ∞,∞
≤ C ∥f ∥L∞ Ḃ∞,∞
t
t
ν
0
– see Lemarié-Rieusset [313]).
If we now look at the square term ⃗γj ⊗ ⃗γj , then we find low frequencies 0, kj − lj and
lj − kj (which are all such that |ξ| ≤ 1) and high frequencies kj + lj and −kj − lj which are
of order 2j . Let P0 be the projection on frequencies less than 2. Then, again, we have
X
−1
≤ C1 δ 2 .
sup ∥(Id − P0 )
B(⃗γj , ⃗γj )(t, .)∥Ḃ∞,∞
t>0
j
Moreover, the frequency 0 may be forgotten, as the constant terms will be killed by applying
the divergence operator (equivalently, as the convolution kernel of Wν(t−s) ∗P div has integral
equal to 0). Thus, we are left with estimating
2 X
⃗ 3 (t, x) = Q
U
22j P0 (B(Wνs ∗ (cos(kj · x)⃗
αj ), Wνs ∗ (cos lj · x)β⃗j ))
N
j∈Λ
2
Q X 2j
2 P0 (B(Wνs ∗ (cos(lj · x)β⃗j ), Wνs ∗ (cos kj · x)⃗
αj ))
N
j∈Λ
Z
Q2 X 2j t −ν(t−s) −νs(22j+1 +1)
=
2 (
e
e
ds) ×
2N
0
+
j∈Λ
⃗j + β⃗j ⊗ α
× P div(cos((kj − lj ) · x)(⃗
αj ⊗ β
⃗ j ))).
We have

0
cos((kj − lj ) · x)(⃗
αj ⊗ β⃗j + β⃗j ⊗ α
⃗ j ) = cos(x2 ) −2−j
−2−j
−2−j
2
2

−2−j
2 
2
The Space BMO−1 and the Koch and Tataru Theorem
and thus
211
 −j

2 sin(x2 )
.
0
P div((cos((kj − lj ) · x)(⃗
αj ⊗ β⃗j + β⃗j ⊗ α
⃗ j ))) = 
−2 sin x2
Writing


V3
⃗3 =  0  ,
U
W3
we find that V3 is small and W3 large:
−1
∥V3 (t, x)∥Ḃ∞,∞
≤C
1 Q2 X −j
2 ≤ C1 δ 2
ν N
j∈Λ
and
−1
∥W3 (t, x)∥Ḃ∞,∞
≤C
1 Q2 X
1 ≤ C1 Q2 .
ν N
j∈Λ
This latter estimate is quite sharp:
−1
∥W3 (t, x)∥Ḃ∞,∞
≈
Z
Q2 X 2j t −ν(t−s) −νs(1+22j+1 )
2
e
e
ds
N
0
j∈Λ
2
=
2j+1
Q 1 −νt X
e
(1 − e−νt2
)
N 2ν
j∈Λ
so that, defining j0 = minj∈Λ j, we find that
−1
C2 Q2 ≤ ∥W3 (t, x)∥Ḃ∞,∞
≤ C3 Q2
with positive constants C2 and C3 independent from Λ, Q, and N , as far as
1
≤ νt ≤ 1.
22j0
−1
⃗ 1 becomes very large in the Ḃ∞,∞
Thus, U
norm in a very short time (t ≈ ν −1 2−2j0 ) and
remains large on a rather long interval (up to t ≈ ν −1 ).
R
⃗ belong to E∞ , then A(F⃗ , G)
⃗ = +∞ Wνs ∗
Remark: In Lemma 9.1, we saw that if F⃗ and G
0
⃗ ds belongs to (BM O)9 and
(F⃗ ⊗ G)
⃗ BM O ≤ Cν ∥F⃗ ∥E ∥G∥
⃗ E .
∥A(F⃗ , G)∥
∞
∞
⃗
⃗ that
In particular, we find that, since B(F⃗ , G)(t,
.) = P div A(10<s<t F⃗ , 10<s<t G),
∥B(Wνs ∗ ⃗u0 , Wνs ∗ ⃗u0 )∥BM O−1 ≤ Cν ∥⃗u0 ∥2BM O−1 .
Thus, we find, for t0 = ν1 ,
⃗ 1 (t0 , .)∥ −1 ≤ C∥U
⃗ 1 (t0 , .)∥BM O−1 ≤ C ′ ∥⃗u0 ∥2
Q2 ≈ ∥U
BM O −1
B∞,∞
while we saw that
∥⃗u0 ∥BM O−1 ≤ C∥⃗u0 ∥B −1 ≈ Q.
∞,2
We have thus clearly established that ∥⃗u0 ∥BM O−1 ≈ Q.
212
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ 2:
Estimates on U
⃗ 2 remains small while
The core idea in the Bourgain and Pavlović proof is to show that U
⃗ 1 becomes very large.
U
⃗ 2 is the solution U
⃗ 2 = ⃗z of the equation
U
⃗ 0 + L(⃗z) − B(⃗z, ⃗z)
⃗z = Z
with
⃗ 0 = B(U
⃗ 0, U
⃗ 1 ) + B(U
⃗ 1, U
⃗ 0 ) − B(U
⃗ 1, U
⃗ 1)
Z
and
⃗ 0 , ⃗z) − B(⃗z, U
⃗ 0 ) + B(U
⃗ 1 , ⃗z) + B(⃗z, U
⃗ 1 ).
L(⃗z) = −B(U
⃗ 2 is a mild solution U
⃗ 2 = ⃗z of
Equivalently, U

⃗1 ⊗ U
⃗1 − U
⃗0 ⊗ U
⃗1 − U
⃗1 ⊗ U
⃗ 0 + ⃗z ⊗ ⃗z)
∂t ⃗z =
ν∆⃗z − div(U



⃗ 0 ⊗ ⃗z + ⃗z ⊗ U
⃗0 − U
⃗ 1 ⊗ ⃗z − ⃗z ⊗ U
⃗ 1 ) − ∇q
⃗
− div(U

div
⃗
z
=
0


⃗z|t=0 = 0
or
(9.12)
⃗ 2 (t, .)∥BM O−1 . Let 0 ≤ T1 ≤ T2 with T1 = 0
We are going to estimate the evolution of ∥U
1
1
for
some
J
∈
Λ,
and
T
=
for
some
J2 ∈ Λ.
1
2
ν22J1
ν22J2
If we define, for a function h on (T1 , T2 ) × R3 ,
p
∥h∥E = sup
t − T1 ∥h(t, .)∥∞
T1 <t<T2
1
+ sup sup
3/4
3
(t
−
T
T1 <t<T2 x0 ∈R
1)
sZ
t
T1
Z
√
B(x0 , t−T1 )
|h(s, y)|2 dy ds
we find, from Lemma 9.1, that
⃗ 2 (T2 , .)∥BM O−1 ≤∥U
⃗ 2 (T1 , .)∥BM O−1 + ∥Z
⃗ 0 (T2 , .) − Z
⃗ 0 (T1 , .)∥BM O−1
∥U
⃗ 2 ∥E (∥U
⃗ 2 ∥E + ∥U
⃗ 1 ∥E + ∥U
⃗ 0 ∥E )
+ Cν ∥U
(9.13)
while, from Theorem 9.1, we know that
⃗ 2 ∥E ≤∥U
⃗ 2 (T1 , .)∥BM O−1 + ∥Z
⃗0 − Z
⃗ 0 (T1 , .)∥E
∥U
⃗ 2 ∥E (∥U
⃗ 2 ∥E + ∥U
⃗ 1 ∥E + ∥U
⃗ 0 ∥E )
+ Cν ∥U
(9.14)
where
⃗ 0 (t, .) − Z
⃗ 0 (T1 , .) =
Z
Z
t
⃗0 ⊗ U
⃗1 + U
⃗1 ⊗ U
⃗0 − U
⃗1 ⊗ U
⃗ 1 ) ds.
Wν(t−s) ∗ P div(U
T1
⃗ 0 (t, .) − Z
⃗ 0 (T1 , .), and find
Again, by Lemma 9.1 and Theorem 9.1, we may estimate Z
⃗ 2 (T2 , .)∥BM O−1 ≤∥U
⃗ 2 (T1 , .)∥BM O−1
∥U
⃗ 1 ∥E (∥U
⃗ 0 ∥E + ∥U
⃗ 1 ∥E )
+ Cν ∥U
⃗ 2 ∥E (∥U
⃗ 2 ∥E + ∥U
⃗ 1 ∥E + ∥U
⃗ 0 ∥E )
+ Cν ∥U
(9.15)
The Space BMO−1 and the Koch and Tataru Theorem
213
and
⃗ 2 ∥E ≤∥U
⃗ 2 (T1 , .)∥BM O−1
∥U
⃗ 1 ∥E (∥U
⃗ 0 ∥E + ∥U
⃗ 1 ∥E )
+ Cν ∥U
⃗ 2 ∥E (∥U
⃗ 2 ∥E + ∥U
⃗ 1 ∥E + ∥U
⃗ 0 ∥E ).
+ Cν ∥U
(9.16)
⃗ 0 ∥E and ∥U
⃗ 1 ∥E . Recall that
Thus, we need to estimate ∥U
X
X
2j
⃗ 0 = √Q
U
2j e−νt2 cos(kj · x)⃗
αj + e−νt cos(lj · x)β⃗j =
⃗γj (t, x).
N j∈Λ
j∈Λ
We split the sum between the indexes j such that j ≤ J2 , those such that J2 < j ≤ J1 and
those such that j > J1 :
X
X
X
⃗=
⃗ =
⃗ =
A
⃗γj (t, x), B
⃗γj (t, x) and C
⃗γj (t, x).
j≤J2
J2 <j≤J1
j>J1
⃗ .) = Wν(t−T ) ∗ A(T
⃗ 1 , .), B(t,
⃗ .) = Wν(t−T ) ∗ B(T
⃗ 1 , .), and
As, on (T1 , T2 ), we have A(t,
1
1
⃗
⃗
C(t, .) = Wν(t−T1 ) ∗ C(T1 , .), we find that
p
⃗ 0 ∥E ≤ Cν ( T2 − T1 ∥A(T
⃗ 1 , .)∥∞ + ∥B(T
⃗ 1 , .)∥BM O−1 + ∥C(T
⃗ 1 , .)∥BM O−1 )
∥U
with
p
p
⃗ 1 , .)∥∞ ≤ T2 ∥A(0,
⃗ .)∥∞
T2 − T1 ∥A(T
X
Q p
≤2 √
T2
2j
N
j∈Λ,j≤J2
Q p
≤4 √
T2 2J2
N
Q
≤Cν √
N
and
⃗ 1 , .)∥BM O−1 ≤C∥C(T
⃗ 1 , .)∥ −1
∥C(T
Ḃ∞,2

1/2
X
≤C ′ 
2−2j ∥⃗γj (T1 , .)∥2∞ 
j∈Λ,j>J1

1/2
X
2j
Q
≤C ′′ √ 
e−2νT1 2 
N j∈Λ,j>J1
Q X −2(16)p 1/2
≤C ′′ √ (
e
)
N p∈N
Q
=C ′′′ √ .
N
214
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ for which we have
The difficult term is, of course, B,
⃗ 1 , .)∥BM O−1 ≤C∥B(T
⃗ 1 , .)∥ −1
∥B(T
Ḃ∞,2

1/2
X
≤C ′ 
2−2j ∥⃗γj (T1 , .)∥2∞ 
j∈Λ,J2 <j≤J1
Q p
≤C ′′ √
#({j ∈ Λ / J2 < j ≤ J1 })
N
If we want this quantity to be small, we need that the ratio
Bourgain and Pavlović’s choice is the ratio
#({j∈Λ / J2 <j≤J1 })
#(Λ)
#({j ∈ Λ / J2 < j ≤ J1 })
1
= 3.
#(Λ)
Q
⃗ 1 , .)∥BM O−1 ≤ CQ−1/2 . As
For that choice, we find that ∥B(T
be small.
(9.17)
√Q
N
≤ Q−1/2 , we find that
⃗ 0 ∥E ≤ C4 Q−1/2 .
∥U
⃗ 1 . Recall that we have split U
⃗ 1 into U
⃗ 3 and U
⃗1 − U
⃗ 3,
Now, we estimate the norms of U
⃗
⃗
⃗
and U3 into P0 U3 and (Id − P0 )U3 . We have
XX
⃗1 − U
⃗3 =
U
B(⃗γj , ⃗γp ).
j̸=p
Let ϵj ∈ {kj , −kj , ll , −lj } and ϵp ∈ {kp , −kp , lp , −lp }. Then
Z
t
2
2
2
e−ν(t−s)|ϵj +ϵp | |ϵj + ϵp |2j e−νs|ϵj | 2p e−νs|ϵp | ds
Z t
2
= 2j 2p |ϵj + ϵp |e−νt|ϵj +ϵp |
e2νsϵj ·ϵp ds
0
0
Let us notice that, if p < j, and due to the lacunarity of Λ, we have 12 2j ≤ |ϵj + ϵp | ≤ 32 2j .
If ϵj · ϵp ≤ 0, we find
Z
t
2
2
2
e−ν(t−s)|ϵj +ϵp | |ϵj + ϵp |2j e−νs|ϵj | 2p e−νs|ϵp | ds
0
2j
1
3
≤ 2j 2p 2j e− 4 νt2 t,
2
while, if p < j and ϵj .ϵp ≥ 0, we find
Z
t
2
2
2
e−ν(t−s)|ϵj +ϵp | |ϵj + ϵp |2j e−νs|ϵj | 2p e−νs|ϵp | ds
0
2
= 2j 2p |ϵj + ϵp |e−νt|ϵj +ϵp |
2
= 2j 2p |ϵj + ϵp |e−νt(|ϵj |
+|ϵp |2 ) 1
e2νtϵj .ϵp − 1
2νϵj .ϵp
− e−2νtϵj .ϵp
2νϵj .ϵp
2j
1
3
≤ 2j 2p 2j e− 4 νt2 t.
2
The Space BMO−1 and the Koch and Tataru Theorem
215
This gives
2
⃗ 1 (t, .) − U
⃗ 3 (t, .)∥∞ ≤C Q
∥U
N
≤C ′
X
2j
1
X
2p t22j e− 4 νt2
j∈Λ p∈Λ,p<j
1 Q2 X
ν N
X
1
2j
1
2j
2p e− 8 νt2 .
j∈Λ p∈Λ,p<j
If σ(j) = sup{p ∈ Λ / p < j}, we find
2
⃗ 1 (t, .) − U
⃗ 3 (t, .)∥∞ ≤C 1 Q
∥U
ν N
X
2σ(j) e− 8 νt2
j∈Λ,j>j0
We then find, for T1 < t(< T2 ),
2
⃗ 1 (t, .) − U
⃗ 3 (t, .)∥∞ ≤C 1 Q
∥U
ν N
≤C ′
2j
1
X
2σ(j) e− 8 ν(t−T1 )2
j∈Λ,j>j0
2
1Q
ν N
X
j∈Λ,j>j0
2σ(j)
p
ν(t − T1 )2j
so that, as 2σ(j) ≤ 2j/2 , we have
⃗ 1 (t, .) − U
⃗ 3 (t, .)∥∞ ≤ Cν
∥U
Q2
1
√
.
N t − Ti
Similarly, we have
2
⃗1 − U
⃗ 3 ∥L2 ((T ,T ),L∞ ) ≤ C 1 Q
∥U
1
2
ν N
X
σ(j)
2
1
2j
e− 4 νt2 dt)1/2
0
j∈Λ,j>j0
so that
+∞
Z
(
2
⃗1 − U
⃗ 3 ∥L2 ((T ,T ),L∞ ) ≤ C 1 Q
∥U
1
2
ν N
X
j∈Λ,j>j0
2σ(j)
√ j
ν2
Finally, we get
2
⃗ 1 (t, .) − U
⃗ 3 (t, .)∥E ≤ C5 Q .
∥U
N
⃗ 3:
A similar estimate holds for (Id − P0 )U
⃗ 3 (t, .)∥∞ ≤ C
∥(Id − P0 )U
1 Q2 X j −2νt22j
2 e
.
ν N
j∈Λ
⃗ 3 ∥E , we split again the sum between the indexes j such
In order to estimate ∥(Id − P0 )U
that j ≤ J2 , those such that J2 < j ≤ J1 and those such that j > J1 :
⃗ 3 (t, .)∥∞ ≤ C 1 (D(t) + E(t) + F (t))
∥(Id − P0 )U
ν
with
D(t) =
Q2
N
X
j∈Λ,j≤J2
2j
2j e−2νt2 ,
E(t) =
Q2
N
X
j∈Λ,J2 <j≤J1
2j e−2νt2
2j
216
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
Q2
N
F (t) =
2j
X
2j e−2νt2 .
j∈Λ,j>J1
Let us write
∥A(t)∥E =
p
sup
T1 <t<T2
We have
T2
Z
A(t)2 dt)1/2 .
t − T1 A(t) + (
T1
⃗ 3 (t, .)∥E ≤ C 1 (∥D(t)∥E + ∥E(t)∥E + ∥∥F (t)∥E .)
∥(Id − P0 )U
ν
2
We have ∥D∥∞ ≤ 2 QN 2J2 and thus
∥D∥E ≤ C
p
T2 − T1
D(t) ≤ 2C
sup
T2 <t<T1
Q2 J2 p
Q2
2
T2 ≤ Cν
.
N
N
To estimate F (t), for T1 < t < T2 , we write
F (t) ≤
Q2
N
≤C
2j
1
X
1
2j e− 8 νT1 2 e− 8 ν(t−T1 )2
2j
j∈Λ,j>J1
Q2
N
X
j∈Λ,j>J1
2j
1
2j
p
e− 8 νT1 2 ;
j
ν(t − T1 )2
on the other hand, we have
∥F ∥L2 (T1 ,T2 ) ≤
Q2
N
Z
2j
1
2j e− 8 νT1 2 (
X
∥F ∥L2 (T1 ,T2 ) ≤ C
Q2
N
1
2j
e− 4 νt2 dt)1/2
0
j∈Λ,j>J1
so that
+∞
X
j∈Λ,j>J1
2j
1
2j
√ j e− 8 νT1 2 .
ν2
Finally, we get
∥F ∥E ≤ Cν
Q2
N
X
e
− 18
22j
22J1
≤ C′
j∈Λ,j>J1
Q2 X − 1 (16)p 1/2
Q2
(
e 8
)
= C ′′
.
N
N
p∈N
For E(t), we write (for t > T1 )
s
X
X
Q2
E(t) ≤
2p 2j e−2ν(t−T1 )22j
N
j∈Λ,J2 <j≤J1 p∈Λ,p≤j
2s
X
Q
≤
2
22j e−2ν(t−T1 )22j
N
j∈Λ,J2 <j≤J1
From this, we find
Q2 p
#({j ∈ Λ / J2 < j ≤ J1 })
N
so that, using inequality (9.17), we find
∥E∥E ≤ Cν
Q1/2
∥E∥E ≤ C ′ √ .
N
The Space BMO−1 and the Koch and Tataru Theorem
217
Collecting those three estimates on D, E and F , we find that
1/2
⃗ 3 ∥E ≤ C6 Q
√
∥(Id − P0 )U
N
⃗ 3 , we have ∥P0 U
⃗ 3 (t, .)∥∞ ≤ CQ2 so that
For the low frequencies P0 U
p
p
⃗ 3 ∥E ≤ Cν T2 − T1 sup ∥P0 U
⃗ 3 (t.)∥∞ ≤ C7 T2 Q2
∥P0 U
t>0
Collecting all those estimates we find
⃗ 2 (T2 , .)∥BM O−1 ≤ ∥U
⃗ 2 (T1 , .)∥BM O−1
∥U
p
p
Q1/2
Q1/2
+C8 (Q2 T2 + √ )(Q−1/2 + Q2 T2 + √ )
N
N
1/2
p
Q
⃗ 2 ∥E (∥U
⃗ 2 ∥E + Q−1/2 + Q2 T2 + √ )
+C8 ∥U
N
(9.18)
⃗ 2 ∥E ≤ ∥U
⃗ 2 (T1 , .)∥BM O−1
∥U
p
p
Q1/2
Q1/2
+C8 (Q2 T2 + √ )(Q−1/2 + Q2 T2 + √ )
N
N
1/2
p
Q
⃗ 2 ∥E (∥U
⃗ 2 ∥E + Q−1/2 + Q2 T2 + √ )
+C8 ∥U
N
(9.19)
and
If we have the bounds
C8 (Q−1/2 + Q2
p
Q1/2
1
T2 + √ ) ≤
4
N
and
⃗ 2 (T1 , .)∥BM O−1 ≤
C8 ∥U
we will find
⃗ 2 ∥E ≤
∥U
1
16
(9.20)
(9.21)
1/2
p
1
⃗ 2 (T1 , .)∥BM O−1 + Q2 T2 + Q
√ )
(4∥U
2
N
and
⃗ 2 ∥E ≤
C8 ∥U
1
.
4
Moreover, we will have
1/2
p
⃗ 2 (T2 , .)∥BM O−1 ≤ 2∥U
⃗ 2 (T1 , .)∥BM O−1 + 1 (Q2 T2 + Q
√ )
∥U
2
N
⃗ 2 (0, .) = 0, we find that if we split the time interval [0, 12j ] into Q3 intervals
As U
ν2 0
[Ti , Ti+1 ] with T0 = 0 and, for i ≥ 1,
Ti =
1
with Ji = jN −i N3
Q
ν22Ji
(recall that Λ = {j0 < j1 < · · · < jN −1 }) so that (9.17) is satisfied, we shall have, provided
that
1
Q1/2
1
C8 (Q−1/2 + Q2 √ j + √ ) ≤ ,
0
4
ν2
N
218
The Navier–Stokes Problem in the 21st Century (2nd edition)
the inequality
1/2
⃗ 2 (Ti , .)∥BM O−1 ≤ (2i − 1) 1 (Q2 √ 1 + Q
√
)
∥U
2
ν2j0
N
as long as
1
Q1/2
1
C8 2i (Q2 √ j + √ ) ≤ .
4
ν2 0
N
We end the proof by fixing Q large enough to ensure that C8 Q−1/2 ≤ 18 , then j0 and N
large enough to ensure that N > Q3 and that
3
Q1/2
1
1
C8 2Q (Q2 √ j + √ ) ≤ .
4
ν2 0
N
We then get, for T =
1
ν22j0
(which we may assume less than δ, by fixing j0 large enough)
⃗ 2 (T, .)∥ −1 ≤ C∥U
⃗ 2 (T, .)∥BM O−1 ≤
∥U
Ḃ∞,∞
⃗ 0 ∥ −1 ≤ δ ≤
(for small enough δ) while ∥U
Ḃ∞,∞
Theorem 9.7 is thus proved.
9.6
1
2δ
1
1
≤
16C8
2δ
⃗ 1 (T, .)∥ −1 ≥ 2 .
and ∥U
Ḃ∞,∞
δ
Further Results on Ill-posedness
Bourgain and Pavlović [52] proved that the Cauchy problem for the Navier–Stokes equa−1
−1
(for 2 < q < +∞) has
. Ill-posedness in the smaller spaces Ḃ∞,q
tions was ill-posed in Ḃ∞,∞
been also discussed by Germain [205] and Yoneda [510]. The proof of Bourgain and Pavlović
is still valid in this case: in their example, their solutions (which belong to L∞ (0, +∞), E2) as
−1
we have seen it in section 9.4) belong to L∞ (Ḃ∞,q
) for 2 ≤ q ≤ +∞; we can take N arbitrar−1 ≈
ily large; as we have ∥⃗u0 ∥Ḃ∞,q
√Q N
N
1
q
−1
≥
and ∥⃗u(T, .)∥Ḃ∞,∞
1
δ
for some T < 1δ , we conclude
−1
−1 .
−1
≤ ∥⃗u(T, .)∥Ḃ∞,q
easily that we have norm inflation in Ḃ∞,q
for q > 2 since ∥⃗u(T, .)∥Ḃ∞,∞
−1
A common assumption was that the problem was well posed in Ḃ∞,q
when 1 ≤ q ≤ 2,
−1
−1
as in that case Ḃ∞,q ⊂ BM O . However, recently, Wang [495] proved that the Cauchy
−1
, 1 ≤ q ≤ +∞.
problem for the Navier–Stokes equations is ill-posed on all Besov spaces Ḃ∞,q
The norm inflation described by Theorem 9.7 occurs in all those spaces, while existence of
−1
with 1 ≤ q ≤ 2. For such data, a global
mild solutions is granted for small data in Ḃ∞,q
−1
⊂ BM O−1
solution will be given by the Koch and Tataru theorem (Theorem 9.2) (as Ḃ∞,q
when 1 ≤ q ≤ 2), and we know by Lemma 9.1 that we will have control on the BM O−1
norm; but we will have no control at all on the Besov norm.
Another approach of ill-posedness can be found in a paper of Bejenaru and Tao [34].
Following their idea, we shall define ill-posedness or well-posedness in the following way:
Definition 9.1.
If ⃗u0 ∈ bmo−1 with div ⃗u0 = 0, let An (⃗u0 ) be the n-th term which appears in the solution
⃗uϵ =
+∞
X
n=1
ϵn An (⃗u0 )
The Space BMO−1 and the Koch and Tataru Theorem
219
of the Navier–Stokes problem
∂t ⃗uϵ = ν∆⃗uϵ − P div(⃗uϵ , ⃗uϵ ),
div ⃗uϵ = 0,
⃗uϵ (0, .) = ϵ⃗u0
for ϵ small enough.
A Banach space Y is adapted to the Navier–Stokes problem if Y ⊂ bmo−1 (continuous embedding) and if, for every T > 0, there exists a constant Cν,T such that the
operators An satisfy, for every n ≥ 1,
n
∥An (⃗u0 )∥L∞ ((0,T ),Y ) ≤ Cν,T
∥⃗u0 ∥nY .
Thus, if ∥⃗u0 ∥Y <
in L∞ ((0, T ), Y ).
1
Cν,T
, the Navier–Stokes problem with initial value ⃗u0 has a solution
Definition 9.2.
The Navier–Stokes equations are well-posed in a Banach space X if X ∩ bmo−1 is
an adapted Banach space and for every T > 0, there exists ϵT > 0 such that if
⃗u0,n ∈ bmo−1 with ∥⃗u0,n ∥bmo−1 + ∥⃗u0,n ∥X < ϵT and limn→+∞ ∥⃗u0,n ∥X = 0 then
the Navier–Stokes problem with initial value ⃗u0,n has a solution ⃗un on (0, T ) and
limn→+∞ ∥⃗un ∥L∞ ((0,T ),X) = 0.
Proposition 9.4.
Let s > −1, 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞. Then the Navier–Stokes equations are well-posed in
s
Bp,q
.
Rt
Proof. Recall that the operator B(⃗u, ⃗v ) = 0 Wν(t−s) ∗ P div(⃗u ⊗ ⃗v ) ds is bounded on the
space ET described in Theorem 9.1:
∥B(u, v)∥ET ≤ C0 ∥u∥ET ∥v∥ET .
Thus, by Theorem 5.1, if ∥⃗u0 ∥bmo−1 < ϵT (so that ∥Wνt ∗ ⃗u0 ∥ET < 4C1 0 ) and div ⃗u0 = 0, the
Navier–Stokes problem with initial value ⃗u0 has a solution ⃗u in ET with
⃗u =
+∞
X
⃗k
U
k=1
⃗ 1 = Wνt ∗ ⃗u0 and U
⃗ k+1 = −
where U
Pk
j=1
⃗ j C, U
⃗ k+1−j ), and we have
B(U
⃗ k ∥E ≤ Ak C k−1 ∥U
⃗ 1 ∥k ≤ C
∥U
ET
T
0
∥⃗u0 ∥bmo−1
ϵT
k
(where Ak is the k-th Catalan number).
By Lemma 9.1, since the Riesz tranfoms are bounded on BMO, we have moreover
⃗ k+1 (t, .)∥BMO−1 ≤ C1
∥U
k
X
⃗ j ∥E ∥U
⃗ k+1−j ∥E ≤ C1 Ak+1 C k ∥U
⃗ 1 ∥k+1 .
∥U
0
T
T
ET
C
0
j=1
220
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, we find, as BMO−1 ⊂ bmo−1 ,
⃗ k ∥L∞ ((0,T ),bmo−1 ) ≤ C
∥U
∥⃗u0 ∥bmo−1
ϵT
k
.
α+s
⃗ k belongs to {U
⃗ / tα/2 U
⃗ ∈ L∞ ((0, T ), Bp,q
We now prove that U
)} for 0 < α < 1 and
s + α > 0. This is based on the following classical estimates:
r
• if u, v ∈ Bp,q
∩ L∞ with r > 0, then
r
r
r )
∥uv∥Bp,q
≤ C(∥u∥∞ ∥v∥Bp,q
+ ∥v∥∞ ∥u∥Bpq
(where C depends only on r, p and q)
r
• if u ∈ Bp,q
∩ L∞ with r ∈ R and if ρ > r, then
ρ
∥Wt ∗ u∥Bp,q
≤ C(1 + t−
ρ−r
2
r
)∥u∥Bp,q
(where C depends only on r − ρ).
Thus, we have, for 0 < t < T (< +∞),
⃗ 1 ∥ s+α ≤ Cν,T ∥⃗u0 ∥B s
tα/2 ∥U
Bp,q
p,q
and
Z
⃗ k+1 ∥ s+α ≤ Cν,T
∥U
Bp,q
t
k
ds X
1
1
2
(t − s) s
0
1+α
2
j=1
⃗ j ∥ α+s sup
sup tα ∥U
Bp,q
0<s<t
√
⃗ k+1−j ∥∞ .
s∥U
0<s<t
⃗ k ∥ s+α , we find that
If Bk = sup0<t<T tα/2 ∥U
Bp,q
Bk+1 ≤ C2
k
X
⃗ 1 ∥k+1−j
Bj Ak+1−j C0k−j ∥U
ET
j=1
and by induction (taking C2 greater than 1), we find that
⃗ 1 ∥k−1 B1 .
Bk ≤ Ak (C2 C0 )k−1 ∥U
ET
Moreover, we have
⃗ k+1 ∥B s ≤ Cν,T
∥U
p,q
Z
0
t
k
ds X
1
(t − s)
1−α
2
s
1+α
2
j=1
⃗ j ∥ α+s sup
sup tα ∥U
Bp,q
0<s<t
√
⃗ k+1−j ∥∞
s∥U
0<s<t
so that
⃗ k+1 ∥B s ≤ C3
sup ∥U
p,q
0<s<t
k
X
⃗ 1 ∥k+1−j ≤ C3 C4k
Bj Ak+1−j C0k−j ∥U
ET
j=1
∥⃗u0 ∥bmo−1
ϵT
k
s .
∥⃗u0 ∥Bp,q
s
Thus, the Navier–Stokes equations are well-posed in Bp,q
.
In order to disprove well-posedness, we shall use the following lemma,of Bejenaru and
Tao [34]:
The Space BMO−1 and the Koch and Tataru Theorem
221
Lemma 9.2. If the Navier–Stokes equations are well-posed in a Banach space X, then
the bilinear operator
Z t
A2 : ⃗u0 7→
Wν(t−s) ∗ P div((Wνs ∗ ⃗u0 ) ⊗ (Wνs ∗ ⃗u0 )) ds
0
maps the unit ball B0 of bmo−1 ∩X to L∞ ((0, T ), bmo−1 ∩X) and is continuous at ⃗u0 = 0 in
the X norm in the following sense: if ⃗un ∈ B0 and ∥⃗un ∥X → 0, then ∥A2 (⃗un )∥L∞ ((0,T ),X) →
0.
Proof. As bmo−1 ∩ X is adapted, we know that
∥A2 (⃗u0 )∥L∞ (bmo−1 ∩X) ≤ C∥⃗u0 ∥2bmo−1 ∩X .
Moreover, there exists a positive ηT such that, if ∥⃗u0 ∥bmo−1 ∩X < ηT , then the Navier–Stokes
problem with initial value ⃗u0 has a solution in L∞ ((0, T ), bmo−1 ∩ X) given by
⃗u =
+∞
X
Ak (⃗u0 )
k=1
where Ak (λ⃗u0 ) = λk Ak (⃗u0 ) and
∥Ak (⃗u0 )∥L∞ ((0,T ),bmo−1 ∩X) ≤ CηT−k ∥⃗u0 ∥kbmo−1 ∩X .
Now, let us consider ⃗u0,n ∈ B0 and ∥⃗u0,n ∥X → 0. For 0 ≤ δ < 1, let ⃗un,δ be the solution
to the Navier–Stokes problem with initial value δ η2T ⃗u0,n . We have
⃗un,δ =
+∞
X
δk
η k
T
2
k=1
Ak (⃗u0,n ).
Since we assume that the Navier–Stokes equations are well-posed in X, we have
limn→+∞ ∥⃗un,δ ∥L∞ ((0,T ),X) = 0. Dividing with δ η2T , we find that
lim ∥A1 (⃗u0,n ) +
n→∞
X
δ k−1
k≥2
η k−1
T
2
Ak (⃗u0,n )∥L∞ ((0,T ),X) = 0.
As we have
X
δ
k−1
k≥2
η k−1
T
2
∥Ak (⃗u0,n )∥L∞ ((0,T ),X)
X 1 k−1
≤
δ
∥⃗u0,n ∥kbmo−1 ∩X ≤ Cδ
2
k≥2
we find that
lim ∥A1 (⃗u0,n )∥L∞ ((0,T ),X) = 0.
n→∞
In particular, we have
lim ∥
n→∞
Dividing with δ η2T
2
X
δ k−1
k≥2
T
2
Ak (⃗u0,n )∥L∞ ((0,T ),X) = 0.
, we find that
lim ∥A2 (⃗u0,n ) +
n→∞
η k−1
X
k≥3
δ k−2
η k−2
T
2
Ak (⃗u0,n )∥L∞ ((0,T ),X) = 0
222
The Navier–Stokes Problem in the 21st Century (2nd edition)
with
X
δ k−2
k≥3
η k−2
T
2
∥Ak (⃗u0,n )∥L∞ ((0,T ),X) ≤
X 1 k−2
δ
∥⃗u0,n ∥kbmo−1 ∩X ≤ Cδ.
2
k≥3
This gives
lim ∥A2 (⃗u0,n )∥L∞ ((0,T ),X) = 0.
n→∞
Wang’s result [495] is then the following one:
Theorem 9.8.
−1
Let 1 ≤ q ≤ ∞. Then the Navier–Stokes equations are ill-posed in B∞,q
.
Proof. The case q > 2 can be dealt with the example of Bourgain and Pavlović [52]: if we
label ⃗u0,Q,N their example given in equation (9.10), we have
1
1
−2
−1 ≤ CQN q
∥⃗u0,Q,N ∥BMO−1 ≤ CQ, ∥⃗u0,Q,N ∥B∞,q
and
∥A2 (⃗u0,Q,N )(
1
1
−1 ≥ c∥A2 (⃗
−1
, .)∥B∞,q
u0,Q,N )( j0 , .)∥B∞,∞
≥ c′ Q2 .
4j0 ν
4 ν
δ
−1 ) ≤ 2Cδ < 1 and
If we fix δ > 0 small enough to get that Q
(∥⃗u0,Q,N ∥BMO−1 l + ∥⃗u0,Q,N ∥B∞,q
1
j0 large enough to have so that 4j0 ν < T , we find that
∥
δ
δ
−1 < 1,
−1 = 0
lim ∥ ⃗u0,Q,N ∥B∞,q
⃗u0,Q,N ∥bmo−1 ∩B∞,q
N →+∞ Q
Q
while
∥A2 (
δ
′ 2
−1
⃗u0,Q,N )∥L∞ ((0,T ),B∞,q
) ≥c δ .
Q
We conclude with Lemma 9.2.
Wang’s idea for dealing with the case q ≤ 2 is to study the operator A2 restricted to
−1
Xk = {u ∈ Ḃ∞,1
/ û(ξ) = 0 for |ξ| < 14 2k or |ξ| > 4 2k }. If u ∈ Xk , then we have, for
−k
−1 ≈ ∥u∥bmo−1 ≈ 2
∥u∥∞ . In particular, we have, for ⃗u0 ∈ Xk ,
1 ≤ q ≤ +∞, ∥u∥B∞,q
−1
∥A2 (⃗u0 )∥L∞ ((0,T )),B∞,∞
u0 )∥L∞ ((0,T )),bmo−1 ) ≤ C ′ ∥⃗u0 ∥2bmo−1 ≤ C ′′ ∥⃗u0 ∥2B −1 .
) ≤ C∥A2 (⃗
∞,q
As the spectrum of A2 (⃗u0 ) is contained in {ξ / |ξ| < 2k+3 }, we have
1/q
′ 1/q
−1
−1
∥A2 (⃗u0 )∥L∞ ((0,T )),B∞,q
∥A2 (⃗u0 )∥L∞ ((0,T )),B∞,∞
∥⃗u0 ∥2B −1 .
) ≤ Ck
) ≤C k
∞,q
Wang has proved that the estimate is sharp:
sup
⃗
u0 ∈Xk ,div ⃗
u0 =0,∥⃗
u0 ∥B −1 ≤1
1/q
−1
∥A2 (⃗u0 )∥L∞ ((0,T ),B∞,q
) ≥ cT k
(9.22)
∞,q
where cT > 0. Due to Lemma 9.2, such an estimate proves that the Navier–Stokes equations
−1
are ill-posed in B∞,q
when q < +∞.
The Space BMO−1 and the Koch and Tataru Theorem
223
In order to prove (9.22), Wang makes a first simplification: he will prove that
sup
⃗
u0 ∈Xk ,div ⃗
u0 =0,∥⃗
u0 ∥B −1 ≤1
1/q 2k
−1 ≥ γ k
∥P div(⃗u0 ⊗ ⃗u0 )∥B∞,q
2
(9.23)
∞,q
−1 ≤ 1, then ∥⃗
where γ > 0. If ⃗u0 ∈ Xk with ∥⃗u0 ∥B∞,q
u0 ∥∞ ≤ 2k , ∥∆⃗u0 ∥∞ ≤ C23k and
Z
∥Wνt ∗ ⃗u0 − ⃗u0 ∥∞ = ν∥
t
Wνs ∗ ∆⃗u0 ds∥∞ ≤ Cνt23k ;
0
this gives the following estimate
Z t
1/q k
−1 ≤ Ctk
∥A2 (⃗u0 ) −
Wν(t−s) ∗ P div(⃗u0 ⊗ ⃗u0 ) ds∥B∞,q
2 νt23k = Cνt2 24k k 1/q .
0
Similarly, we have
t
Z
Wνs ∗ ∆(⃗u0 ⊗ ⃗u0 ) ds∥∞ ≤ Cνt24k ;
∥Wνt ∗ (⃗u0 ⊗ ⃗u0 ) − ⃗u0 ⊗ ⃗u0 ∥∞ = ν∥
0
this gives the following estimate
Z t
1/q
−1 ≤ Ctk
∥tP div(⃗u0 ⊗ ⃗u0 ) −
Wν(t−s) ∗ P div(⃗u0 ⊗ ⃗u0 ) ds∥B∞,q
νt24k = Cνt2 24k k 1/q .
0
−2k
< T , we find, assuming that (9.23) is true,
sup
−1
∥A2 (⃗u0 )∥L∞ ((0,T ),B∞,q
)
For t0 = η2
⃗
u0 ∈Xk ,div ⃗
u0 =0,∥⃗
u0 ∥B −1 ≤1
∞,q
≥
sup
⃗
u0 ∈Xk ,div ⃗
u0 =0,∥⃗
u0 ∥B −1 ≤1
−1
∥A2 (⃗u0 )(t0 , .)∥B∞,q
∞,q
≥t0
sup
⃗
u0 ∈Xk ,div ⃗
u0 =0,∥⃗
u0 ∥B −1 ≤1
2 4k 1/q
−1 − 2Cνt 2
∥P div(⃗u0 ⊗ ⃗u0 )∥B∞,q
k
0
∞,q
2
≥(γη − 2Cνη )k
1/q
and thus, taking η small enough, we get (9.22).
We may now describe Wang’s example which leads to estimate (9.23). We take Φ ∈
S(R3 ), Φ(0) ̸= 0 and such that its Fourier transform Φ̂ is supported in {ξ / |ξ| √
≤ 1}. For
k
k
k ≥ 10, we define Nk = {l ∈ 4N / 100
≤ l ≤ 10
}. Let αk = 2k ( 13 , 23 , 23 ), βl = 2l (ϵ, ϵ, 1 − 2ϵ2 )
and xl = (0, 0, 2l ). We define Ψ as
X
Ψ = 4λ
Φ(x − xl ) cos(αk · (x − xl )) cos(βl · (x − xl ))
l∈Nk
and
⃗u0 = (−∂2 Ψ, ∂1 Ψ, 0)
where λ > 0 does not depend on k. We have
X
l
Ψ̂ = λ
e−i2 ξ3 (Φ̂(ξ − αk − βl ) + Φ̂(ξ − αk + βl ) + Φ̂(ξ + αk + βl ) + Φ̂(ξ + αk − βl )).
l∈Nk
l
The function e−i2 ξ3 Φ̂(ξ − αk − βl ) is supported in
{ξ ∈ R3 / |ξ − αk − βl | ≤ 1}
224
The Navier–Stokes Problem in the 21st Century (2nd edition)
hence in the area where
|ξ| ≥ |αk | − |βl | − 1 = 2k − 2l − 1 ≥
1 k
2
2
|ξ| ≤ |αk | + |βl | + 1 = 2k + 2l + 1 ≤
3 k
2 .
2
and
Moreover,
X
∥Ψ∥∞ ≤ 4λ∥
|Φ(x − (0, 0, 2l ))∥∞ ≤ Cλ
l∈4N
so that
′
−1 ≤ C λ.
∥⃗u0 ∥Ḃ∞,q
For λ fixed small enough,
have ⃗u0 ∈ Xk .
we
−1
−i2l ξ3
Let Φl,ϵ1 ,ϵ2 = F
e
Φ̂(ξ − ϵ1 αk − ϵ2 βl ) . If we look at the support of the Fourier
transform of
l
m
2
∂p F −1 e−i2 ξ3 Φ̂(ξ − ϵ1 αk − ϵ2 βl ) ∂q,r
F −1 e−i2 ξ3 Φ̂(ξ − ϵ3 αk − ϵ4 βm )
(with ϵi ∈ {−1, 1}), it is contained in
{ξ ∈ R3 / |ξ − (ϵ1 + ϵ3 )αk − ϵ2 βl − ϵ4 βm | ≤ 2}.
• if ϵ1 = ϵ3 , we find 12 2k+1 ≤ |ξ| ≤ 32 2k+1
• if ϵ1 ̸= ϵ3 and l ̸= m (hence sup(l, m) ≥ inf(l, m) + 4), we find that 34 2sup(l,m) ≤ |ξ| ≤
5 sup(l,m)
42
• if ϵ1 ̸= ϵ3 , l = m and ϵ2 ̸= ϵ4 , we find |ξ| ≤ 2
• if ϵ1 ̸= ϵ3 , l = m and ϵ2 = ϵ4 , we find 74 2l ≤ |ξ| ≤ 94 2l .
The spectral domains are well separated and we find that
−1
∥P div(⃗u0 ⊗ ⃗u0 )∥B∞,q
1
≥ λ
C
!1/q
X
2−lq ∥P⃗vl ∥q∞
l∈Nk
where
P

∂ (∂ Φ
∂ Φ
) − ∂2 (∂1 Φl,ϵ1 ,ϵ2 ∂2 Φl,−ϵ1 ,ϵ2 )
P(ϵ1 ,ϵ2 )∈{−1,1}2 1 2 l,ϵ1 ,ϵ2 2 l,−ϵ1 ,ϵ2
⃗vl =  (ϵ1 ,ϵ2 )∈{−1,1}2 ∂2 (∂1 Φl,ϵ1 ,ϵ2 ∂1 Φl,−ϵ1 ,ϵ2 ) − ∂1 (∂2 Φl,ϵ1 ,ϵ2 ∂1 Φl,−ϵ1 ,ϵ2 ) .
0
The Fourier transform of ⃗vl is supported in
{ξ ∈ R3 / |ξ − 2βl | ≤ 2} ∪ {ξ ∈ R3 / |ξ + 2βl | ≤ 2}.
In particular, we have 74 2l ≤ |ξ| ≤ 94 2l , 74 2l ≤ |ξ3 | ≤ 94 2l and 74 ϵ2l ≤ |ξ1 |, |ξ2 | ≤ 49 ϵ2l (k
large enough for a fixed ϵ). If θ ∈ D(R) is equal to 1 on 7/4 ≤ |t| ≤ 9/4 and is supported in
1 ≤ t ≤ 3 and if θ0 (t) = tθ(t) and θ1 (t) = 1t θ(t), we control easily the Riesz ransforms Rj
(1 ≤ j ≤ 3) of the components vl,1 , vl,2 of ⃗vl :
ξ3
|ξ|
R3 vl,p = F −1 iθ0 ( l )θ1 ( l )v̂l,p
2
2
The Space BMO−1 and the Koch and Tataru Theorem
225
and, for j = 1 or j = 2,
ξj
|ξ|
Rj vl,p = F −1 iϵθ0 ( l )θ1 ( l )v̂l,p .
2ϵ
2
As θ0 ∈ F(L1 (R)) and θ1 (|ξ|) ∈ F(L1 (R3 )), we find that
∥P⃗vl − ⃗vl ∥∞ ≤ Cϵ∥⃗vl ∥∞
and thus (for ϵ small enough)
−1
∥P div(⃗u0 ⊗ ⃗u0 )∥B∞,q
λ
≥
2C
!1/q
X
2−lq ∥⃗vl ∥q∞
.
l∈Nk
Similarly, for 1 ≤ p, q ≤ 2, the function ∂p Φl,ϵ1 ,ϵ2 ∂q Φl,−ϵ1 ,ϵ2 is supported in |ξ1 −
ξ1 −ξ2
9 l
2
2ϵ2 2l |, |ξ2 − 2ϵ2 2l | ≤ 2, hence in 74 ϵ2l ≤ | ξ1 +ξ
2 | ≤ 4 ϵ2 , | 2 | ≤ 2.
We write
P

∂ Φ
− ∂1 Φl,ϵ1 ,ϵ2 ∂2 Φl,−ϵ1 ,ϵ2
2 ∂ Φ
∂1 + ∂2 P(ϵ1 ,ϵ2 )∈{−1,1} 2 l,ϵ1 ,ϵ2 2 l,−ϵ1 ,ϵ2

⃗vl =
(ϵ1 ,ϵ2 )∈{−1,1}2 ∂1 Φl,ϵ1 ,ϵ2 ∂1 Φl,−ϵ1 ,ϵ2 − ∂2 Φl,ϵ1 ,ϵ2 ∂1 Φl,−ϵ1 ,ϵ2
2
0
P

∂ Φ
+ ∂1 Φl,ϵ1 ,ϵ2 ∂2 Φl,−ϵ1 ,ϵ2
2 ∂ Φ
∂1 − ∂2 P (ϵ1 ,ϵ2 )∈{−1,1} 2 l,ϵ1 ,ϵ2 2 l,−ϵ1 ,ϵ2

+
(ϵ1 ,ϵ2 )∈{−1,1}2 −∂1 Φl,ϵ1 ,ϵ2 ∂1 Φl,−ϵ1 ,ϵ2 − ∂2 Φl,ϵ1 ,ϵ2 ∂1 Φl,−ϵ1 ,ϵ2
2
0
∂1 + ∂2 ⃗
∂1 − ∂2 ⃗
=
Vl +
Wl
2
2
with
∥
∂1 + ∂2 ⃗
ϵ
⃗l ∥∞ and ∥ ∂1 − ∂2 W
⃗ l ∥∞ ≤ C∥W
⃗ l ∥∞ ≤ C ′ 22k .
Vl ∥∞ ≥ 2l ∥V
2
C
2
We have
Φl,ϵ1 ,ϵ2 = Φ(x − xl )eiϵ1 αk ·(x−xl ) eiϵ2 βl ·(x−xl )
so that, for p = 1 or p = 2,
|∂p Φl,ϵ1 ,ϵ2 − iϵ1
2k p
Φl,ϵ1 ,ϵ2 | ≤ Cϵ2l
3
hence we have, for
X
Vl,1 =
∂2 Φl,ϵ1 ,ϵ2 ∂2 Φl,−ϵ1 ,ϵ2 − ∂1 Φl,ϵ1 ,ϵ2 ∂2 Φl,−ϵ1 ,ϵ2 ,
(ϵ1 ,ϵ2 )∈{−1,1}2
the estimate
|Vl,1 − 2
22k
9
X
Φl,ϵ1 ,ϵ2 Φl,−ϵ1 ,ϵ2 | = |Vl,1 − 4
(ϵ1 ,ϵ2 )∈{−1,1}2
22k
Φ(x − xl )2 cos βl · (x − xl ))| ≤ Cϵ22k .
9
Thus,
∥Vl,1 ∥∞ ≥ |Vl,1 (xl )| ≥ 4
22k
Φ(0)2 − Cϵ22k .
9
226
The Navier–Stokes Problem in the 21st Century (2nd edition)
We may conclude that
−1
∥P div(⃗u0 ⊗ ⃗u0 )∥B∞,q
!1/q
λ
≥
2C
λ
≥ ′
C
X
2−lq ∥⃗vl ∥q∞
l∈Nk
!1/q
X
⃗l ∥q∞
∥V
!1/q
′
−C λ
l∈Nk
X
2
−lq
⃗ l ∥q∞
∥W
l∈Nk
λ
22k
#(Nk )1/q (4
Φ(0)2 − Cϵ22k ) − C ′′ λ22k
′
C
9
≥C1 λk 1/q 22k (1 − C2 ϵ) − C3 λ22k .
≥
Thus, we get (9.23) and we prove the theorem.
Further, this result has been generalized by Cui [135] to the setting of logarithmically
−1
improved Besov spaces. Recall that the Besov spaces Ḃ∞,q
(1 ≤ q ≤ +∞) are characterized
by the homogeneous Littlewood–Paley decomposition as
X
−1
f ∈ Ḃ∞,q
⇔f =
∆j f in S′ and (2−j ∥∆j f ∥∞ )j∈Z ∈ lq .
j∈Z
Homogeneity is important mainly for existence of global solutions (compare the Koch and
Tataru theorem for bmo−1 [local solutions] and BMO−1 [global solutions]). The (non−1
(1 ≤ q ≤ +∞) is characterized by the Littlewood–Paley
homogeneous) Besov space B∞,q
decomposition as
−1
f ∈ B∞,q
⇔ S0 f ∈ L∞ and (2−j ∥∆j f ∥∞ )j∈N ∈ lq .
−1
We have B∞,q
⊂ bmo−1 when 1 ≤ q ≤ 2.
Definition 9.3.
−1,σ
is defined by
For σ ≥ 0, the logarithmically improved Besov space B∞,q
−1,σ
f ∈ B∞,q
⇔ S0 f ∈ L∞ and (2−j (j + 1)σ ∥∆j f ∥∞ )j∈N ∈ lq .
Proposition 9.5.
Let σ ≥ 0, 1 ≤ q ≤ ∞ and 0 < T < +∞. Then the following assertions are equivalent
−1,σ
(A) f ∈ B∞,q
dt
σ
q
−1,σ
(B) t1/2 (ln( eT
t )) ∥Wνt ∗ f ∥∞ ∈ L ((0, T ), t ) and the norms ∥f ∥B∞,q ,
t1/2 (ln(
eT σ
)) ∥Wνt ∗ f ∥∞
t
Lq ((0,T ), dt
t )
and
sup t1/2 (ln(
0<t<T
are equivalent.
eT σ
eT
)) ∥Wνt ∗ f ∥∞ + t1/2 (ln( ))σ ∥Wνt ∗ f ∥∞
t
t
Lq ((0,T ), dt
t )
The Space BMO−1 and the Koch and Tataru Theorem
227
Proof.
P
(A) =⇒ (B): We write f = S0 f + j≥0 ∆j f with S0 f ∈ L∞ and ∥∆j f ∥∞ = 2j (1 + j)−σ ϵj
with (ϵj )j∈N ∈ lq . We have ∥Wνt ∗ S0 f ∥∞ ≤ ∥S0 f ∥∞ ; on the other hand, we have
∥Wνt ∗ ∆j f ∥∞ = ∥∆Wνt ∗
1
∆j f ∥∞ ≤ C min(1, (νt)−1 2−2j )∥∆j f ∥∞ .
∆
Now, if t ≤ T , we choose j0 so that 1 ≤ 4j0 Tt < 4 and we write
t1/2 (ln(
eT σ
)) ∥Wνt ∗ f ∥∞
t
X
ϵj
ϵj
+
2(j0 −j)
)
σ
(1 + j)
(1 + j)σ
j>j0
j≤j0
X (j−j0 )
X
j0
2 2 ϵj +
≤ Cν,T,σ (2− 2 ∥S0 f ∥∞ +
2(j0 −j) ϵj ).
≤ Cν,T (1 + j0 )σ (2−j0 ∥S0 f ∥∞ +
X
j≤j0
2(j−j0 )
j>j0
(B) =⇒ (A): Using the integrability of the kernel of the convolution operator e−νT ∆ S0 ,
we write, for T /2 ≤ t ≤ T ,
′
∥S0 f ∥∞ ≤ Cν,T ∥WνT f ∥∞ ≤ Cν,T
(2t)1/2 (ln(
eT σ
)) ∥Wνt f ∥∞
t
and get S0 f ∈ L∞ .
Similarly, we write, for j ≥ 0 and 1 ≤ 4j Tt ≤ 4, using the integrability of the kernel of
−4νT ∆
e
∆0 ,
2−j (1 + j)σ ∥∆j f ∥∞ ≤ Cν,T t1/2 (ln(
eT σ
eT
)) ∥W4ν Tj ∗ f ∥∞ ≤ Cν,T t1/2 (ln( ))σ ∥Wνt ∗ f ∥∞ .
4
t
t
σ
Thus, we have the equivalence of (A) and (B). The control of sup0<t<T t1/2 (ln( eT
t )) ∥Wνt ∗
−1,σ
−1,σ
f ∥∞ is then a consequence of the embeddding B∞,q ⊂ B∞,∞ .
−1,σ
−1
If the case q = 2, we have B∞,2
⊂ B∞,2
⊂ bmo−1 . Thus, we know that we can solve
−1,σ
the Navier–Stokes problem on (0, T ) × R3 with initial value ⃗u0 if the norm of ⃗u0 in B∞,2
is small enough. Cui’s result reads as:
Theorem 9.9.
−1,σ
• for σ ≥ 21 , the Cauchy problem is locally well posed for small data in B∞,2
: for every
T > 0,
lim
sup
sup ∥⃗u(t, .)∥B −1,σ = 0.
δ→0
∥⃗
u0 ∥
−1,σ <δ
B∞,2
0<t<T
∞,2
−1,σ
• for 0 ≤ σ < 12 , the Cauchy problem is ill-posed in B∞,q
.
Proof. The proof follows Yoneda [510] for well-posedness and Wang for ill-posedness [495].
Case σ ≥ 1/2:
−1,σ
As f ∈ B∞,2
⇔ Wνt ∗ f ∈ XT , where
∥F ∥XT = sup t1/2 (ln(
0<t<T
eT σ
eT
)) ∥F (t, .)∥∞ + t1/2 (ln( ))σ ∥F (t, .)∥∞
t
t
,
L2 ((0,T ), dt
t )
228
The Navier–Stokes Problem in the 21st Century (2nd edition)
we just need to prove that
t
Z
Wν(t−s) ∗ P div(⃗u ⊗ ⃗v ) ds
B(⃗u, ⃗v ) =
0
−1,σ
maps boundedly XT × XT to XT ∩ L∞ ((0, T ), B∞,2
).
We start from the inequalities
Z t
1
p
∥B(⃗u, ⃗v )(t, .)∥∞ ≤ C
∥⃗u(s, .)∥∞ ∥⃗v (s, .)∥∞ ds
ν(t − s)
0
and
t
Z
1
p
∥⃗u(s, .)∥∞ ∥⃗v (s, .)∥∞ ds.
ν(t + θ − s)
∥Wνθ ∗ B(⃗u, ⃗v )(t, .)∥∞ ≤ C
0
Thus, defining
ω(t) = ln(
eT
),
t
∥f ∥YT = sup t1/2 (ω(t))σ |f (t)| + ∥ω σ f ∥L2 ((0,T ),dt) ,
0<t<T
Z
J(f, g)(t) =
t
√
0
and
Z
Kt (f, g)(θ) =
0
t
√
1
f (s)g(s) ds,
t−s
1
f (s)g(s) ds,
t+θ−s
we are going to prove
∥J(f, g)∥YT ≤ CT ∥f ∥YT ∥g∥YT
(9.24)
and
sup ∥ω σ (θ)Kt (f, g)(θ)∥L2 ((0,T ),dθ) ≤ CT ∥f ∥YT ∥g∥YT .
(9.25)
0<t<T
For 0 < s < t < T , we have
1 ≤ ω σ (t) ≤ ω σ (s) ≤ ω 2σ (s)
so that
t
1
√
ω (t)|J(f, g)(t)| ≤
ω σ (s)|f (s)|ω σ (s)|g(s)| ds
t
−s
0
r Z t/2
Z t/2
2
≤
(
ω 2σ f 2 ds)1/2 (
ω 2σ g 2 ds)1/2
t 0
0
σ
Z
+ ( sup s1/2 ω σ (s)|f (s)|)( sup s1/2 ω σ (s)|g(s)|)
0<s<t
0<s<t
Z
t
t/2
√
1 ds
t−s s
1
≤C √ ∥f ∥YT ∥g∥YT .
t
On the other hand, we have
r
Z t/2
2 σ
σ
ω (t)|J(f, g)(t)| ≤
ω (t)
|f (s)||g(s)| ds
t
0
Z t
1/2 σ
√
+ ( sup s ω (s)|f (s)|)
0<s<t
t/2
1
√ ω σ (s)|g(s)| ds.
t−s s
(9.26)
The Space BMO−1 and the Koch and Tataru Theorem
229
Rt
1√
The operator h 7→ t/2 √t−s
h(s) ds maps boundedly L1 ((0, T ), dt) to L1 ((0, T ), dt) and
s
L∞ ((0, T ), dt) to L∞ ((0, T ), dt), hence L2 ((0, T ), dt) to L2 ((0, T ), dt). Thus, the second
term in the right-hand side of (9.26) is well controlled in L2 ((0, T ), dt).
For the first term, we define
r
1 σ
A(t, s) = 1s<t
ω (t)|f (s)||g(s)|
t
and write
r
2 σ
ω (t)
t
Z
t/2
√ Z
|f (s)||g(s)| ds ≤ 2
0
A(t, s) ds.
0
The Minkowski inequality then gives
Z T Z T
Z
(
(
A(t, s) ds)2 dt)1/2 ≤
0
T
0
T
Z
0
Z
T
A(t, s)2 dt)1/2 ds
(
0
T
=
Z
(
0
T
ω 2σ (t)
s
dt 1/2
) |f (s)||g(s)| ds
t
Z T
1
1
=√
ω σ+ 2 (s)|f (s)||g(s)| ds
2σ + 1 0
Z T
1
≤√
ω 2σ (s)|f (s)||g(s)| ds
2σ + 1 0
(since 1 ≤ ω and 1/2 ≤ σ). Thus, inequality (9.24) is proved.
In order to estimate Kt (f, g)(θ), we write
Z t
1
1
√
|f (s)|g(s)| ds + 1t>θ
|f (s)|g(s)| ds.
t−s
θ−s
0
θ
Rθ 1
We know that we can control ω σ (θ) 0 √θ−s
|f (s)|g(s)| ds in L2 ((0, T ), dθ). For the second
term, we define
1
At (θ, s) = 1θ<s √
ω σ (θ)|f (s)||g(s)|
t−s
Z
θ
|Kt (f, g)(θ)| ≤
√
and write
σ
Z
ω (θ)
θ
t
1
√
|f (s)|g(s)| ds =
t−s
Z
t
At (θ, s) ds.
0
The Minkowski inequality then gives
Z T Z t
Z t Z T
(
(
At (θ, s) ds)2 dθ)1/2 ≤
(
At (θ, s)2 dθ)1/2 ds
0
0
0
0
Z t Z s
1
=
(
ω 2σ (θ)dθ)1/2 √
|f (s)||g(s)| ds
t
−s
0
0
Z t
√
1
≤C
ω σ (s) s √
|f (s)||g(s)| ds
t
−s
0
Z t
√
√
1
ds
√
≤C
ω σ (s) s|f (s)ω σ (s) s||g(s)| √
s
t−s
0
≤πC( sup s1/2 ω σ (s)|f (s)|)( sup s1/2 ω σ (s)|g(s)|).
0<s<t
Thus, inequality (9.25) is proved.
0<s<t
230
The Navier–Stokes Problem in the 21st Century (2nd edition)
Case σ < 1/2:
If we look at the example of Wang, we have ⃗u0 ∈ Xk with ∥⃗u0 ∥∞ ≈ 2k . In particular,
∥⃗u0 ∥B −1,σ ≈ k σ .
∞,2
We begin with getting rid of the time, as in Wang’s proof: we write
Z t
∥A2 (⃗u0 ) −
Wν(t−s) ∗ P div(⃗u0 ⊗ ⃗u0 ) ds∥B −1,σ ≤ Ctk σ+1/2 2k νt23k = Cνt2 24k k σ+1/2 .
∞,2
0
and
t
Z
∥tP div(⃗u0 ⊗ ⃗u0 ) −
0
Wν(t−s) ∗ P div(⃗u0 ⊗ ⃗u0 ) ds∥B −1,σ ≤ Ctk σ+1/2 νt24k = Cνt2 24k k σ+1/2 .
∞,2
For t0 = η2−2k < T , we find
∥A2 (⃗u0 )(t0 , .)∥B −1,σ ≥ η2−2k ∥P div(⃗u0 ⊗ ⃗u0 )∥B −1,σ − 2Cνη 2 k σ+1/2 .
∞,2
∞,2
Moreover, we have
∥P div(⃗u0 ⊗ ⃗u0 )∥B −1,σ
∞,2
1
≥
C
!1/2
X
2σ −2l
l 2
∥P⃗vl ∥q∞
l∈Nk
1
≥ ′ kσ
C
!1/2
X
−2l
2
∥P⃗vl ∥q∞
l∈Nk
and we have seen that
!1/2
X
−2l
2
∥P⃗vl ∥2∞
≥ C1 k 1/2 22k (1 − C2 ϵ) − C3 22k .
l∈Nk
Thus, we have ∥⃗u0 ∥B −1,σ ≈ k σ and
∞,2
∥A2 (⃗u0 )∥L∞ ((0,T ),B −1,σ ) ≥ γk 2σ k 1/2−σ .
∞,2
−1,σ
As σ < 1/2, we conclude that the Navier–Stokes equations are ill-posed in B∞,2
.
9.7
Large Data for Mild Solutions
We saw by Theorem 9.2 that we have a global mild solution to
⃗ + div F
∂t ⃗u = ν∆⃗u − div(⃗u ⊗ ⃗u) − ∇p
with div ⃗u = 0 and ⃗u|t=0 = ⃗u0 , provided that ⃗u0 is small enough in BM O−1 (and F small
enough in F∞ ). Theorem 9.7 shows that there is little hope that the sole smallness of
−1
∥⃗u0 ∥Ḃ∞,∞
(instead of ∥⃗u0 ∥BM O−1 ) should be sufficient for such a result.
On the other hand, we might wonder if global mild solutions exist for some large initial
−1
values in BM O−1 (and even in Ḃ∞,∞
). The answer is obviously positive, as can be seen by
the example of ⃗u0 ∈ E2 (Theorem 9.6): for such initial value, there is no restriction at all
on the size of ⃗u0 .
Other examples of “two-dimensional” vector fields that lead to global mild solutions
with no restriction on their size will be given in Chapter 10:
The Space BMO−1 and the Koch and Tataru Theorem
231
ˆ Majda and Bertozzi’s two-and-a-half dimensional flows [40] (Proposition 10.1); those
initial values may be perturbated by a small enough vector field that belongs to
Ḣ 1/2 (R3 ), as shown by Gallagher [195] (Theorem 10.2)
ˆ regular enough axisymmetric flows with no swirl, as shown by Ladyzhenskaya [295],
Uchovskii and Yudovich [486] and Leonardi, Malek, Nečas, and Pokorný [326] (Theorem 10.4)
ˆ vector fields with helical symmetry, as shown by Mahalov, Titi, and Leibovich in [347]
(Theorem 10.7)
ˆ strong Beltrami flows [40, 129], also known as Trkalian flows [300] as they were first
studied by Trkal [477] (Theorem 10.16)
In a series of papers, Chemin and Gallagher presented more genuinely three-dimensional
−1
examples of vector fields that are large in Ḃ∞,∞
and lead to global mild solutions. The series
began with [107], followed by [108], [109], [111] (joint paper with Paicu) and [112] (joint
paper with Zhang). This is currently an active field of research, as it can be seen in the
recent papers of Kukavica, Rusin and Ziane [288], Paicu and Zhang [391] or Wong [505].
We present here (a slightly simplified version of) Chemin and Gallagher’s result [108]:
Theorem 9.10.
Let ω ∈ D(R) a non-negative even function supported in [−1, 1] (with ω =
̸ 0), and let
Ω ∈ S(R) the inverse Fourier transform of ω: FΩ = ω. For ϵ ∈ (0, 1), let Φϵ be the function
Φϵ (x) = | ln(ϵ)|1/5 Ω(x1 )Ω(
and let
x2
ϵ
1
2
) cos(
x3
)Ω(x3 )
ϵ
  2   
0
−∂2 Φϵ
αϵ
⃗ ∧  0  = ∂1 ∂2 Φϵ  =  βϵ  .
=∇
−∂2 Φϵ
0
0

⃗u0,ϵ
−1
:
Then div ⃗u0,ϵ = 0, ⃗u0,ϵ is large in Ḃ∞,∞
−1
≈ | ln(ϵ)|1/5
∥⃗u0,ϵ ∥Ḃ∞,∞
and, for ϵ small enough, the Cauchy problem for the Navier–Stokes equations with initial
value ⃗u0,ϵ (and forcing term f⃗ = 0) has a global mild solution ⃗u such that ⃗u ∈ L2 L∞ .
Proof. In light of Theorem 8.16 and its Corollary 8.1, we will work mainly in the setting of
the frequency variable ξ. Recall that we have the following embedings between Fourier-Herz
spaces and the Besov spaces
−1
−1
−1
−1
FB1,1
⊂ FB1,2
⊂ Ḃ∞,2
⊂ BM O−1 ⊂ Ḃ∞,∞
.
If A is the Wiener space A = FL1 , we have, for the Littlewood–Paley decomposition of a
function h,
Z
X
dξ
∥h∥F B−1 ≈
2−j ∥∆j h∥A ≈ |ĥ(ξ)|
1,1
|ξ|
j∈Z
while
−1
∥h∥Ḃ∞,∞
≈
X
j∈Z
2−j ∥∆j h∥∞ .
232
The Navier–Stokes Problem in the 21st Century (2nd edition)
The function Φϵ has a precise frequency localization:
1
5
5
Φ̂ϵ (ξ) = | ln(ϵ)|1/5 ϵ1/2 ω(ξ1 )ω(ϵ 2 ξ2 )(ω(ξ3 + ) + ω(ξ3 − ))
ϵ
ϵ
so that
Φ̂ϵ (ξ) ̸= 0 ⇒ ξ ∈ [−1, 1] × [−
1
ϵ
1
2
,
1
5
5
5
5
] × ([−1 − , 1 − ] ∪ [−1 + , 1 + ]),
ϵ
ϵ
ϵ
ϵ
ϵ
1
2
and thus (for ϵ small enough)
| |ξ| −
5
1
1
|≤2+ 1 ≤ .
ϵ
ϵ
2
ϵ
Thus, we have
Z
1
∥βϵ ∥F B−1 = ∥∂1 ∂2 Φϵ ∥F B−1 ≈ ϵ
1,1
1,1
|ξ1 ξ2 | |Φ̂ϵ (ξ)| dξ = 2ϵ 2 | ln ϵ|1/5 ∥ω∥1 ∥sω∥21
(9.27)
|ξ22 | |Φ̂ϵ (ξ)| dξ = 2| ln ϵ|1/5 ∥ω∥21 ∥s2 ω∥1 .
(9.28)
and
∥αϵ ∥F B−1 = ∥∂22 Φϵ ∥F B−1 ≈ ϵ
1,1
Z
1,1
Similarly, we have
−1
∥βϵ ∥Ḃ∞,∞
≈ ϵ∥βϵ ∥∞
ϵ
≤
(2π)3
Z
1
|β̂ϵ | dξ ≤ Cϵ 2 | ln ϵ|1/5
and, since α̂ϵ is non-negative,
−1
∥αϵ ∥Ḃ∞,∞
≈ ϵ∥αϵ ∥∞ =
ϵ
(2π)3
Z
|α̂ϵ | dξ ≈ | ln ϵ|1/5 .
Thus, we get that
−1
∥⃗u0,ϵ ∥Ḃ∞,∞
≈ ∥⃗u0,ϵ ∥B−1 ≈ | ln ϵ|1/5 .
1,∞
(9.29)
Another useful estimate will be
1
∥⃗u0,ϵ ∥Ḣ −1 ≈ ϵ∥⃗u0,ϵ ∥2 ≈ ϵ 4 | ln ϵ|1/5 .
(9.30)
We shall now look for the mild solution ⃗u in L2 A = L2 FL1 . We define the bilinear
operator B as
Z t
B(⃗v , w)
⃗ =
Wν(t−s) ∗ P div(⃗v ⊗ w)
⃗ ds
0
and we look for a solution of
⃗u = Wνt ∗ ⃗u0,ϵ − B(⃗u, ⃗u).
We know by Theorem 8.16 that
∥B(⃗v , w)∥
⃗ L2 A ≤ C0 ∥⃗v ∥L2 A ∥w∥
⃗ L2 A
but we cannot proceed directly with the Banach contraction principle as
∥Wνt ∗ ⃗u0,ϵ ∥L2 A ≈ ∥⃗u0,ϵ ∥F B−1 ≈ | ln ϵ|1/5 .
1,2
The Space BMO−1 and the Koch and Tataru Theorem
233
⃗ 0 = Wνt ∗⃗u0,ϵ , U
⃗ 1 = B(U
⃗ 0, U
⃗ 0 ) and let V
⃗ = ⃗u − U
⃗ 0. V
⃗ must be solution to the equation
Let U
⃗ =U
⃗ 1 − B(U
⃗ 0, V
⃗ ) − B(V
⃗ ,U
⃗ 0 ) − B(V
⃗ ,V
⃗ ).
V
(9.31)
⃗ 1 ∥L2 A is much smaller than C0 ∥U
⃗ 0 ∥|2 2 ≈ | ln ϵ|2/5 . We have
We begin by checking that ∥U
L A
seen (in Theorem 8.16 ) that
⃗ 1 ∥L2 A ≤ C1 ∥ √
∥U
1
⃗0 ⊗ U
⃗ 0 )∥L1 A
div(U
−∆
We have
  



γ1
Wνt ∗ αϵ
∂1 (γ12 ) + ∂2 (γ1 γ2 )
⃗ 0 = γ2  =  Wνt ∗ βϵ  and div(U
⃗0 ⊗ U
⃗ 0 ) = ∂1 (γ1 γ2 ) + ∂2 (γ22 ) .
U
0
0
0
We have
1
∥√
∂j (γ1 γ2 )∥L1 A ≤
−∆
and thus
∥√
+∞
Z
∥γ1 ∥A ∥γ2 ∥A ds ≤ ∥γ1 ∥L2 A ∥γ2 ∥L2 A
0
1
1
∂j (γ1 γ2 )∥L1 A ≤ C∥αϵ ∥F B−1 ∥βϵ ∥F B−1 ≤ C ′ ϵ 2 | ln ϵ|2/5 .
1,2
1,2
−∆
Similarly, we have
∥√
For the last term
1
∂2 (γ22 )∥L1 A ≤ C∥βϵ ∥2F B−1 ≤ C ′ ϵ| ln ϵ|2/5 .
1,2
−∆
√ 1 ∂1 (γ 2 ),
1
−∆
Eϵ = [−2, 2] × [−
2
ϵ
1
2
,
we notice that the spectral support of γ12 is contained in
2
5
5
5
5
] × ([−2 − , 2 − ] ∪ [−2, 2] ∪ [−2 + , 2 + ]).
ϵ
ϵ
ϵ
ϵ
ϵ
1
2
1
Thus, we have (splitting the integral on ξ between |ξ| > R and |ξ| ≤ R with R << ϵ− 2 ),
∥√
1
∂1 (γ12 )∥L1 A =(2π)3
−∆
≤(2π)3
Z
+∞
Z
+∞
Z
0
Z
Eϵ
|ξ1 |
|γ̂1 ∗ γ̂1 | dξ
|ξ|
∥γ̂1 ∗ γ̂1 ∥∞ dξ
0
Z
ξ∈Eϵ ,|ξ|<R
+∞ Z
2
|γ̂1 ∗ γ̂1 | dξ
0
ξ∈Eϵ ,|ξ|>R R
2
≤(2π)3 (|Eϵ ∩ B(0, R)|∥γ̂1 ∥2L2 L2 + ∥γ̂1 ∥2L2 L1 )
R
+ (2π)3
1
with |Eϵ ∩ B(0, R)| ≤ 32R, ∥γ̂1 ∥2L2 L2 ≤ C∥αϵ ∥2Ḣ −1 ≤ C ′ ϵ 2 | ln ϵ|2/5 and ∥γ̂1 ∥2L2 L1 ≤
1
C∥αϵ ∥2F B−1 ≤ C ′ | ln ϵ|2/5 . Taking R = ϵ− 4 , we find that
1,2
∥√
Thus, we have
1
1
∂1 (γ12 )∥L1 A ≤ Cϵ 4 | ln ϵ|2/5 .
−∆
1
⃗ 1 ∥L2 A ≤ C1 ϵ 4 | ln ϵ|2/5 .
∥U
(9.32)
234
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ of Equation (9.31). As
We may now proceed to the construction of the solution V
2
⃗
2
∥U0 ∥L A is large, we must change the norm on L A. Let λ > 0 and
∥U ∥λ = ∥µλ (t)U ∥L2 A , where µλ (t) = e−λ
We have obviously, for Aλ = eλ
R +∞
0
⃗ 0 (s,.)∥2
∥U
A
ds
Rt
0
⃗ 0 (s,.)∥2 ds
∥U
A
.
,
∥U ∥λ ≤ ∥U ∥L2 A ≤ Aλ ∥U ∥λ .
Thus,
⃗ ,W
⃗ )∥λ ≤ ∥B(V
⃗ ,W
⃗ )∥L2 A ≤ C0 A2λ ∥V
⃗ ∥λ ∥W
⃗ ∥λ
∥B(V
and
⃗ 1 ∥λ ≤ C1 ϵ1/4 | ln ϵ|2/5 .
∥U
⃗ 0, V
⃗ ) and B(V
⃗ ,U
⃗ 0 ), we define
In order to estimate B(U
Z tZ
2
Z(V, W ) =
e−ν(t−s)|ξ| |ξ| |V (s, η)| |W (s, ξ − η)| dη ds.
0
⃗ 0 (t, ξ)| and V (t, ξ) = |F V
⃗ (t, ξ)|, so that
We write U0 (t, ξ) = |F U
⃗ 0, V
⃗ ) + B(V
⃗ ,U
⃗ 0 ))| ≤ 2(2π)3 Z(U0 , V )(t, ξ).
|F(B(U
Thus,
⃗ 0, V
⃗ ) + B(V
⃗ ,U
⃗ 0 )∥λ ≤ 2(2π)3 ∥µλ (t)Z(U0 , V )(t, ξ)∥L2 L1 .
∥B(U
We write
Z tZ
µλ (t)Z(U0 , V )(t, ξ) =
2
e−ν(t−s)|ξ| |ξ|
0
µλ (t)
U0 (s, η)µλ (s)V (s, ξ − η) dη ds.
µλ (s)
As µλ (t) ≤ µλ (s), we find
∥µλ (t)Z(U0 , V )(t, ξ)∥L1 L1 (|ξ| dξ) ≤ Z(U0 , µλ V ) ≤ C0 ∥U0 ∥L2 L1 ∥µλ V ∥L2 L1 .
2
As e−ν(t−s)|ξ| ≤ 1, we find
∥µλ (t)Z(U0 , V )(t, ξ)∥L1 ( dξ )
|ξ|
2
Z t
µλ (t)
≤(
∥U0 (s, .)∥21 ds)1/2 ∥µλ V ∥L2 L1
µλ (s)
0
with
Z t
0
µλ (t)
µλ (s)
2
∥U0 (s, .)∥21
t
Z
e−2λ
ds =
Rt
s
∥U0 (σ,.)∥21 dσ
∥U0 (s, .)∥21 ds
0
"
=
e−2λ
Rt
s
∥U0 (σ,.)∥21 dσ
−2λ
#s=t
s=0
1
≤ .
2λ
This gives
r
∥µλ (t)Z(U0 , V )(t, ξ)∥L∞ L1 ( dξ ) ≤
|ξ|
1
∥µλ V ∥L2 L1
2λ
and finally
s
∥µλ (t)Z(U0 , V )(t, ξ)∥L2 L1 ≤
For λ = C4 ∥U0 ∥2L2 L1 , we find
C0 ∥U0 ∥L2 L1
√
∥µλ V ∥L2 L1 .
2λ
The Space BMO−1 and the Koch and Tataru Theorem
235
⃗ ∥λ
⃗ 0, V
⃗ ) + B(V
⃗ ,U
⃗ 0 )∥λ ≤ 1 ∥V
ˆ ∥B(U
2
⃗ 1 ∥λ ≤ C1 ϵ1/4 | ln ϵ|2/5
ˆ ∥U
⃗ ,W
⃗ )∥λ ≤ ∥B(V
⃗ ,W
⃗ )∥L2 A ≤ C0 A2 ∥V
⃗ ∥λ ∥W
⃗ ∥λ
ˆ ∥B(V
λ
Thus, the Picard iterative scheme will work in (L2 A, ∥ ∥λ ), provided that
C1 ϵ1/4 | ln ϵ|2/5 ≤
We have
1
.
16C0 A2λ
4
2
1
= e−2λ∥U0 ∥L2 L1 = e−2C4 ∥U0 ∥L2 L1
2
Aλ
with
∥U0 ∥L1 L2 ≈ ∥⃗u0,ϵ ∥F B−1 ≈ | ln ϵ|1/5 .
1,2
As
ϵ1/4 | ln ϵ|2/5 = o(e−C5 | ln ϵ|
4/5
)
as ϵ → 0, we have proven the theorem.
As proven by Chemin and Gallagher, we have found a non-linear smallness criterion for
−1
global existence: there exists a constant C0 such that, if ⃗u0 ∈ FB1,2
and
4
∥B(Wνt ∗ ⃗u0 , Wνt ∗ ⃗u0 )∥L2 A ≤
−1
1 −C0 ∥⃗u0 ∥F B1,2
,
e
C0
then we have global existence of a mild solution.
Chemin and Gallagher’s example relies on the frequency anisotropy of their initial data.
The role of this anisotropy has been commented by Chemin, Gallagher and Mullaert [110].
9.8
Stability of Global Solutions
When ⃗u0 is an (regular enough) initial value that generates a global mild solution for the
Navier–Stokes problem (with no forcing term), then small (in BM O−1 norm) perturbations
of ⃗u0 still lead to initial values of global mild solutions. This stability of global soutions has
been studied by many authors including Kawanago [257], Gallagher, Iftimie and Planchon
[198] or Auscher, Dubois and Tchamitchian [10]. In particular, we have:
Theorem 9.11.
Let C be the space of (smooth) vector fields ⃗u on (0, +∞) × R3 such that
• for all 0 < T1 < T2 < +∞, supT1 <t<T2 ∥⃗u(t, .)∥∞ < +∞
• div ⃗u = 0
• ∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u)
• ⃗u is *-weakly continuous from [0, +∞) to BM O−1
236
The Navier–Stokes Problem in the 21st Century (2nd edition)
Let ⃗u ∈ C such that moreover
• ⃗u0 = ⃗u(0, .) belongs to the closure of L2 ∩ BM O−1 in BM O−1
• limt→0+ ∥⃗u(t, .) − ⃗u0 ∥BM O−1 = 0.
Then ⃗u satisfies the following properties:
• ⃗u is strongly continuous from [0, +∞) to BM O−1
• limt→+∞ ∥⃗u(t, .)∥BM O−1 = 0
• there exists a positive ϵ0 (which depends on ⃗u) such that, for all divergence-free vector
field ⃗v0 with ∥⃗v0 − ⃗u0 ∥BM O−1 < ϵ0 , there exists ⃗v ∈ C such that ⃗v (0, .) = ⃗v0 .
Proof. Continuity in BM O−1 -norm:
The continuity of t 7→ ⃗u is quite obvious: the continuity at initial point t = 0 is given by
the assumption limt→0+ ∥⃗u(t, .) − ⃗u0 ∥BM O−1 = 0; for t > 0, we have that ⃗u is bounded with
all its derivatives on every compact subset [T0 , T1 ] of (0, +∞) (with values in L∞ ), so that
∂t ⃗u is bounded on [T0 , T1 ] with values in BM O−1 , and hence that ⃗u is locally Lipschitzian
from (0, +∞) to BM O−1 .
Decay at t = +∞:
We now prove that limt→+∞ ∥⃗u(t, .)∥BM O−1 = 0. Recall that we defined the space ET , for
0 < T ≤ +∞, as the space of measurable functions h on (0, T ) × R3 such that
√
sup
t ∥h(t, .)∥∞ < +∞
0<t<T
and
s
sup
x∈R3 , 0<t<T
ZZ
1
t3/2
√
(0,t)×B(x, t)
|h(s, y)|2 ds dy < +∞.
By Theorem 9.1, we know that the bilinear operator
Z t
⃗
⃗
⃗ ds
B(F , G) =
Wν(t−s) ∗ P div(F⃗ ⊗ G)
0
is bounded on the space ET for every T ∈ (0, +∞]:
⃗ E ≤ C0 ∥F⃗ ∥E ∥G∥
⃗ E
∥B(F⃗ , G)∥
T
T
T
(9.33)
for a constant C0 which depends on ν but not on T .
Moreover, we know that
∥Wνt ∗ F⃗ ∥ET ≤ C1 ∥F⃗ ∥BM O−1
(for a constant C1 which depends on ν but not on T ).
As ⃗u0 is the limit of ⃗u(t, .) in BM O−1 and as ⃗u(t, .) is in L∞ for t > 0, we have
lim ∥Wνt ∗ ⃗u0 ∥ET = 0
T →0+
so that, for some T0 > 0, we have
∥Wνt ∗ ⃗u0 ∥ET0 <
1
.
8C0
(9.34)
The Space BMO−1 and the Koch and Tataru Theorem
237
By strong continuity of t 7→ ⃗u, we find that, for some T1 > 0, we have
sup ∥Wνt ∗ ⃗u(τ, .)∥ET0 ≤
0≤τ ≤T1
1
.
8C0
Thus, we may construct, by Picard’s iterative scheme, a solution ⃗vτ (t, x) of the Navier–
Stokes equations on (0, T0 ) × R3 with initial value ⃗vτ (0, .) = ⃗u(τ, .): ⃗vτ = limn→+∞ ⃗vτ,n
with
⃗vτ,0 =
Wνt ∗ ⃗u(τ, .)
⃗vτ,n+1 = ⃗vτ,0 − B(⃗vτ,n , ⃗vτ,n )
and
∥⃗vτ,n+1 − ⃗vτ,n ∥ET0 ≤
1 1
.
2n 4C0
∞
Moreover, for τ > 0, we have ⃗vτ,0 ∈ L∞
and
t L
sup ∥⃗vτ,n+1 (t, .) − ⃗vτ,n (t, .)∥∞
0<t<T0
Z
≤ C sup
0<t<T0
0
t
1
1 √
p
√ s∥⃗vτ,n (s, .) − ⃗vτ,n−1 (s, .)∥∞ ×
ν(t − s) s
×(∥⃗vτ,n (s, .)∥∞ + ∥⃗vτ,n−1 (s, .)∥∞ ) ds
1
1
≤ C′ √ n
sup (∥⃗vτ,n (t, .)∥∞ + ∥⃗vτ,n−1 (t, .)∥∞ )
ν 2 C0 0<t<T0
so that
sup ∥⃗vτ,n+1 (t, .) − ⃗ττ,n (t, .)∥∞ ≤ C
0<t<T0
1
∥⃗u(τ, .)∥∞ .
2n
(where C depends on ν and C0 ). Thus, ⃗vτ ∈ L∞ ((0, T0 ), L∞ ). By uniqueness of mild solutions in L∞ L∞ , we have
⃗vτ (t, s) = ⃗u(τ + t, x).
Letting τ go to 0, we find that
∥⃗u∥ET0 ≤ lim inf ∥⃗vτ ∥ET0 ≤
τ →0
1
.
4C0
As ∥⃗v0 ∥ET0 ≤ 4C1 0 , we find that ⃗u = ⃗v0 on (0, T0 ), by uniqueness of small mild solutions2 .
We now use the fact that ⃗u0 belongs to the closure of L2 ∩ BM O−1 in BM O−1 : for
1
1
⃗0
0 < δ ≤ 32
u0 = α
⃗0 + β
C0 C1 (where δ > 0 may be taken as small as we like), we write ⃗
with β⃗0 ∈ L2 ∩ BM O−1 and ∥⃗
α0 ∥BM O−1 ≤ δ. (We may assume that div α
⃗ 0 = 0, as the
Leray projection operator P is bounded on both L2 and BM O−1 ). As ∥Wνt ∗ α
⃗ 0 ∥E+∞ ≤
1
C1 δ ≤ 32C
,
we
may
construct
a
global
mild
solution
α
⃗
of
the
Navier–Stokes
equations
on
0
(0, +∞) × R3 with initial value α
⃗ (0, .) = α
⃗ 0 . This solution satisfies ∥⃗
α∥E+∞ ≤ 2C1 δ, so that,
by Lemma 9.1, sup0<t ∥⃗
α(t, .)∥BM O−1 ≤ C2 δ. We shall assume that δ is small enough to
ensure that C2 δ ≤ 8C11 C0 .
Let β⃗ = ⃗u − α
⃗ . It is a mild solution of
⃗ − B(β,
⃗ α
⃗ β).
⃗
β⃗ = Wνt ∗ β⃗0 − B(⃗
α, β)
⃗ ) − B(β,
2 This uniqueness argument goes back to an old paper of Brezis [66]; see Miura for the use of this argument
in the context of bmo−1 [364].
238
The Navier–Stokes Problem in the 21st Century (2nd edition)
It may be constructed on a small interval (0, T1 ) as the limit of the Picard iterative scheme
⃗ = limn→+∞ ⃗γn with
β
⃗γ0 =
Wνt ∗ β⃗0
⃗γn+1 = ⃗γ0 − B(⃗
α, ⃗γn ) −B(⃗γn , α
⃗ ) − B(⃗γn , ⃗γn )
If we take δ small enough to get 24C12 C0 δ < 1 and T1 small enough to have ∥⃗u∥ET1 < C1 δ,
we find that ∥⃗γ0 ∥ET1 ≤ 3C1 δ, and ∥⃗γn ∥ET1 ≤ 4C1 δ.
Moreover, we have
sup ∥⃗γn+1 (t, .) − ⃗γn (t, .)∥2
0<t<T1
Z
≤ C sup
0<t<T1
0
t
1
p
ν(t − s)
1
√ ∥⃗γn (s, .) − ⃗γn−1 (s, .)∥2 ×
s
√
× s(∥⃗γn (s, .)∥∞ + ⃗γn−1 (s, .)∥∞ + ∥⃗
α(s, .)∥∞ ) ds
≤ C3 C1 δ sup ∥⃗γn (t, .) − ⃗γt−1 (s, .)∥2
0<t<T1
If C3 C1 δ < 1, we find convergence of ⃗γn in L∞ L2 as well.
This proves that β⃗ ∈ L∞ ((0, T1 ), L2 ) for T1 small enough. As α
⃗ and β⃗ belong to
∞
∞
L ((T1 , T2 ), L ) for every T2 > T1 , we find (reiterating the contraction argument on
⃗ belongs to L∞ ((T1 , T2 ), L2 ). Moreover,
small enough intervals with fixed length) that β
⃗ ∈ L∞ ((0, T1 ), Ḣ −1 ) so that β⃗ − Wν(t−T ) ∗ β(T
⃗ 1 , .) ∈
we have div(⃗
α ⊗ β⃗ + β⃗ ⊗ α
⃗ + β⃗ ⊗ β)
1
2
1
∞
2
2
⃗
L ((T1 , T2 ), H ). Finally, we conclude that β belongs to L ((T1 , T2 ), L ) ∩ L ((T1 , T2 ), H 1 )
while ∂t β⃗ ∈ L2 ((T1 , T2 ), H −1 ). We may write
Z
2
⃗
⃗ t β⃗ dx.
∂t ∥β(t, .)∥2 = 2 β.∂
As div ⃗u = 0, we have
Z
⃗ α + β).
⃗ ∇
⃗ dx = 0
⃗ β)
β.((⃗
so that
⃗ .)∥2 = −2ν∥∇
⃗ 2+2
⃗ ⊗ β∥
∂t ∥β(t,
2
2
Z
⃗∇
⃗ dx ≤ −ν∥∇
⃗ 2 + 1 ∥⃗
⃗ 2.
⃗ β)
⃗ ⊗ β∥
α
⃗ .(β.
α∥2∞ ∥β∥
2
2
ν
This gives
⃗ .)∥2
∥β(t,
2
and
≤
⃗ 1 , .)∥2
∥β(T
2
e
1
ν
2 δ2
C1
s
T1
Rt
ds
=
⃗ 1 , .)∥2
∥β(T
2
t
2 2
⃗ 2 ds ≤ 1 ∥β(T
⃗ 1 , .)∥2 (1 + 1 C12δ
⃗ ⊗ β∥
∥∇
2
2
ν
ν C1νδ2
T1
T1
Z
Z
t
s
t
T1
2 δ2
C1
ν
T1
By interpolation, we find that
⃗ L4 ((T ,T ),L3 ) ≤ C4 ∥β(T
⃗ 1 , .)∥2
∥β∥
1
2
If δ is small enough to have
C12 δ 2
2ν
< 14 , this gives
⃗ .)∥3 = 0
lim inf ∥β(t,
t→+∞
T2
T1
C12νδ2
2 2
1δ
C2ν
.
ds
).
s
The Space BMO−1 and the Koch and Tataru Theorem
239
and thus
⃗ .)∥BM O−1 = 0
lim inf ∥β(t,
t→+∞
⃗ 3 , .)∥BM O−1 < C2 δ. This gives that
Thus, we may find a time T3 such that ∥β(T
∥⃗u(T3 , .)∥BM O−1 < 2C2 δ. Hence, we have ∥Wν(t−T3 ) ∗ ⃗u(T3 , .)∥E+∞ ≤ 2C1 C2 δ ≤ 4C1 0 ; it
means that we may control ⃗u on (T3 , +∞) by
∥⃗u(T3 + t, .)∥E+∞ ≤ 4C1 C2 δ
and thus
sup ∥⃗u(t, .)∥BM O−1 ≤ 2C2 δ + C4 (4C1 C2 δ)2 .
t>T3
As δ may be taken arbitrarily small, we find that limt→+∞ ∥⃗u(t, .)∥BM O−1 = 0.
Stability:
We know that we have
⃗ E ≤ C0 ∥F⃗ ∥E ∥G∥
⃗ E
∥B(F⃗ , G)∥
T
T
T
(9.35)
⃗ L∞ BM O−1 ≤ C0 ∥F⃗ ∥E ∥G∥
⃗ E
∥B(F⃗ , G)∥
T
T
(9.36)
and
for a constant C0 which depends on ν but not on T . Moreover, for a constant C1 , we have
⃗ νt ∗ w
∥W
⃗ 0 ∥ET ≤ C1 ∥w
⃗ 0 ∥BM O−1
Similarly, we have
√
⃗ νt ∗ w
∥W
⃗ 0 ∥ET ≤ C2 T ∥w
⃗ 0 ∥∞ .
⃗0 and ⃗γ0 satisfy div(⃗
One may conclude that, if α
⃗ 0, β
α0 + β⃗0 ) = div ⃗γ0 = 0, ∥⃗
α0 ∥BM O−1 ≤
1
1
⃗
γ0 ∥BM O−1 ≤ 16C0 C1 , then
16C0 C1 , ∥β0 ∥∞ ≤ M and ∥⃗
∥Wνt ∗ (⃗
α0 + β⃗0 )∥ET ≤ 8C1 0 , so that the Navier–Stokes equations
with initial value α
⃗ 0 + β⃗0 have a mild solution w
⃗ on (0, T ) × R3 , with ∥w∥
⃗ ET ≤ 4C1 0
• for T =
1
(16C2 C0 M )2 ,
1
, the Navier–Stokes equations with initial value α
⃗ 0 + β⃗0 + ⃗γ0
• as ∥Wνt ∗ ⃗γ0 ∥ET ≤ 16C
0
3
then have a mild solution w
⃗ + ⃗γ on (0, T ) × R , with ∥⃗γ ∥ET ≤ 2C1 ∥⃗γ0 ∥BM O−1
• moreover, we have ∥⃗γ (T, .)∥BM O−1 ≤ (1 + 54 C1 )∥⃗γ0 ∥BM O−1 .
Now, recall that ⃗u is strongly continuous from [0, +∞) to the closure of L∞ ∩BM O−1 in
BM O−1 and that limt→+∞ ∥⃗u(t, .)∥BM O−1 = 0. Thus, we can find a time T0 > 0 such that
∥⃗u(T0 , .)∥BM O−1 ≤ 8C10 C1 . By compactness of ⃗u([0, T0 ]) in BM O−1 , we can find a M < +∞
⃗t with ∥β⃗t ∥∞ ≤ M and
such that, for every t ∈ [0, T0 ], we may decompose ⃗u(t, .) into α
⃗t + β
1
1
1
∥⃗
αt ∥BM O−1 < 16C0 C1 . Let N ∈ N such that N T0 < (16C2 C0 M )2 . If
5
1
∥⃗v0 − ⃗u0 ∥BM O−1 < (1 + C1 )−N
4
16C0 C1
k
then we may construct the mild solution ⃗v inductively on [ N
T0 , k+1
N T0 ] for 0 ≤ k < N :
k
indeed, we will have ⃗u([ N T0 , .) = α
⃗ k T0 + β⃗ k T0 with ∥β⃗ k T0 ∥∞ ≤ M and ∥⃗
α k T0 ∥BM O−1 <
N
N
N
N
1
16C0 C1 and
5
∥⃗u( k T0 , .) − ⃗v ( k T0 , .)∥BM O−1 ≤(1 + C1 )k ∥⃗u0 − ⃗v0 ∥BM O−1
N
N
4
5
1
≤(1 + C1 )k−N
.
4
16C0 C1
240
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, we may construct ⃗v on [0, T0 ] and we have
∥⃗v (T0 , .)∥BM O−1 ≤ ∥⃗u(T0 , .)∥BM O−1 + ∥⃗v (T0 , .) − ⃗u(T0 , .)∥BM O−1 ≤
3
.
16C0 C1
Thus, ⃗v (T0 , .) is small enough to grant existence of a mild solution ⃗v on [T0 , +∞) as well.
9.9
Analyticity
Let ⃗u be a mild solution of the Navier–Stokes problem generated from an initial value
⃗u0 in the closure of L∞ in BM O−1 (or bmo−1 ) (in absence of a force f⃗). We have seen that
such a solution exists on a small interval (0, T ) and is locally bounded; more precisely,
√
t∥⃗u(t, .)∥∞ < +∞.
sup
0<t<T
This solution may be prolongated as long as it does not blow up, as existence time for
t > t0 is bounded by below by C∥⃗u(t10 ,.)∥2 . If T ∗ is the maximal existence time of the mild
∞
solution ⃗u, then ⃗u is actually analytical in the time and space variables on (0, T ∗ ) × R3 .
Analyticity was first proven by Kahane [249] and Masuda [351]. A simple proof (in case
of solutions in Sobolev spaces) was given by Foias and Temam [181] (see Section 8.7 for a
short presentation).
Spatial analyticity in the context of Lebesgue spaces has been studied by Grujić and
Kukavica [218] and by Lemarié-Rieusset [312, 314]. Spatial analyticity has been more recently studied in the context of BM O−1 (Germain, Pavlović and Staffilani [206], Miura and
Sawada [365], Guberović [219]), Besov spaces (Bae, Biswas and Tadmor [13]) or modulation
spaces (Guo, Wang and Zhao [224]).
We follow in this section the proof of Cannon and Knightly [80] for time and space
analyticity:
Analyticity
Theorem 9.12.
Let ⃗u be a mild solution of the Navier–Stokes equations on (T0 , T1 ) × R3 :
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u)
(9.37)
that is bounded on (T0 , T1 ) × R3 :
⃗u ∈ L∞ ((T0 , T1 ) × R3 ).
Then ⃗u is analytical in the time and space variables.
Proof. If C0 is a constant such that, for bounded vector fields ⃗v and w
⃗ and for T > 0, we
have
√
∥B(⃗v , w)(t,
⃗
.)∥∞ ≤ C0 T sup ∥⃗v (t, .)∥∞ sup ∥w(t,
⃗ , .)∥∞
0<t<T
where
Z
0<t<T
t
Wν(t−s) ∗ P div(⃗v ⊗ w)
⃗ ds,
B(⃗v , w)
⃗ =
0
The Space BMO−1 and the Koch and Tataru Theorem
241
we know by the proof of Theorem 5.1 (Oseen’s method) that for t0 ∈ (T0 , T1 ) and for
t ∈ (t0 , t0 + (4C0 ∥⃗u(t10 ,.)∥∞ )2 ), ⃗u(t, .) is given by
⃗u(t, .) =
+∞
X
⃗k (t, .)
V
(9.38)
k=0
with
⃗0 = Wν(t−t ) ∗ ⃗u(t0 , .)
V
0
and
⃗k+1 = (−1)k+1
V
Z
t
k
X
⃗j ⊗ V
⃗k−j ) ds.
Wν(t−s) ∗ P div(
V
t0
j=0
⃗k is analytical in time and space variables
The idea of the proof is then to show that V
and is defined as a holomorphic function on a neighborhood Ω of (t0 , t1 ) × R3 in C × C3
(with t1 = t0 + (4C1 ∥⃗u(t10 ,.)∥∞ )2 for some positive constant C1 ) and that the expansion (9.38)
converges uniformly on every compact subset of Ω, so that ⃗u is still holomorphic on Ω.
Let us recall the results of Theorem 4.6. The heat kernel Wνt = (νt)13/2 W ( √xνt ) is given
by
|x|2
1
W (x) =
e− 4
3/2
(4π)
and the Oseen tensor which is the matrix convolution kernel of the operator Wνt ∗ P is the
tensor (Oj,k (νt, x))1≤j,k≤3 given by
1
x
Oj,k ( √ )
(νt)3/2
νt
Oj,k (νt, x) = δj,k Wνt + G ∗ ∂j ∂k Wνt =
where the functions Oj,k are determined through Oseen’s formula:
Oj,k (x) = δj,k W (x) + 2∂j ∂k
Z
1
|x|
e
(4π)3/2 |x|
!
2
− s4
ds ,
(9.39)
0
which may be rewritten as
1
Z
Oj,k (x) = δj,k W (x) + 2∂j ∂k
W (θx) dθ
(9.40)
0
or as
Oj,k (x) =∂j ∂k
1
4π|x|
− 2∂j ∂k
+ δj,k W (x)
1
(4π)3/2 |x|
Z
∞
2
e
− s4
!
(9.41)
ds .
|x|
Let us now fix γ > 0 and let
Ωγ = {(τ, z) ∈ C × C3 / 0 < ℜ(τ ), |ℑ(τ )| < γℜ(τ ) and |ℑ(z)| < γ(ν|τ |)1/2 }.
√
On Ωγ , the function τ is well defined and holomorphic. Thus, the functions Wνt (x) and
Oj,k (νt, x) have holomorphic extensions Wντ (z) and Oj,k (ντ, z) to Ωγ . We now define
Z
⃗0 (τ, z) = Wν(τ −t ) (z − y)⃗u(t0 , y) dy
V
0
242
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
⃗k+1 (τ, z) = (−1)k+1
V
Z
τ
Z
O(ν(τ − t0 − σ), z − y) div(
t0
k
X
⃗j (σ, y) ⊗ V
⃗k−j (σ, y)) dσ dy.
V
j=0
On t0 + Ωγ , we have, writing τ − t0 = t + iσ and z = α + iβ (real and imaginary parts),
|Wν(τ −t0 ) (z − y)| =
t|α−y|2
t|β|2
2σβ.(α−y)
1
1
−
−
e 4ν|t+iσ|2 e 4ν|t+iσ|2 e 4ν|t+iσ|2
3/2
3/2
(4πν) |t + iσ|
p
As t ≤ |t + iσ| ≤ t 1 + γ 2 , we find
|Wν(τ −t0 ) (z − y)| ≤
|α−y|2
1
γ |α−y|
γ2
1
√
− 4(1+γ
2)
νt
νt .
4 e2
e
e
(4πνt)3/2
⃗0 is holomorphic on t0 + Ωγ and that
Thus, we may conclude that V
⃗0 ∥L∞ (t +Ω ) ≤ Cγ ∥⃗u(t0 , .)∥∞ .
∥V
0
γ
⃗j is holomorphic and (locally) bounded on t0 + Ωγ for 0 ≤ j ≤ k, we
Assuming now that V
⃗
may estimate Vk+1 . Indeed, we have, for (τ, z) ∈ Ωγ ,
⃗ ⊗ O(ντ, z)| =
|∇
Let Z =
√z .
ντ
1
⃗ ⊗ O|( √z ).
|∇
(ν|τ |)2
ντ
We have, for Z 2 = z12 + z22 + z32 ,
ℜ(Z 2 ) ≥ −γ 2 + ℜ(
and
(ℜ(z)2
ℜ(τ )
1 (ℜ(z))2
) = −γ 2 + (ℜ(z))2
≥ −γ 2 +
2
ντ
ν|τ |
1 + γ 2 νℜ(τ )
|ℜ(z)|
|Z| ≤ γ + p
.
νℜ(τ )
From
Z
∂l Oj,k (Z) = δj,k ∂l W (Z) + 2∂l ∂j ∂k
1
W (θZ) dθ
(9.42)
0
we find that
|∂l Oj,k (Z)| ≤ C(|Z|e−
(ℜ(Z 2 )
4
Z
+
1
(1 + θ3 |Z|3 )e−θ
2
2 ℜ(Z )
4
)
0
and thus
|∂l Oj,k (Z)| ≤ Cγ .
Moreover, if |Z| > γ(1+
of Z. We have
p
1 + γ 2 ), we have ℜ(Z 2 ) > 0 and (Z 2 )1/2 is a holomorphic function
1
∂l Oj,k (Z) =∂l ∂j ∂k
+ δj,k ∂l W (Z)
4π(Z 2 )1/2
Z ∞
2
1
−θ 2 Z4
− 2∂l ∂j ∂k
e
dθ .
(4π)3/2 1
Thus, we get that
|Z|4 |∂l Oj,k (Z)| ≤ Cγ .
(9.43)
The Space BMO−1 and the Koch and Tataru Theorem
243
Writing again, on t0 + Ωγ , τ − t0 = t + iσ and z = α + iβ (real and imaginary parts), we
⃗ (τ, z) and W
⃗ (τ, z) that are holomorphic on t0 + Ωγ and
find that, for two vector fields V
bounded on each t0 + Ωγ,M = {(τ, z) ∈ t0 + Ωγ / |ℜ(τ ) − t0 | < M }, we have
Z τZ
⃗ ⊗ O(ν(τ − t0 − σ), z − y).(V
⃗ (σ, y) ⊗ W
⃗ (σ, y)) dσ dy|
|
(∇
t0
Z τZ
⃗ ⊗ O(ν(τ − t0 − σ), z − y| |V
⃗ (σ, y) ⊗ W
⃗ (σ, y))| dσ dy
≤
|∇
t0
≤ Cγ
⃗ (σ, .)∥∞
∥V
sup
|σ−t0 |<|τ −t0 |
=
Cγ′
⃗ (σ, .)∥∞
∥W
sup
Z
|σ−t0 |<|τ −t0 |
p
ℜ(τ ) − t0
sup
Z
dt dα
t2 + |α|4
t0
⃗ (σ, .)∥∞
∥V
|σ−t0 |<|τ −t0 |
ℜτ
sup
⃗ (σ, .)∥∞
∥W
|σ−t0 |<|τ −t0 |
⃗k is holomorphic on t0 + Ωγ and satisfies
Thus, we find by induction on k, that V
⃗k (τ, z)| ≤ αk |ℜ(τ ) − t0 |k/2 ∥⃗u(t0 , .)∥k+1
|V
∞
where the constant αk does not depend on τ , z, or ⃗u(t0 , .). We have
α0 ≤ Cγ
and
αk+1 ≤ Cγ
k
X
αj αk−j .
j=0
If β0 = 1 and βk+1 =
that
Pk
j=0
βj βk−j and F is the formal series F (z) =
P
k∈N
βk z k+1 , we find
F 2 (z) = F (z) − z
and thus
√
1
(1 − 1 − 4z).
2
The radius of convergence of the Taylor expansion of F is
negative, thus we find
+∞ k
X
1
1
βk = .
4
2
F (z) =
1
4
and its coefficients are non-
k=0
We thus find
αk ≤ Cγ1+2k βk
and thus
⃗k (τ, z)| ≤ C 1+2k βk |ℜ(τ ) − t0 |k/2 ∥⃗u(t0 , .)∥k+1
|V
γ
∞
The series
P
k∈N
⃗k (τ, z) will converge normally on t0 + Ωγ,M if we choose M such that
V
√
4Cγ2 ∥⃗u(t0 , .)∥∞ M < 1.
If
M0 =
1
2
8Cγ2 ∥⃗u∥L∞ ((T0 ,T1 )×R3 )
we find that ⃗u has a holomorphic extension to ∪T0 <t0 <T1 t0 + Ωγ,M0 , thus ⃗u is analytic on
(T0 , T1 ) × R3 with respect to the time and space variables.
244
The Navier–Stokes Problem in the 21st Century (2nd edition)
9.10
Small Data
If ⃗u0 is small enough in BM O−1 , we know that the Cauchy problem for the Navier–
Stokes equations with initial value ⃗u0 and forcing term f⃗ = 0 will have a global mild
solution ⃗u that can be constructed by the Picard iterative scheme. If ⃗u0 belongs moreover
s
to some Besov space Ḃp,q
with s > −1, the sequence of the Picard iterates will converge in
∞
s
s
L ((0, +∞), Ḃp,q ) as well, without any restriction on the size of ⃗u in Ḃp,q
nor on p, q, s. It
means that not only ⃗u keeps its regularity at all times but also the Picard iterates behave well
at all times. This property has been described as persistency by Furioli, Lemarié-Rieusset,
Zahrouni, and Zhioua [188] (see a generalization to multilinear equations [315]).
Theorem 9.13.
Let ⃗u0 ∈ BM O−1 with div ⃗u0 = 0. Let B be the bilinear operator
Z t
⃗ ,W
⃗ )=
⃗ (s, .) ⊗ W
⃗ (s, .)) ds.
B(V
Wν(t−s) ∗ P div(V
0
⃗ n be the sequence of Picard iterates defined by U
⃗ 0 = Wνt ∗ ⃗u0 and U
⃗ n+1 = U
⃗0 −
Let U
⃗
⃗
B(Un , Un ). Assume that we have
√
P
⃗
⃗
•
n∈N supt>0 t∥Un+1 (t, .) − Un (t, .)∥∞ < +∞
⃗n
(for instance, assume that ∥⃗u0 ∥BM O−1 is small enough). Then the limit ⃗u = limn→+∞ U
satisfies the Navier–Stokes equations
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u)
and we have
√
sup t∥⃗u(t, .)∥∞ < +∞.
t>0
σ
with 1 ≤ p, q ≤ +∞ and −1 < σ ≤ 0,
If moreover ⃗u0 belongs to some Besov space Ḃp,q
then
sup ∥⃗u(t, .)∥Ḃ σ < +∞.
p,q
t>0
We have more precisely
X
n∈N
⃗ n+1 (t, .) − U
⃗ n (t, .)∥ σ < +∞.
sup ∥U
Ḃ
p,1
t>0
Proof. Case −1 < σ < 0:
We introduce the path space
⃗ / t−σ/2 ∥U
⃗ (t, .)∥p ∈ Lq ( dt )}.
X = {U
t
√
⃗ ∈ X and supt>0 t∥W
⃗ (t, .)∥∞ < +∞, then B(V
⃗ ,W
⃗ ) and
We easily check that, if V
⃗ ,V
⃗ ) still belong to X. Indeed, we have
B(W
Z
∥
0
t
⃗ ) ds∥p ≤ Cp
Wν(t−s) ∗ P div(⃗v ⊗ W
Z
0
t
√
√
1
⃗ (s, .)∥p ds sup s∥W
⃗ (s, .)∥∞ .
√ ∥V
t−s s
s>0
The Space BMO−1 and the Koch and Tataru Theorem
245
Thus, we must only check that, for 0 < α < 1/2,
Z t
1
tα
√
F 7→
√ α F (s) ds
t−s ss
0
is bounded on Lq ( dt
t ) for 1 ≤ q ≤ +∞ ; this is obvious for q = +∞ (direct estimation) and
for q = 1 (first integrate on t, by Fubini’s theorem), and thus for all q by interpolation.
⃗ n belongs to X for every n (since U
⃗ 0 ∈ X because
Thus, by induction, we have that U
σ
⃗u0 ∈ Ḃp,q ). Moreover, writing
⃗ n+1 − U
⃗ n = −B(U
⃗ n, U
⃗n − U
⃗ n−1 ) − B(U
⃗n − U
⃗ n−1 , U
⃗ n−1 )
U
⃗ −1 = 0), we find, for
for all n ∈ N (with the convention that U
√
⃗n − U
⃗ n−1 ∥X and ϵn = sup t∥U
⃗ n (t, .) − U
⃗ n−1 (t, .)∥∞ ,
αn = ∥U
t>0
that
αn+1 ≤ C0 ϵn
n
X
αk
k=0
where the constant C0 depends only on ν, s and p. Thus, if
Mn =
n
X
αk ,
k=0
we have
⃗ 0 ∥X
Mn+1 ≤ (1 + C0 ϵn )Mn ≤ ∥U
n
Y
(1 + C0 ϵk ).
k=0
Q
P
< +∞, we find that k∈N (1 + C0 ϵk ) < +∞, so that k∈N αk < +∞.
−1
⃗ ∈ Ḃp,∞
⃗ ∈ Ḃ σ with
Now, if f ∈ Lp , we have ∇f
and thus (as s > −1) Wνt ∗ ∇f
p,1
1
σ
σ
⃗ ∥ σ ≤C
∥Wνt ∗ ∇f
∥f
∥
.
Thus,
since
Ḃ
⊂
Ḃ
,
and
thus
p
(1+σ)/2
p,q
p,∞
Ḃ
(t−s)
As
P
k∈N ϵk
p,1
σ
⃗ 0 (t, .)∥p +
sup t− 2 ∥U
t>0
+∞
X
k=0
σ
⃗ k+1 (t, .) − U
⃗ k (t, .)∥p = M < +∞,
sup t− 2 ∥U
t>0
we find
⃗ n+1 (t, .) − U
⃗ n (t, .)∥ σ ≤ C
∥U
Ḃ
p,1
Z
0
t
1
dsM ϵn = C ′ M ϵn
(t − s)(1+σ)/2 s(1−σ)/2
and finally
X
n∈N
⃗ n+1 (t, .) − U
⃗ n (t, .)∥ σ < +∞.
sup ∥U
Ḃ
t>0
p,1
Remark: this proves that, in contrast with the case p = +∞, the Cauchy problem is well
σ
posed in Ḃp,q
for 3 < p < +∞ and σ = −1 + p3 (for small data).
Case σ = 0:
−1/2
−1/2
0
−1
⃗n
We have Ḃp,q
∩ Ḃ∞,∞
⊂ Ḃ2p,∞ . Thus, our analysis of the case ⃗u0 ∈ Ḃ2p,∞ shows that U
will converge in the path space
⃗ / t1/4 ∥U
⃗ (t, .)∥2p ∈ L∞ ( dt )}.
X = {U
t
246
The Navier–Stokes Problem in the 21st Century (2nd edition)
More precisely, we proved that
X
⃗n − U
⃗ n−1 ∥X < +∞.
∥U
n∈N
We write again
⃗ n+1 − U
⃗ n = −B(U
⃗ n, U
⃗n − U
⃗ n−1 ) − B(U
⃗n − U
⃗ n−1 , U
⃗ n−1 )
U
and consider the action of B on X × X. We have
⃗ ⊗W
⃗ )∥ 0 ≤ C∥V
⃗ ∥p ∥W
⃗ ∥p
∥Wν(t−s) ∗ P div(V
Ḃ
p,1
1
(ν(t − s))1/2
so that
⃗ ,W
⃗ )(t, .)∥ 0 ≤ C∥V
⃗ ∥X ∥W
⃗ ∥X .
∥B(V
Ḃ
p,1
Thus,
X
n∈N
⃗ n+1 (t, .) − U
⃗ n (t, .)∥ 0 ≤ C sup ∥U
⃗ n ∥X
sup ∥U
Ḃ
p,1
t>0
n∈N
X
⃗n − U
⃗ n−1 ∥X < +∞.
∥U
n∈N
0
0
Remark: As Ḃp,1
⊂ Lp ⊂ Ḃp,∞
, this proves that we have persistency for the Lp norm as
well.
Theorem 9.14.
Let ⃗u0 ∈ BM O−1 with div ⃗u0 = 0. Let B be the bilinear operator
Z t
⃗
⃗
⃗ (s, .) ⊗ W
⃗ (s, .)) ds.
B(V , W ) =
Wν(t−s) ∗ P div(V
0
⃗ n be the sequence of Picard iterates defined by U
⃗ 0 = Wνt ∗ ⃗u0 and U
⃗ n+1 = U
⃗0 −
Let U
⃗ n, U
⃗ n ). Assume that we have
B(U
√
P
⃗ n+1 (t, .) − U
⃗ n (t, .)∥∞ < +∞
•
supt>0 t∥U
n∈N
⃗ n (t, .)∥ −1 < +∞
• supn∈N supt>0 ∥U
Ḃ∞,∞
⃗n
(for instance, assume that ∥⃗u0 ∥BM O−1 is small enough). Then the limit ⃗u = limn→+∞ U
satisfies the Navier–Stokes equations
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u)
and we have
√
sup t∥⃗u(t, .)∥∞ < +∞.
t>0
σ
If moreover ⃗u0 belongs to some Besov space Ḃp,q
with 1 ≤ p, q ≤ +∞ and σ > 0, then
sup ∥⃗u(t, .)∥Ḃ σ < +∞.
t>0
p,q
We have more precisely
X
n∈N
⃗ n+1 (t, .) − U
⃗ n (t, .)∥ σ < +∞.
sup ∥U
Ḃ
t>0
p,1
The Space BMO−1 and the Koch and Tataru Theorem
247
σ
−1
σ
Proof. We would like to control a product f g when f ∈ Ḃp,q
∩ Ḃ∞,∞
and g ∈ Ḃp,q
∩ L∞ .
Using the Littlewood–Paley decomposition, we write
f g = π(f, g) + ρ(f, g)
where
π(f, g) =
X
Sj+3 g∆j f and ρ(f, g) =
j∈Z
X
Sj−2 f ∆j g
j∈Z
We have
σ
p
∥π(f g)∥Ḃ∞,q
≤ C∥g∥∞ ∥f ∥Bp,q
and
1
⃗
σ .
∥Wν(t−s) ∗ ∇(π(f,
g))∥Ḃ σ ≤ C p
∥g∥∞ ∥f ∥Bp,q
p,1
ν(t − s)
σ
If λ ∈ ( 1+σ
, 1), we have
∥f ∥Ḃ σ(1−λ)−λ ,∞ ≤ C∥f ∥1−λ
∥f ∥λḂ −1
Ḃ σ
p
1−λ
p,q
∞,∞
and
∥g∥Ḃ λσ
≤ C∥g|λḂ σ ∥g∥1−λ
∞ ;
p ,∞
p,q
λ
as σ(1 − λ) − λ < 0, we find that
p
∥Sj+2 f ∥ 1−λ
≤ C∥f ∥Ḃ σ(1−λ)−λ ,∞ 2−j(σ(1−λ)−λ)
p
1−λ
and thus
∥ρ(f, g)∥Ḃp,∞
∥g∥Ḃ λσ
σ−λ ≤ C∥f ∥ σ(1−λ)−λ
Ḃ
,∞
p ,∞
p
1−λ
λ
and
⃗
∥Wν(t−s) ∗ ∇(ρ(f,
g))∥Ḃ σ ≤ C
p,1
1
λ
λ
∥g∥1−λ
∥f ∥1−λ
σ .
σ ∥f ∥B −1
∞ ∥g∥Ḃp,q
Bp,q
∞,∞
ν(t − s))(1+λ)/2
We write again
⃗ n+1 − U
⃗ n = −B(U
⃗ n, U
⃗n − U
⃗ n−1 ) − B(U
⃗n − U
⃗ n−1 , U
⃗ n−1 )
U
⃗ −1 = 0), we find, for
for all n ∈ N (with the convention that U
⃗ 0∥ ∞ s ,
α0 = ∥U
L Ḃ
p,q
⃗ n+1 − U
⃗ n ∥ ∞ σ , Mn =
αn+1 = ∥U
L Ḃ
n
X
p,1
αk ,
k=0
√
⃗ k ∥ ∞ −1 and ϵn = sup t∥U
⃗ n (t, .) − U
⃗ n−1 (t, .)∥∞ ,
βn = sup ∥U
L Ḃ∞,∞
t>0
k≤n
that
αn+1 ≤ C0 (ϵn Mn + ϵ1−λ
αnλ Mn1−λ βnλ )
n
where the constant C0 depends only on ν, σ, λ, and p. Young’s inequality gives
αn+1 ≤
λ
1
1−λ
αn + C1 β∞
ϵn Mn .
2
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The Navier–Stokes Problem in the 21st Century (2nd edition)
This gives (as Mn−1 ≤ 2Mn − Mn−1 )
λ
1
1
1−λ
Mn+1 − Mn ≤(1 + C1 β∞
ϵn )Mn − Mn−1
2
2
λ
λ
1
1
1−λ
1−λ
≤(1 + C1 β∞ ϵn )(Mn − Mn−1 ) + C1 β∞
ϵn Mn−1
2
2
λ
1
1−λ
≤(1 + 2C1 β∞
ϵn )(Mn − Mn−1 )
2
We get finally
+∞
Y
λ
1
1
1−λ
Mn ≤ Mn+1 − Mn = (M1 − M0 )
(1 + 2C1 β∞
ϵk ) < +∞.
2
2
k=1
Thus, the theorem is proved.
Remark:
σ
(i) The homogeneous Besov spaces Ḃp,q
are not well
P defined as Banach spaces of distributions, as we have not the convergence of j∈Z ∆j f to f in S ′ if σ > 3/p (or
σ
σ = 3/p and q > 1). However, we work here with BM O−1 ∩ Ḃp,q
so that the infrared
divergence is avoided.
3
(ii) As the homogeneous Sobolev spaces Ẇ −1+ p ,p for 1 < p < 3 satisfies −1 +
−1+ 3
and Ḃp,1 p
3
−1+ p
,p
Ẇ
⊂ Ẇ
3
−1+ p
,p
⊂
−1+ 3
Ḃp,∞ p
3
p
> 0,
⊂ BM O−1 , the Cauchy problem is well posed in
for 1 < p < 3 (for small data).
Chapter 10
Special Examples of Solutions
10.1
Symmetries for the Navier–Stokes Equations
In this section, we consider the Navier–Stokes equations
⃗ u − ∇p
⃗ + f⃗
∂t ⃗u = ν∆⃗u − ⃗u.∇⃗
(10.1)
with div ⃗u = 0, defined on (T0 , T1 ) × R3 . We assume that p is vanishing at infinity, in the
sense that, if Sj is the Littlewood–Paley operator Sj f = 23j ϕ(2j x) ∗ p(x, t), then
lim ∥Sj p∥∞ = 0.
j→−∞
(10.2)
We consider especially the case f⃗ = 0 and say that in that case we have ⃗u ∈ N S 0 .
We list there some known symmetries for the Navier–Stokes equations and the associated
transforms on N S 0 . Such transforms have been discussed one century ago by Wilczynski
[502].
ˆ time translation: assume we change the origin of time (new coordinates T = t −
t0 , X = x). Then we obtain for ⃗u ∈ N S 0 another solution ⃗v ∈ N S 0 (defined on
(T0 − t0 , T1 − t0 ) × R3 ) given by
⃗v (T, X) = ⃗u(T + t0 , X)
associated to the pressure
q(T, X) = p(T + t0 , X)
The fact that ⃗v obeys the same equation as ⃗u comes from the fact that the coefficients
in Equation (10.1) are constant.
Of course, if we assume the force f⃗ =
̸ 0, then we must modify the force f⃗ in (10.1)
into ⃗g with
⃗g (T, X) = f⃗(T + t0 , X).
ˆ space translation: assume we change the origin of space coordinates (new coordinates
T = t, X = x − x0 ). Then we obtain for ⃗u ∈ N S 0 another solution ⃗v ∈ N S 0 given by
⃗v (T, X) = ⃗u(T, X + x0 )
associated to the pressure
q(T, X) = p(T, X + x0 )
The fact that ⃗v obeys the same equation as ⃗u comes again from the fact that the
coefficients in Equation (10.1) are constant.
If we assume the force f⃗ ̸= 0, then we must modify the force f⃗ in (10.1) into ⃗g with
⃗g (T, X) = f⃗(T, X + x0 ).
DOI: 10.1201/9781003042594-10
249
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The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ space rotation: assume we change the reference axes by a rotation: I⃗ = R⃗i, J⃗ =
⃗ = R⃗k. The new coordinates are then T = t, X = R−1 x. Then we obtain for
R⃗j, K
⃗u ∈ N S 0 another solution ⃗v ∈ N S 0 given by
⃗v (T, X) = R−1 ⃗u(T, RX)
associated to the pressure
q(T, X) = p(T, RX)
The fact that ⃗v obeys the same equation as ⃗u comes from the fact that we have
modelized the evolution of an isotropic fluid.
If we assume the force f⃗ ̸= 0, then we must modify the force f⃗ in (10.1) into ⃗g with
⃗g (T, X) = R−1 f⃗(T, RX).
ˆ change of Galilean frame: if we change the reference frame into another one moving
⃗ , the laws of Newtonian physics do not change. We get the
with uniform velocity U
⃗ . Then we obtain for ⃗u ∈ N S 0 another solution
new coordinates T = t, X = x − tU
⃗v ∈ N S 0 given by
⃗)−U
⃗
⃗v (T, X) = ⃗u(T, X + T U
associated to the pressure
⃗ ).
q(T, X) = p(T, X + T U
If we assume the force f⃗ =
̸ 0, then we must modify the force f⃗ in (10.1) into ⃗g with
⃗ ).
⃗g (T, X) = f⃗(T, X + T U
ˆ change of scale: if we want to change the scales of times and of space, and keep the
viscosity coefficient constant (remember that the kinematic viscosity ν has the same
dimension as U L, where U is a velocity and L a length), we use the new coordinates
T = t/λ2 , X = x/λ for some positive λ. Then we obtain for ⃗u ∈ N S 0 another solution
⃗v ∈ N S 0 given by
⃗v (T, X) = λ⃗u(λ2 T, λX)
associated to the pressure
q(T, X) = λ2 p(λ2 T, λX).
If we assume the force f⃗ =
̸ 0, then we must modify the force f⃗ in (10.1) into ⃗g with
⃗g (T, X) = λ3 f⃗(λ2 T, λX).
ˆ change of orientation: if we work with the new coordinates T = t, X = −x, we still
find solutions in N S 0 : we obtain for ⃗u ∈ N S 0 another solution ⃗v ∈ N S 0 given by
⃗v (T, X) = −⃗u(T, −X)
associated to the pressure
q(T, X) = p(T, −X).
If we assume the force f⃗ =
̸ 0, then we must modify the force f⃗ in (10.1) into ⃗g with
⃗g (T, X) = −f⃗(T, −X).
Special Examples of Solutions
251
Those are the only symmetries for the Navier–Stokes equations (see Bytev [73] and Lloyd
[341]). We did not consider the generalized symmetries described by Boisvert [49] that we
obtain by dropping the request on the control (10.2) of p at large and larger scales. The
tranforms we shall not consider in the following are then the following ones:
ˆ uniform change of pressure: with the same coordinates t, x, the same velocity ⃗u and
the same force f⃗, just change p into q(t, x) = p(t, x) + ϖ(t), where ϖ is arbitrary. The
new pressure q satisfies Sj (q(t, x)) = Sj p(t, x) + ϖ(t), and thus satisfies (10.2) if and
only if ϖ = 0.
ˆ motion of the observer with no rotation of the axes: in the new frame, we have
coordinates T = t and X = x − m(t), where m(t) is an arbitrary (C 2 ) function
of t. We then change ⃗u, p, f⃗ into ⃗v , q, ⃗g with ⃗v (T, X) = ⃗u(T, X + m(t)) − ṁ(t),
q(T, X) = p(T, X + m(t)) + m̈(t).X and ⃗g (T, X) = f⃗(T, X + m(t)). The new pressure
q satisfies Sj (q(T, X)) = (Sj p)(T, X + m(t)) + m̈(t).X = 0, and thus satisfies (10.2)
if and only if m̈(t) = 0 (hence the change of coordinates amounts to a mere space
translation followed by a change of Galilean reference frame).
10.2
Two-and-a-Half Dimensional Flows
In this section, we consider the Navier–Stokes problem with the following symmetry
property: ⃗u is invariant under the action of space translations parallel to the x3 axis.
Thus, ⃗u does not depend on x3 and may be seen as a (time-dependent) bivariate function:
⃗u(t, x1 , x2 , x3 ) = ⃗v (t, x1 , x2 ). As this is a three-dimensional bivariate vector field, Bertozzi
and Majda label those vector fields as “two-and-a-half dimensional flows” [40].
Thus, if we consider the Cauchy problem, we start with a data ⃗u0 (x1 , x2 , x3 ) = ⃗v0 (x1 , x2 )
and a force f⃗(t, x1 , x2 , x3 ) = ⃗g (t, x1 , x2 ).
If we may construct the solution by the Picard iterative scheme, then the symmetry of
the Navier–Stokes equations gives us that the solution (⃗u, p⃗) will satisfy the same symmetry
as ⃗u0 and f⃗: ⃗u(t, x1 , x2 , x3 ) = ⃗v (t, x1 , x2 ) and p(t, x1 , x2 , x3 ) = q(t, x1 , x2 ).
For the time being, let us assume that f⃗ = 0. We know (by Theorem 8.3) that the
problem
⃗ u − ∇p
⃗
∂t ⃗u = ν∆⃗u − ⃗u.∇⃗
(10.3)
with div ⃗u = 0 and ⃗u(0, .) = ⃗u0 has a mild solution when two conditions are satisfied:
ˆ ⃗u0 belongs to the Morrey space Ṁ 2,3 :
Z
1
sup
|⃗u0 (x)|2 dx < +∞
R>0,x0 ∈R3 R B(x0 ,R)
ˆ ⃗u0 satisfies:
lim t1/2 ∥Wνt ∗ ⃗u0 ∥∞ = 0
t→0
In that case, we know that we have a solution in the space {⃗u ∈ L∞ ((0, T0 ), Ṁ 2,3 ) / sup0<t<T0
√
t∥⃗u(t, .)∥∞ < +∞} for some positive T0 .
If we assume moreover that ⃗u0 (x1 , x2 , x3 ) = ⃗v0 (x1 , x2 ), then we have
⃗u0 ∈ Ṁ 2,3 ⇔ ⃗v0 ∈ L2 (R2 )
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The Navier–Stokes Problem in the 21st Century (2nd edition)
and the condition limt→0 t1/2 ∥Wνt ∗ ⃗u0 ∥∞ = 0 is automatically fulfilled, as L∞ ∩ L2 is dense
in L2 . Thus, a natural assumption on ⃗v0 is the fact that ⃗v0 ∈ L2 .
If we assume that ⃗u0 and f⃗ do not depend on x3 and if we search for a solution ⃗u and
a pressure p which do not depend on x3 , we find two equations. More precisely, writing
u1 (t, x) = v1 (t, x1 , x2 ), u2 (t, x) = v2 (t, x1 , x2 ), u3 (t, x) = w(t, w1 , x2 ), p(t, x) = q(t, x1 , x2 ),
f1 (t, x) = g1 (t, x1 , x2 ), f2 (t, x) = g2 (t, x1 , x2 ) and f3 (t, x) = h(t, x1 , x2 ), we have to solve
the following equations
ˆ ⃗v satisfies a 2D Navier–Stokes equation:

⃗ v = ν∆⃗v + ⃗g − ∇p
⃗
 ∂t⃗v + (⃗v .∇)⃗
⃗v (0, x1 , x2 ) = (u0,1 (x), u0,2 (x))

div ⃗v = 0
ˆ w satisfies a linear advection-diffusion scalar equation:
⃗
∂t w + (⃗v .∇)w
= ν∆w + h
w(0, x1 , x2 ) = u0,3 (x)
(10.4)
(10.5)
The study of 2D Navier–Stokes equations with initial value ⃗v0 ∈ L2 (R2 ) and ⃗g ∈ L2 H −1
was initiated by the works of Leray [327, 328, 329], and fully developed by Ladyzhenskaya,
Lions and Prodi [293, 339].
2D Navier–Stokes equations
Theorem 10.1.
If ⃗v0 ∈ (L2 (R2 ))2 with div ⃗v0 = 0 and let ⃗g ∈ L2 ((0, T ), (H −1 (R2 )2 ), then there exists
a unique solution ⃗v of equation

⃗ v = ν∆⃗v + ⃗g − ∇p
⃗
∂t⃗v + (⃗v .∇)⃗
(10.6)
⃗v (0, .) = ⃗v0

div ⃗v = 0
2 1
2
such that ⃗v ∈ L∞
t Lx ∩ Lt Hx (if T < +∞).
Proof. Local existence:
We rewrite (10.6) into
⃗v =
(2)
Wνt
Z
t
∗ ⃗v0 +
0
(2)
Wν(t−s) ∗ P(2) (⃗g − div(⃗v ⊗ ⃗v )) ds
(2)
(10.7)
where Wt (x1 , x2 ) is the 2D heat kernel and P(2) is the 2D Leray projection operator. We
are going to look for a solution ⃗v ∈ L4 ((0, T0 ), L4 (R2 )) for T0 small enough.
Indeed, we have the following estimates:
(2)
∥Wνt ∗ ⃗v0 ∥L∞ L2 = ∥⃗v0 ∥2
1
(2)
∥Wνt ∗ ⃗v0 ∥L2 Ḣ 1 = √ ∥⃗v0 ∥2
2ν
Special Examples of Solutions
253
√
1
(2)
Wν(t−s) ∗ P(2)⃗g ds∥L∞ ((0,T0 ),L2 ) ≤ ( T 0 + √ )∥⃗g ∥L2 H −1
2ν
t
Z
∥
0
Z
∥
0
t
(2)
Wν(t−s) ∗ P(2)⃗g ds∥L2 ((0,T0 ),Ḣ 1 ) ≤ (CT0 +
1
)∥⃗g ∥L2 H −1
ν
Moreover, we have the Sobolev embedding
H 1/2 (R2 ) ⊂ L4 .
(2)
⃗0 = Wνt
Thus, we find that V
∗ ⃗v0 +
⃗
limt→0 ∥V0 ∥L4 ((0,T ),L4 ) = 0.
Rt
0
(2)
Wν(t−s) ∗ P(2)⃗g ds belongs to L4 L4 . Moreover,
0
Now, if ⃗u and ⃗v belong to L4 L4 , we have that div(⃗u ⊗ ⃗v ) belong to L2 Ḣ −1 , so that
R t (2)
B(⃗u, ⃗v ) = 0 Wν(t−s) ∗ P div(⃗u ⊗ ⃗v ) ds belongs to L∞ L2 ∩ L2 Ḣ 1 ∩ L4 L4 with
∥B(⃗u, ⃗v )∥L4 ((0,T0 ),L4 ≤ Cν ∥⃗u∥L4 ((0,T0 ),L4 ∥⃗v ∥L4 ((0,T0 ),L4 .
⃗0 ∥L4 ((0,T ),L4 < 1 , we shall find a solution ⃗v through
If T0 is small enough, so that ∥V
0
4Cν
Picard’s iterative process. The process will converge in L∞ L2 ∩ L2 Ḣ 1 ∩ L4 L4 .
Global existence:
If the solution ⃗v belongs to L∞ ((0, T0 ), L2 )∩L2 ((0, T0 ), Ḣ 1 ) then we find an estimate slightly
better than just ⃗v ∈ L∞ L2 . As a matter of fact, we have
Z t
(2)
(2)
(2)
⃗ ds = Wνt
⃗)
⃗v = Wνt ∗ ⃗v0 +
W
∗V
∗ ⃗v0 + L(V
0
ν(t−s)
⃗ = P(2) (⃗g − div(⃗v ⊗ ⃗v )) belongs to L2 H −1 ; the operator L maps L2 H −1 to L∞ L2
where V
2 2
and L H to Lip L2 ; as L2 H 2 is dense in L2 H −1 , we find that L actually maps L2 H −1 to
C([0, T0 ], L2 ).
If T ∗ is the maximal time of existence of the solution ⃗v , so that ⃗v belongs to
∞
L ((0, T0 ), L2 ) ∩ L2 ((0, T0 ), Ḣ 1 ) for every T0 < T ∗ , we find that T ∗ = T unless that ⃗v
does not belong to L∞ ((0, T ∗ ), L2 ) ∩ L2 ((0, T ∗ ), Ḣ 1 ): if ⃗v belonged to L∞ ((0, T ∗ ), L2 ) ∩
L2 ((0, T ∗ ), Ḣ 1 ) with T ∗ < T , then it would belong to C([0, T ∗ ], L2 ) and we could solve the
Cauchy problem for the Navier–Stokes equations on some interval [T ∗ , T ∗ + T0 ] with initial
value ⃗v (T ∗ , .).
Thus, in order to show that we have a global solution, we only need to control the sizes
⃗ v on [0, T ∗ ). As ⃗v belongs to L2 H 1 and ∂t⃗v belongs to L2 H −1 on every compact
of ⃗v and ∇⊗⃗
interval of [0, T ∗ ), we may write
d
⃗ ⊗ ⃗v (t, .)∥22 + 2⟨⃗v (t, .)|⃗g (t, .)⟩H 1 ,H −1
∥⃗v (t, .)∥22 = 2⟨⃗v (t, .)|∂t⃗v (t, .)⟩H 1 ,H −1 = −2ν∥∇
dt
so that
d
⃗ ⊗ ⃗v (t, .)∥22 + ν∥⃗v (t, .)∥22 + 1 ∥⃗g (t, .)∥2 −1 .
∥⃗v (t, .)∥22 ≤ ν∥∇
H
dt
ν
By Grönwall’s lemma, we get
Z t
Z t
⃗ ⊗ ⃗v (s, .)∥22 ds ≤ eνt (∥⃗v0 ∥22 + 1
∥⃗v (t, .)∥22 + ν
∥|∇
∥⃗g (s, .)∥2H −1 ds).
ν
0
0
Hence, T ∗ = T : we have a global solution.
254
The Navier–Stokes Problem in the 21st Century (2nd edition)
We may now solve the Navier–Stokes equations with initial value ⃗u0 = (⃗v0 , w0 ) and
forcing term f⃗ = (⃗g , h):
Proposition 10.1.
If (⃗v0 , w0 ) ∈ (L2 (R2 ))3 with div ⃗v0 = 0 and (⃗g , h) ∈ (L2 ((0, T ), H −1 (R2 )))3 then the Equations (10.4) and (10.5) have global solutions ⃗v and w in L∞ L2 ∩ L2 H 1 (if T < +∞).
Proof. The existence of ⃗v has been proved in Theorem 10.1. For the existence of w, we write
w as a fixed point of the transform
Z t
(2)
(2)
(2)
ω 7→ Wνt ∗ w0 +
Wν(t−s) ∗ (h − div(ω⃗v )) ds = Wνt ∗ w0 + L(w).
0
L is a bounded linear operator on L∞ ((0, T0 ), L2 ) ∩ L2 ((0, T0 ), L2 ) and satisfies (uniformly
in T0 )
∥L(w)∥L∞ ((0,T0 ),L2 )∩L2 ((0,T0 ),L2 ) ≤ C0 ∥⃗v ∥L4 ((0,T0 ),L4 ∥w∥L∞ ((0,T0 ),L2 )∩L2 ((0,T0 ),L2 ) .
Thus, L is a contraction as soon as T0 is small enough to grant that C0 ∥⃗v ∥L4 ((0,T0 ),L4 < 1.
Global existence of w is then proved by splitting [0, T ] into a finite union of intervals
[Tj , Tj+1 ] with C0 ∥⃗v ∥L4 ((Tj ,Tj+1 ),L4 < 1: once w is constructed on [0, Tj+1 ], one constructs
w on [Tj+1 , Tj+2 ] by considering the Cauchy problem with initial value w(Tj+1 , .) at t =
Tj+1 .
Thus, we have global existence of unique solutions to the Navier–Stokes problem when
the initial value ⃗u0 depends only on the first two variables (x1 , x2 ) and when the force
f⃗ depends only on t and (x1 , x2 ) and when ⃗u0 ∈ L2 (R2 ) and f⃗ ∈ L2 ((0, T ), H −1 (R2 ))
(whatever their sizes). The case of ⃗u0 ∈ L2uloc (R2 ) has been discussed by Basson [24] (when
f⃗ = 0). He found global existence as well in this case.
The stability of the solutions decribed in Theorem 10.1 and in Proposition 10.1 under
small enough 3D perturbations has been discussed by Gallagher [195] and Iftimie [239]. We
are going to present Gallagher’s result in the following theorem:
3D perturbation of the 2D Navier–Stokes equations
Theorem 10.2.
Let ⃗v0 ∈ (L2 (R2 ))3 with ∂1 v1 + ∂2 v2 = 0 and let ⃗g ∈ L2 ((0, T ), (H −1 (R2 )3 ) (where
T < +∞), and let ⃗v be the associated solution of the Navier–Stokes equations such
that ⃗v ∈ (L∞ ((0, T ), L2 (R2 ) ∩ L2 ((0, T ), H 1 (R2 ))3 . Then there exists a positive ϵ which
depends on T , ⃗v0 and ⃗g such that the Navier–Stokes equations

⃗ u = ν∆⃗u + f⃗ − ∇p
⃗
∂t ⃗u + ⃗u.∇⃗



⃗u(0, .) = ⃗v0 + w
⃗0
(10.8)
⃗
⃗

f
=
⃗
g
+
h


div ⃗u = 0
with ∥w
⃗ 0 ∥Ḣ 1/2 < ϵ and
1/2
RT
∥⃗h∥2Ḣ −1/2
0
3/2
L2t Ḣx .
ds < ϵ2 , have a global solution ⃗u = ⃗v + w
⃗ with
w
⃗ ∈ L∞
t ((0, T ), Ḣx ) ∩
The same result holds for T
= +∞ provided that we have ⃗g
L2 ((0, +∞), (Ḣ −1 (R2 )3 ) or ⃗g ∈ L1 ((0, +∞), (L2 (R2 )3 ).
∈
Special Examples of Solutions
255
Proof. We first construct our solution w
⃗ on (0, T0 ) with T0 small enough.
w
⃗ is a solution of the fixed-point problem
Z t
w
⃗ = Wνt ∗ w
⃗0 +
Wν(t−s) ∗ P(⃗h − div(⃗v ⊗ w
⃗ +w
⃗ ⊗ ⃗v + w
⃗ ⊗ w))
⃗ ds.
0
We write
⃗ 0 − L(w)
w
⃗ =W
⃗ − B(w,
⃗ w)
⃗
⃗ 0 = Wνt ∗ w
with W
⃗0 +
Rt
0
Wν(t−s) ∗ P⃗h ds,
Z
t
L(w)
⃗ =
Wν(t−s) ∗ P div(⃗v ⊗ w
⃗ +w
⃗ ⊗ ⃗v ) ds
0
and
Z
B(w
⃗ 1, w
⃗ 2) =
t
Wν(t−s) ∗ P div(w
⃗1 ⊗ w
⃗ 2 ) ds
0
1/2
3/2
2
1/2
Let YT0 = L∞
(R3 ) and β ∈
t ((0, T ), Ḣx ) ∩ Lt Ḣx . We have, for α ∈ Ḣ
2
−1/2
3
L ((0, T0 ), Ḣ
(R ))
∥Wνt ∗ α∥YT0 ≤ C0 ∥α∥Ḣ 1/2
(10.9)
and
t
Z
∥
Wν(−s)t ∗ β(s.) ds∥YT0 ≤ C0 ∥β∥L2 Ḣ −1/2
0
(10.10)
where C0 does not depend on T0 .
We thus get
Z
⃗ 0 ∥Y ≤ C1
∥W
T0
∥w
⃗ 0 ∥Ḣ 1/2 + (
T0
0
!
∥⃗h(s, .)∥2Ḣ −1/2 ds)1/2
and, for w
⃗ 1, w
⃗ 2 ∈ YT0 ,
∥B(w
⃗ 1, w
⃗ 2 )∥YT0 ≤ C2 ∥w
⃗ 1 ∥L4 Ḣ 1 ∥w
⃗ 2 ∥L4 Ḣ 1 ≤ C2 ∥w
⃗ 1 ∥YT0 ∥w
⃗ 2 ∥YT0
t
t
Moreover, we may see from inequality (10.9) that we have, for β ∈ L1 ((0, T0 ), Ḣ 1/2 (R3 ))
t
Z
∥
Wν(−s)t ∗ β(s.) ds∥YT0 ≤ C3 ∥β∥L1 Ḣ 1/2
0
(10.11)
Complex interpolation between (10.10) and (10.11) gives then
Z
∥
0
t
Wν(−s)t ∗ β(s.) ds∥YT0 ≤ C4 ∥β∥L4/3 L2
(10.12)
We have
⃗ w)∥
∥P(⃗v .∇
⃗ L4/3 L2 ≤ ∥⃗v ∥L4t L4x
t
x
1 ,x2
⃗ ⊗ w∥
∥∇
⃗ L2t L2x
3
L4x
1 ,x2
with
⃗ ⊗ w∥
∥∇
⃗ L2t L2x
3
L4x
1 ,x2
⃗ ⊗ w∥
≤ C∥∇
⃗ L2 L2
1/2
x3 Ḣx1 ,x2
t
≤ C∥w∥
⃗ Y T0
while
⃗ v )∥ 4/3 2 ≤ ∥∇
⃗ ⊗ ⃗v ∥L2 L2
∥P(w.
⃗ ∇⃗
L
L
t x
t
x
1 ,x2
∥w∥
⃗ L4t L2x
3
L∞
x ,x
1
2
256
The Navier–Stokes Problem in the 21st Century (2nd edition)
with
∥w∥
⃗ L4t L2x
3
L∞
x ,x
1
2
≤ C′
≤ C∥w∥
⃗ L4 L2
t
1
x3 (Ḃ2,1 )x1 ,x2
q
∥w∥
⃗ L∞ L2
1/2
x3 Ḣx1 ,x2
t
∥w∥
⃗ L2 L2
t
3/2
x3 Ḣx1 ,x2
so that
∥w∥
⃗ L4t L2x
3
L∞
x ,x
1
2
≤ C ′ ∥w∥
⃗ YT0 .
Hence, we have
∥L(w)∥
⃗ YT0 ≤ C5 (∥⃗v ∥L4t L4x
1 ,x2
⃗ ⊗ ⃗v ∥L2 L2
+ ∥∇
t x
1 ,x2
)∥w∥
⃗ Y T0 .
Thus, if T0 is small enough to grant that
C5 (∥⃗v ∥L4t L4x
1 ,x2
⃗ ⊗ ⃗v ∥L2 L2
+ ∥∇
t x
1 ,x2
)≤
1
2
and if w
⃗ 0 and T0 are small enough to grant that
Z
∥w
⃗ 0 ∥Ḣ 1/2 + (
T0
0
∥⃗h(s, .)∥2Ḣ −1/2 ds)1/2 <
1
16C1 C2
then we shall find a fixed-point w
⃗ with
Z
∥w∥
⃗ YT0 < 4C1 (∥w
⃗ 0 ∥Ḣ 1/2 + (
0
T0
∥⃗h(s, .)∥2Ḣ −1/2 ds)1/2 ).
We now turn to the global existence. If w
⃗ is defined on (0, T1 ) with T1 < T and if
sup0<t<T1 ∥w(t,
⃗ .)∥Ḣ 1/2 < 16C11 C2 , we find that the behavior of w
⃗ in Ḣ 1/2 is controlled by
⃗ 0 , L(w)
the behaviors of W
⃗ and B(w,
⃗ w);
⃗ due to the fact that smooth functions are dense
2 −1/2
in L Ḣ
and in L3/2 L2 , we find that w
⃗ ∈ C([0, T1 ], Ḣ 1/2 ); we can then reiterate the
construction of w
⃗ from the departure time t = T1 and see that w
⃗ may be defined on a larger
interval. Thus, in order to check the existence of a global solution, we just have to check
that the Ḣ 1/2 norm of w
⃗ is controlled.
We may write
√
√
d
∥w∥
⃗ 2Ḣ 1/2 = 2⟨ −∆w|∂
⃗ t w⟩
⃗ = 2 ⟨ −∆w|ν∆
⃗
w
⃗ + ⃗h − div(⃗v ⊗ w
⃗ +w
⃗ ⊗ ⃗v + w
⃗ ⊗ w)⟩
⃗
dt
so that
√
d
∥w∥
⃗ 2Ḣ 1/2 = −2ν∥(−∆)3/4 w∥
⃗ 22 + 2⟨ −∆w|
⃗ ⃗h⟩H 1/2 ,H −1/2
dt
√
√
⃗w
⃗ v ⟩L2 ,L2 − 2⟨ −∆w|
−2⟨ −∆w|⃗
⃗ v .∇
⃗ + w.
⃗ ∇⃗
⃗ div(w
⃗ ⊗ w)⟩
⃗ H 1/2 ,H −1/2
2
≤ −2ν∥w∥
⃗ 3/2 + 2 ∥w∥
⃗ H 3/2 ∥⃗h∥H −1/2
H
⃗ ⊗ ⃗v ∥L2 (R2 ) ∥w∥
+C∥w∥
⃗ Ḣ 1 (∥⃗v ∥L4 (R2 ) ∥w∥
⃗ Ḣ 3/2 + ∥∇
⃗ Ḣ 1 ) + C∥w∥
⃗ H 3/2 ∥w∥
⃗ 2Ḣ 1
1/2
3/2
≤ −2ν∥w∥
⃗ 2H 3/2 + 2 ∥w∥
⃗ H 3/2 ∥⃗h∥H −1/2 + C∥w∥
⃗ Ḣ 1/2 ∥⃗v ∥L4 (R2 ) ∥w∥
⃗ Ḣ 3/2
⃗ ⊗ ⃗v ∥L2 (R2 ) ∥w∥
+C∥∇
⃗ Ḣ 1/2 ∥w∥
⃗ Ḣ 3/2 + C∥w∥
⃗ 2H 3/2 ∥w∥
⃗ Ḣ 1/2
ν
1
≤ − ∥w∥
⃗ 2H 3/2 + ∥⃗h∥2H −1/2
2
ν
1
1 ⃗
2
+C6 ∥w∥
⃗ Ḣ 1/2 ( 4 ∥⃗v ∥L4 (R2 ) + 2 ∥∇
⊗ ⃗v ∥2L2 (R2 ) ) + C7 ∥w∥
⃗ 2H 3/2 ∥w∥
⃗ Ḣ 1/2
ν
ν
Special Examples of Solutions
257
As long as 2C7 ∥w∥
⃗ Ḣ 1/2 < ν, we find that
1
+
ν
∥w∥
⃗ Ḣ 1/2 ≤ (∥w
⃗ 0 ∥Ḣ 1/2
Z
0
T
C
∥⃗h∥2H −1/2 ds) e 6
Rt
1
0 ν4
⃗ v ∥2 2 2 ds
∥⃗
v ∥L4 (R2 ) + ν12 ∥∇⊗⃗
L (R )
Thus, we have global existence on (0, T ), provided that
∥w
⃗ 0 ∥Ḣ 1/2 +
1
ν
T
Z
∥⃗h∥2H −1/2 ds <
0
RT
1
ν
−C
,
)e 6 0
16C1 C2 2C7
min(
1
ν4
⃗ v ∥2 2 2 ds
∥⃗
v ∥L4 (R2 ) + ν12 ∥∇⊗⃗
L (R )
.
If we want to get a criterion for existence on (0, +∞), we need that
Z
+∞
0
1
1 ⃗
∥⃗v ∥L4 (R2 ) + 2 ∥∇
⊗ ⃗v ∥2L2 (R2 ) ds < +∞.
ν4
ν
This is the case if ⃗g ∈ L2 Ḣ −1 or ⃗g ∈ L1 L2 : we start from the energy balance
d
⃗ ⊗ ⃗v ∥22 + 2⟨⃗v |⃗g ⟩.
∥⃗v ∥22 = −2ν∥∇
dt
• if ⃗g ∈ L2 Ḣ −1 , we get
d
⃗ ⊗ ⃗v ∥2 + 1 ∥⃗g ∥2 −1
∥⃗v ∥22 ≤ −ν∥∇
2
Ḣ
dt
ν
so that
∥⃗v (t, .)∥22 + ν
Z
t
⃗ ⊗ ⃗v ∥22 ds ≤ ∥⃗v0 ∥22 +
∥∇
0
1
ν
Z
0
t
∥⃗g ∥2Ḣ −1 ds
• if ⃗g ∈ L1 L2 , we get
d
∥⃗v ∥22 ≤ 2∥⃗g ∥2 ∥⃗v ∥2
dt
so that
Z
t
∥⃗v (t, .)∥2 ≤ ∥⃗v0 ∥2 +
∥⃗g ∥2 ds
0
and
Z
2ν
0
t
⃗ ⊗ ⃗v ∥2 ds ≤ (∥⃗v0 ∥2 +
∥∇
2
2
Z
t
∥⃗g ∥2 ds)2 .
0
A special example of two-and-a-half dimensional flow is the parallel flow : assume that
⃗ u = ν∆⃗u − ∇p,
⃗ div ⃗u = 0 that depends only on t, x1 , x3 and
⃗u is a solution of ∂t ⃗u + ⃗u.∇⃗
that moreover u3 = 0. Then the condition div ⃗u = 0 gives ∂1 u1 = 0, so that u1 = u1 (t, x3 ),
⃗ u = u1 ∂1 ⃗u = (0, u1 ∂1 u2 , 0) and div(⃗u · ∇⃗
⃗ u) = 0,
while u2 = u2 (t, x1 , x3 ). Moreover, ⃗u · ∇⃗
so that the pressure is equal to 0. Finally, the Navier—Stokes system is transformed into
a linear heat equation ∂t u1 = ν∂32 u1 and a linear advection-diffusion equation ∂t u2 =
ν(∂12 + ∂32 )u2 − u1 (t, x3 )∂1 u2 , which are easily solved.
258
The Navier–Stokes Problem in the 21st Century (2nd edition)
10.3
Axisymmetrical Solutions
In this section, we consider the Navier–Stokes problem with the following symmetry
property: ⃗u is invariant under the action of rotations around the x3 axis (i.e., ⃗u is axisymmetric). In order to describe those solutions, we shall use the cylindrical coordinates:
r > 0, θ ∈ (−π, π), z ∈ R, with x1 = r cos θ, x2 = r sin θ and x3 = z. We then write
⃗u(x1 , x2 , x3 ) = Ur ⃗er + Uθ ⃗eθ + Uz ⃗ez , with ⃗er = (cos θ, sin θ, 0), ⃗eθ = (− sin θ, cos θ, 0) and
⃗ez = (0, 0, 1). Let us remark however that this change of coordinates is degenerated on the
axis r = 0.
Our hypothesis is that ⃗u is axisymmetric: it means that Ur , Uθ and Uz do not depend
on θ, or equivalently:
∂θ ⃗u = ⃗ez ∧ ⃗u.
(10.13)
If we want for a locally square integrable axisymmetric vector field ⃗u = Ur ⃗er +Uθ ⃗eθ +Uz ⃗ez
to belong to Ṁ 2,3 , we may suppose that
ZZ
|Ur (r, z)|2 + |Uθ (r, z)|2 + |Uz (r, z)|2 dr dz < +∞
(10.14)
(0,+∞)×R
⃗ ∈ (L2 ((0, +∞) × R))3 , or equivalently that
i.e., U
1
⃗u
(x21 +x22 )1/4
∈ L2 (R3 ). Indeed, if we
integrate |⃗u|2 on B(x0 , R), with x0 = (r0 cos θ0 , r0 sin θ0 , z0 ), we find:
ˆ if r0 < 9R,
Z
|⃗u|2 dx ≤
B(x0 ,R)
ZZ
⃗ (r, z)|2 r dr dz ≤ 10R∥U
⃗ ∥2
|U
2
0<r<10 R,|z−z0 |<R
π 9R
0
ˆ if r0 > 9R and |x − x0 | < R, then 8r90 < r < 10r
9 , |z − z0 | < R and |θ − θ0 | < 2 8r0 so
that
Z
ZZ
9R
⃗ ∥22
⃗ (r, z)|2 r dr dz ≤ 5π R∥U
|⃗u|2 dx ≤ π
|U
10r
8r0
4
B(x0 ,R)
r< 9 0 ,|z−z0 |<R
1
x21 +x22
Thus, if ⃗u0 ∈ L2 ( √
Ṁ
dx) and is axisymmetric (with ⃗ez as symmetry axis), then ⃗u0 ∈
2,3
.
Conversely, if ⃗u is a regular (axisymmetric) field (⃗u0 ∈ Ḣ 1/2 (R3 ), we use the embedding
1/2
1/2
Ḣ 1/2 ⊂ L2x3 Ḣx1 ,x2 and the Hardy inequality Ḣx1 ,x2 ⊂ L2 ( 1r dx1 dx2 ), to get that ⃗u0 ∈
L2 ( √ 21 2 dx).
x1 +x2
1
x21 +x22
Thus, the assumption ⃗u0 ∈ L2 ( √
dx) is quite natural for the study of axisymmetric
fields. Since the smooth function that are compactly supported in (0, +∞) × R are dense in
L2 ((0, +∞) × R), we see that smooth compactly supported axisymmetric fields are dense in
the spaces of axisymmetric fields that belong to L2 ( √ 21 2 dx); for such vector fields ⃗u0 ,
x1 +x2
√
we have ⃗u0 ∈ Ṁ 2,3 and limt→0+ t∥Wνt ∗ ⃗u0 ∥∞ = 0. When f⃗ = 0, this ensures the local
existence of a mild solution of the Navier–Stokes equations (and global existence if the Ṁ 2,3
norm of ⃗u0 is small enough), by Theorem 8.3.
As underlined by Gallagher, Ibrahim and Majdoub [197], one may prove directly that
the Picard algorithm works in the frame of weighted Lebesgue spaces, using the theory of
Muckenhoupt weights [248, 448]:
Special Examples of Solutions
259
Definition 10.1.
Let (X, ρ, µ) be a space of homogeneous type (see Definition 5.1) and 1 < p < +∞. A
positive function w on X belongs to the Muckenhoupt class Ap if it satisfies the reverse
Hölder inequality:
Z
Z
1
1
(
w(x) dµ)1/p (
w(x)− p−1 dµ)1−1/p < +∞.
sup
µ(B(x
,
r)
0
x0 ∈X,r>0
B(x0 ,r)
B(x0 ,r)
Then the theory of singular integrals on spaces of homogeneous type [125, 313] allows
one to prove the following facts:
ˆ if w ∈ Ap (X), then the Hardy–Littlewood maximal operator is bounded on Lp (w dµ):
∥Mf ∥Lp (w dµ) ≤ C∥f ∥Lp (w dµ) .
ˆ if T is a bounded Calderón–Zygmund operator on L2 (X, dµ), then it can be extended
as a bounded operator on Lp (X, w dµ)
ˆ if T is a bounded Calderón–Zygmund operator from Lq (X, dµ, Lp0 (X0 , dµ0 )) to
Lq (X, dµ, Lp1 (X1 , dµ1 )) for some 1 < q < +∞ (where X0 and X1 are locally compact σ-compact metric spaces and µ0 and µ1 are regular Borel measures on X0
and X1 ), it can be extended as a bounded operator on Lp (X, w dµ, Lp0 (X0 , dµ0 ))
to Lp (X, w dµ, Lp1 (X1 , dµ1 ))
We shall use as well a variant of Hedberg’s inequality (see Lemma 5.3):
Lemma 10.1.
−σ
⃗ ∈ L1 , then we have the pointwise inequality
If σ > 0, f ∈ Ḃ∞,∞
(R3 ) and if ∇f
loc
1
σ
1+σ
|f (x)| ≤ C(M∇f
⃗ (x)) 1+σ ∥f ∥Ḃ −σ .
∞,∞
Proof. We write
Z +∞
Z
f =−
∆Wt ∗ f dt =
0
R
Z
+∞
∆Wt ∗ f dt +
0
∆Wt ∗ f dt = AR (x) + BR (x).
R
We have
Z
R
|AR (x)| ≤
⃗ )| dt ≤ C
|div(Wt ∗ ∇f
0
Z
R
0
√
1
√ M∇f
⃗ (x)
⃗ (x) dt = 2C RM∇f
t
and
Z
+∞
|BR (x)| ≤
Z
+∞
|(∆Wt ) ∗ f | dt ≤ C
R
R
1
σ
t1+ 2
−σ
∥f ∥Ḃ∞,∞
dt =
2C 1
σ ∥f ∥Ḃ −σ .
∞,∞
σ R2
We end the proof by taking
R=
−σ
∥f ∥Ḃ∞,∞
2
! 1+σ
M∇f
⃗ (x)
.
Then the study of axisymmetric solutions ⃗u ∈ L∞ ((0, T ), L2 ( √
1
x21 +x22
by noticing that
√ 21 2
x1 +x2
3
belongs to A2 (R ) and that
−1
∥⃗u∥Ḃ∞,∞
≤ C∥⃗u∥Ṁ 2,3 ≤ C ′ ∥⃗u∥L2 ( √
1
2
x2
1 +x2
dx) .
dx) can be done
260
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proposition 10.2.
Let w be a weight on R3 such that w ∈ A2 (R3 , dx). Then:
• |Wνt ∗ f (x)| ≤ Mf (x) for f ∈ L2 (w dx)
2
⃗
• if f ∈ L2 (w dx), then Wνt ∗ f belongs to L∞
t L (w dx), and ∇Wνt ∗ f belongs to
2 2
Lt L (w dx):
sup ∥Wνt ∗
t>0
f (t, .)∥2L2 (w dx)
Z
+∞
2
⃗ νt ∗ f ∥2 2
∥∇W
L (w dx) dt ≤ C∥f ∥L2 (w dx)
+ν
(10.15)
0
R
⃗ = t ∇W
⃗ ν(t−s) ∗ g(s, .) ds, then G
⃗ belongs to
• if g ∈ L2 ((0, +∞), L2 (w dx) and G
0
∞ 2
2 2
⃗
⃗
Lt L (w dx) and ∇ ⊗ G belongs to Lt L (w dx):
2
⃗ .)∥2 2
ν sup ∥G(t,
L (w dx) + ν
t>0
Z
+∞
2
⃗ ⊗ G(t,
⃗ .)∥2 2
∥∇
L (w dx) dt ≤ C∥g∥L2 (w dx)
(10.16)
0
−1
2 2
4 4
⃗
• if g belongs to L∞
t Ḃ∞,∞ and ∇g belongs to Lt L (w dx), then g belongs to Lt L (w dx)
and
q
⃗ L2 L2 (w dx)
−1
∥g∥L4 L4 (w dx) ≤ C ∥g∥L∞ Ḃ∞,∞
∥∇g∥
(10.17)
Proof. From the inequality |Wνt ∗ f (x)| ≤ Mf (x) (Lemma 7.4), we get
sup ∥Wνt ∗ f (t, .)∥L2 (w dx) ≤ ∥Mf ∥L2 (w dx) ≤ C∥f ∥L2 (w dx) .
t>0
√
⃗ νt ∗ f ∥L2 L2 (w dx) , we first remark that with no loss of generality we
To estimate ν∥|∇W
may assume that ν = 1 (changing t into τ = νt). We then write L2t L2 (w dx) = L2 (w dx)L2t .
⃗ t ∗ f )t>0 is bounded from L2 (R3 ) to L2 (dx, L2 (dt)). MoreThe mapping f 7→ (∇W
Rt
R
2
⃗ ν(t−s) (x)h(s) ds∥L2 (dt) ≤ +∞ |∇W
⃗ νt (x)| dt∥h∥2 =
over, we have, for h ∈ L (R), ∥ −∞ ∇W
0
−3
C|x| ∥h∥2 and, for i = 1, . . . , 3,
Z
t
∥
⃗ ν(t−s) (x)h(s) ds∥L2 (dt) ≤
∂i ∇W
−∞
Z
+∞
⃗ νt (x)| dt∥h∥2 = C|x|−4 ∥h∥2
|∂i ∇W
0
Thus, we may apply the theory of singular integrals with values in L2 (dt) and we find that
⃗ t ∗ f )t>0 is bounded from L2 (w dx) to L2 (w dx, L2 (dt)). Inequality (10.15) is
f 7→ (∇W
proved.
⃗ .)∥L2 (w dx) , we use the fact that L2 (w dx) is the dual of L2 (w−1 dx)
Now, to estimate ∥G(t,
−1
and that w ∈ A2 as well. If f ∈ L2 (w−1 dx), we have
Z
⃗ x)f (x) dx = −
G(t,
Z
t
⃗ (ν(t−s) ∗ f (x) dx ds
g(s, x)∇W
0
so that
Z
|
⃗ x)f (x) dx| ≤∥g∥L2 L2 (w dx) ∥∇W
⃗ (νt ∗ f ∥L2 L2 (w−1 dx)
G(t,
t
t
1
≤C √ ∥g∥L2t L2 (w dx) ∥f ∥L2 (w−1 dx)
ν
Special Examples of Solutions
261
⃗ ⊗ G∥
⃗ L2 L2 (w dx) , we shall use the theory of the maximal regularity
Now, to estimate ∥∇
t
for the heat kernel [313]. As the Riesz transforms are bounded on L2 (w dx), we just have to
Rt
Rt
estimate ∥ 0 ∆Wν(t−s) ∗ g(s, .) ds∥L2t L2 (w dx) . The operator h 7→ −∞ ∆Wν(t−s) ∗ h(s, .) ds
may be viewed as a Calderón–Zygmund operator on the parabolic space R × R3 , which is
a space of homogeneous type with quasi-metric ρ((t, x), (s, y)) = ((t − s)2 + |x − y|)1/4 and
measure dµ = dt dx. The boundedness of this operator on L2 (R × R3 ) is obvious, as this is
2
a convolution in R4 with a kernel K(t, x) whose Fourier transform is K̂(τ, ξ) = iτ−|ξ|
−|ξ|2 . As
a function of (t, x) (independent of t), w still is a Muckenhoupt weight: w(x) ∈ A2 (R × R3 ).
Thus, we have
Z t
∥
∆Wν(t−s) ∗ h(s, .) ds∥L2 (w(x) dt dx) ≤ C∥h∥L2 (w(x) dt dx) .
−∞
We conclude by taking h = 1t>0 g, since L2 (w(x) dt dx) = L2t L2 (w dx).
−1
2 2
⃗
Let us now consider g ∈ L∞
t Ḃ∞,∞ with ∇g ∈ Lt L (w dx). Hedberg’s inequality (Lemma
10.1) gives
1/2
(x))1/2
|g(t, x)| ≤ C∥g(t, .)∥Ḃ −1 (M∇g(t,.)
⃗
∞,∞
with
Z
((M∇g(t,.)
(x))1/2 )4 w(x) dx ≤ C
⃗
Z
⃗
|∇g(t,
x)|2 w(x) dx.
Navier–Stokes equations and Muckenhoupt weights
Theorem 10.3.
Let w ∈ A2 (R3 ) be such that, for axisymmetric vector fields ⃗u in L2 (w dx), we have the
inequality
−1
≤ C∥⃗u∥L2 (w dx) .
∥⃗u∥Ḃ∞,∞
Let ⃗u0 ∈ L2 (w dx) be an axisymmetric vector field with div ⃗u0 = 0 and let f⃗ be an
√
axisymmetric forcing term that can be written as f⃗ = −∆F⃗ , with F⃗ ∈ L2 (w dx).
Then there exists a time T > 0 such that the problem

⃗ u = ν∆⃗u + f⃗ − ∇p
⃗
∂t ⃗u + ⃗u.∇⃗
(10.18)
⃗u(0, .) = ⃗u0

div ⃗u = 0
2
⃗ u∈
has a unique axisymmetric solution ⃗u on (0, T ) × R3 with ⃗u ∈ L∞
t L (w dx) and ∇ ⊗ ⃗
2 2
Lt L (w dx).
Moreover, there exists a positive ϵ (which does not depend on ν, nor on ⃗u0 nor
⃗
f ), such that, when ∥⃗u0 ∥L2 (w dx) < ϵν and ∥F⃗ ∥L2t L2 (w dx) < ϵν 3/2 , then the solution is
global.
Proof. We are going to solve the problem in the space L4t L4 (w dx). Indeed, using (10.15)
and (10.17), we find
∥Wνt ∗ ⃗u0 ∥L4 L4 (w dx) ≤ Cν −1/4 ∥⃗u0 ∥L2 (w dx)
and, using (10.16) and (10.17), we find
Z t
√
−∆Wν(t−s) ∗ F⃗ ds∥L4 L4 (w dx) ≤ Cν −3/4 ∥F⃗ ∥L2 L2 (w dx)
∥
0
262
The Navier–Stokes Problem in the 21st Century (2nd edition)
Moreover, if B is the bilinear operator
Z t
B(⃗u, ⃗v ) =
Wν(t−s) ∗ div(⃗u ⊗ ⃗v ) ds
0
we find
∥B(⃗u, ⃗v )∥L4 L4 (w dx) ≤
C
ν 3/4
∥⃗u ⊗ ⃗v ∥L2 L2 (w dx) ≤
C0
∥⃗u∥L4 L4 (w dx) ∥⃗v ∥L4 L4 (w dx)
ν 3/4
Thus, we can see that we get a solution of the fixed point problem on (0, T0 ) provided that,
R
√
⃗ 0 = Wνt ∗ ⃗u0 + t Wν(t−s) ∗ P −∆F⃗ ds, we have
defining U
0
Z
T0
Z
⃗ 0 (t, x)|4 dx dt <
|U
0
As
Z
T0
Z
⃗ 0 (t, x)|4 dx dt ≤
|U
0
ν3
.
256 C04
C14 1
1
( ∥⃗u0 ∥4L4 (w dx) + 3 ∥F⃗ ∥4L2 L2 (w dx) )
2 ν
ν
we get global existence for
∥⃗u0 ∥L2 (w dx) <
ν
4C1 C0
and
∥F⃗ ∥L2 L2 (w dx) <
ν 3/2
.
4C1 C0
Theorem 10.3 gives then the result of [197]:
Locally square integrable axisymmetric solutions
Corollary 10.1.
Let ⃗u0 ∈ L2 ( √
1
x21 +x22
dx) with div ⃗u0 = 0 be an axisymmetric vector field. If
∥⃗u0 ∥L2 (w dx) < ϵν (where the constant ϵ > 0 does not depend on ν, nor on ⃗u0 ), then the
problem

⃗ u = ν∆⃗u − ∇p
⃗
∂t ⃗u + ⃗u.∇⃗
(10.19)
⃗u(0, .) = ⃗u0

div ⃗u = 0
2 √
has a unique axisymmetric solution ⃗u on (0, +∞) × R3 with ⃗u ∈ L∞
t L (
⃗ ⊗ ⃗u ∈ L2t L2 ( √
∇
1
x21 +x22
1
x21 +x22
dx) and
dx).
In order to describe axisymmetric flows, it is convenient to use cylindrical coordinates:
x1 = r cos θ, x2 = r sin θ and x3 = z. A scalar function A(x) is axisymmetrical if ∂θ A = 0.
⃗ (x) is axisymmetrical if ∂θ V
⃗ = ⃗ez ∧ V
⃗ . The swirl of the vector field is the
A vector field V
⃗ = Vr ⃗er + Vθ ⃗eθ + Vz ⃗ez .
component Vθ , where V
Vector calculus in cylindrical coordinates
In the open set Ω = {x ∈ R3 / (x1 , x2 ) ̸= (0, 0)}, we have the following formula for
⃗, W
⃗:
scalar functions A and vector fields V
Special Examples of Solutions
• ∂1 = cos θ ∂r − sin θ 1r ∂θ , ∂2 = sin θ ∂r + cos θ
⃗ = ⃗er ∂r + 1 ⃗eθ ∂θ + ⃗ez ∂z
∇
r
1
r ∂θ ,
263
∂3 = ∂z so that, formally,
⃗ = ∂r A ⃗er + 1 ∂θ A ⃗eθ + ∂z A ⃗ez
• ∇A
r
• ∆A = ∂r2 A + 1r ∂r A +
1 2
r 2 ∂θ A
+ ∂z2 A
⃗ = ∂r Vr + 1 Vr + 1 ∂θ Vθ + ∂z Vz
• div V
r
r
⃗ ⊗V
⃗ |2 = |∂r Vr |2 + |∂r Vθ |2 + |∂r Vz |2 + |∂z Vr |2 + |∂z Vθ |2 + |∂z Vz |2 + 12 (|∂θ Vr −
• |∇
r
2
Vθ | + |∂θ Vθ + Vr |2 + |∂θ Vz |2 )
⃗ ∧V
⃗ = ( 1 ∂θ Vz − ∂z Vθ )⃗er + (∂z Vr − ∂r Vz )⃗eθ + (∂r Vθ − 1 ∂θ Vr + 1 Vθ )⃗ez
• ∇
r
r
r
⃗ .∇)A
⃗
• (V
= Vr ∂r A + 1r Vθ ∂θ A + Vz ∂z A
⃗ .∇)
⃗ W
⃗ = (Vr ∂r Wr + 1 Vθ ∂θ Wr +Vz ∂z Wr − 1 Vθ Wθ )⃗er +(Vr ∂r Wθ + 1 Vθ ∂θ Wθ +
• (V
r
r
r
Vz ∂z Wθ + 1r Vθ Wr )⃗eθ + (Vr ∂r Wz + 1r Vθ ∂θ Wz + Vz ∂z Wz )⃗ez
⃗ = (∂r2 Vr + 1 ∂r Vr + 12 ∂ 2 Vr + ∂z2 Vr − 12 Vr − 2 12 ∂θ Vθ )⃗er + (∂r2 Vθ + 1 ∂r Vθ +
• ∆V
θ
r
r
r
r
r
1 2
1
2
eθ + (∂r2 Vz + 1r ∂r Vz + r12 ∂θ2 Vz + ∂z2 Vz )⃗ez
r 2 ∂θ Vθ + ∂z Vθ − r 2 Vθ )⃗
For axisymmetric functions or vector fields, the ∂θ terms vanish and we find:
⃗ = ∂r A ⃗er + ∂z A ⃗ez
ˆ ∇A
ˆ ∆A = ∂r2 A + 1r ∂r A + ∂z2 A
⃗ = ∂r Vr + 1 Vr + ∂z Vz
ˆ div V
r
⃗ ⊗V
⃗ |2 = |∂r Vr |2 + |∂r Vθ |2 + |∂r Vz |2 + |∂z Vr |2 + |∂z Vθ |2 + |∂z Vz |2 +
ˆ |∇
1
2
r 2 (|Vr |
+ |Vθ |2 )
⃗ ∧V
⃗ = −∂z Vθ ⃗er + (∂z Vr − ∂r Vz )⃗eθ + (∂r Vθ + 1 Vθ )⃗ez
ˆ ∇
r
⃗ .∇)A
⃗
ˆ (V
= Vr ∂r A + Vz ∂z A
⃗ .∇)
⃗ W
⃗ = (Vr ∂r Wr + Vz ∂z Wr − 1 Vθ Wθ )⃗er + (Vr ∂r Wθ + Vz ∂z Wθ + 1 Vθ Wr )⃗eθ +
ˆ (V
r
r
(Vr ∂r Wz + Vz ∂z Wz )⃗ez
⃗ = (∂r2 Vr + 1 ∂r Vr + ∂z2 Vr −
ˆ ∆V
r
1
2
ez
r ∂r Vz + ∂z Vz )⃗
1
er
r 2 Vr )⃗
+ (∂r2 Vθ + 1r ∂r Vθ + ∂z2 Vθ −
1
eθ
r 2 Vθ )⃗
+ (∂r2 Vz +
Thus, if we consider the axisymmetric solution ⃗u of the Navier–Stokes problem (with
axisymmetric forcing term f⃗ and axisymmetric pressure p), we find the following evolution
equation for the swirl uθ of ⃗u:
1
1
∂t uθ =ν(∂r2 uθ + ∂r uθ + ∂z2 uθ − 2 uθ ) + fθ
r
r
1
− (ur ∂r uθ + uz ∂z uθ + uθ ur )
r
(10.20)
Thus, if the force f⃗ has no swirl (fθ = 0) and the initial value ⃗u(0, .) has no swirl (uθ (0, .) =
0), then the solution ⃗u will still have no swirl:
uθ = 0
(10.21)
264
The Navier–Stokes Problem in the 21st Century (2nd edition)
and in this case the vorticity ω
⃗ is very simple:
ω
⃗ = ωθ ⃗eθ = (∂z ur − ∂r uz )⃗eθ .
(10.22)
2
2 3
We are going to consider a solution ⃗u that is locally in time L∞
t H ∩ Lt H . Under the
condition that ⃗u is divergence free and axisymmetrical without swirl, we find that:
ˆ the norm ∥⃗u∥Ḣ 1 is equivalent to ∥⃗
ω ∥2 = ∥ωθ ∥2
q
⃗ ⊗ω
ˆ the norm ∥⃗u∥Ḣ 2 is equivalent to ∥∇
⃗ ∥2 = ∥∂r ωθ ∥22 + ∥∂z ωθ ∥22 + ∥ 1r ωθ ∥22
ˆ the norm ∥⃗u∥Ḣ 3 is equivalent to ∥∆⃗
ω ∥2 = ∥∂r2 ωθ + ∂z2 ωθ + ∂r ( ωrθ )∥2
Moreover, we have:
ˆ (∂z2 ωθ )⃗eθ = ∂32 ω
⃗ , hence ∥∂z2 ωθ ∥2 ≤ ∥⃗
ω ∥Ḣ 2
ˆ (∂r2 ωθ )⃗eθ = (cos2 θ∂12 + sin2 θ∂22 + 2 cos θ sin θ∂1 ∂2 )⃗
ω hence ∥∂r2 ωθ ∥2 ≤ ∥⃗
ω ∥Ḣ 2
ˆ in particular, we get that ∥∂r ( ωrθ )∥2 ≤ 3∥⃗
ω ∥Ḣ 2
ˆ We find that
∥
ωθ
r1+γ
ωθ
r
belongs to L2x3 Hx11 ,x2 , hence for every γ ∈ (0, 1), we have
∥2 ≤ Cγ
ωθ
r
1−γ
L2x3 L2x1 ,x2
ωθ
r
γ
L2x3 Ḣx11 ,x2
≤ Cγ ∥⃗
ω ∥1−γ
∥⃗
ω ∥γḢ 2 .
Ḣ 1
(10.23)
In the case of axisymmetric flows with no swirl, Ladyzhenskaya [295], Uchovskii and
Yudovich [486] proved global existence under regularity assumptions on ⃗u0 and f⃗ but without any size requirements on the data. We shall follow the very simple proof proposed by
Leonardi, Malek, Nečas, and Pokorný [326]. (Abidi [1] proved global existence with very
1
weak (close to optimality) regularity requirements: ⃗u0 ∈ H 1/2 and f⃗ ∈ L2 H 4 +ϵ in Abidi’s
paper, while we assume ⃗u0 ∈ H 2 and f⃗ ∈ L2 H 1 . The regularity on ⃗u0 is unessential: if
⃗u0 belongs to H 1/2 and f⃗ ∈ L2 H s with s ≥ 1/2, then the mild solution ⃗u that belongs
to L∞ ((0, T ), H 1/2 ) for every T < T ∗ will actually belong to L∞ ((T0 , T ), H s+1 for every
0 < T0 < T < T ∗ ).
Global existence of axisymmetrical solutions without swirl
Theorem 10.4.
Let ⃗u0 ∈ (H 2 (R3 ))3 with div ⃗u0 = 0 be an axisymmetric vector field without swirl, and
let f⃗ ∈ L2 ((0, T ), (H 1 (R3 )3 ) be axisymmetric without swirl. Then the problem

⃗ u = ν∆⃗u + f⃗ − ∇p
⃗
∂t ⃗u + ⃗u.∇⃗
(10.24)
⃗u(0, .) = ⃗u0

div ⃗u = 0
2
2 3
has a unique global axisymmetric solution ⃗u on (0, T ) × R3 with ⃗u ∈ L∞
t H ∩ L H (if
T < +∞).
Special Examples of Solutions
265
Proof. Due to Theorem 7.3, we know that there exists a time T ∗ < T and a unique solution
⃗u that belongs to ∩0<T0 <T ∗ L∞ ((0, T0 ), H 2 ) ∩ L2 ((0, T0 ), H 3 ). This solution will be axisymmetric with no swirl. Moreover, if T ∗ is the maximal existence time, then, if T < T ∗ , we
have sup0<t<T ∗ ∥⃗u∥H 1 = +∞.
Due to div ⃗u = 0, we have
Z
⃗ u dx = 0
⃗u.(⃗u.∇)⃗
and
Z
⃗ =0
⃗u.∇p
so that the nonlinearity disappears in the energy balance and we find
Z
2
⃗ ⊗ ⃗u∥22 + ∥f⃗∥22 + ∥⃗u∥22
∂t ∥⃗u(t, .)∥2 = 2 (ν∆⃗u + f⃗).⃗u dx ≤ −2ν∥∇
and
∥⃗u(t, .)∥22
Z
+ 2ν
0
t
∥⃗u(s, .)∥2Ḣ 1
ds ≤ e
t
(∥⃗u0 ∥22
Z
+
t
∥f⃗(s, .)∥22 ds).
0
In the case of axisymmetric flows without swirl, we have another energy inequality. Let
ω
⃗ = curl ⃗u. We know that ω
⃗ is solution of:
⃗ u − (⃗u.∇)⃗
⃗ ω + curl f⃗.
∂t ω
⃗ = ν∆⃗
ω + (⃗
ω .∇)⃗
⃗ ⊗ω
ω
⃗ belongs, for any T0 < T ∗ , to L∞ ((0, T0 ), H 1 ) ∩ L2 ((0, T0 ), H 2 ). Thus, ∇
⃗ belongs to
∞
2
L ((0, T0 ), L ). We have ω
⃗ = ωθ ⃗eθ = (∂z ur − ∂r uz )⃗eθ and
|ωθ |2
|⃗
ω |2
2
2
=
|∂
ω
|
+
|∂
ω
|
+
.
r
θ
z
θ
r2
r2
R 2
Ladyzhenskaya’s key observation is that we have a uniform control of |⃗ωr2| dx. This is
based on the identity:
Z
dx
⃗ u − (⃗u.∇)⃗
⃗ ω ).⃗
((⃗
ω .∇)⃗
ω 2 = 0.
(10.25)
r
⃗ ⊗ω
|∇
⃗ |2 = |∂r ωθ |2 + |∂z ωθ |2 +
⃗ ⊗ω
Before proving this identity, we notice that the integral is well defined: we have ∇
⃗ ∈
⃗ ⊗ ⃗u
L∞ ((0, T0 ), L2 ) ∩ L2 ((0, T0 ), H 1 ), hence 1r ω
⃗ belongs to L∞ L2 ∩ L2 L6 ⊂ L4 L3 , while ∇
⃗ ⊗ ⃗u| 12 ; similarly, we
belongs to L2 ((0, T0 ), H 2 ) ⊂ L2 L3 , so that we may integrate |⃗
ω ||∇
r
1
1
2
6
2
3
∞
2
⃗ ⊗ω
have ∇
⃗ ∈ L L , rω
⃗ ∈ L L and r ⃗u ∈ L L .
In order to prove (10.25), we introduce a function α which is smooth on (0, +∞), equal
to 0 on (0,1) and to 1 on (2, +∞), and αϵ (r) = α(r/ϵ). We have
Z
Z
dx
αϵ (r) 1 2 ⃗
αϵ (r)
⃗ u − (⃗u.∇)⃗
⃗ ω ).⃗
⃗ u).⃗
((⃗
ω .∇)⃗
ω αϵ (r) 2 = ((⃗
ω .∇)⃗
ω 2 + |⃗
ω | (⃗u.∇)
dx
r
r
2
r2
Z
αϵ (r) 1
αϵ (r)
= ωθ2 ur 3 + ωθ2 ur ∂r
dx
r
2
r2
Z
1
dx
1
=
ωθ2 ur α′ (r/ϵ) 2
2
ϵ
r
with
|
1
2
Z
1
dx
ωθ2 ur α′ (r/ϵ) 2 | ≤ ∥α′ ∥∞
ϵ
r
Z
ϵ<r<2ϵ
|ωθ |2 |ur |
dx
.
r3
266
The Navier–Stokes Problem in the 21st Century (2nd edition)
R
∈ L4 L3 and urr ∈ L2 L3 , we have limϵ→0 ϵ<r<2ϵ |ωθ |2 |ur | dx
r 3 = 0, and (10.25) is
As ωrθ
proved.
R
We may now estimate
|⃗
ω |2
r2
dx. A direct proof would need
Z T0 Z
ωθ2
dx ds = 0
lim
4
ϵ→0 0
ϵ<r<2ϵ r
ωθ
r2
∈ L2 L2 , or at least
but we do not have such an estimate on ω
⃗ . A weaker property is that, for 0 < η < 1,
Z T0 Z
ω2
rη 4θ dx ds = 0.
lim
(10.26)
ϵ→0 0
r
ϵ<r<2ϵ
This is a consequence of rη/2 ωr2θ ∈ L2 L2 for 0 < η < 2 by inequality (10.23).
We thus follow [326] and replace in our computations the function αϵ by rη αϵ , with
0 < η < 1. We write
Z
Z
αϵ (r)
|⃗
ω |2 η
∂t
r αϵ (r)dx =2 ∂t ω
⃗ .⃗
ω rη 2 dx
r2
r
Z
Z
αϵ (r)
dx
=2 (ν∆⃗
ω + curl f⃗).⃗
ω rη 2 dx + ωθ2 ur ∂r (rη α(r/ϵ)) 2
r
r
with
Z
Z
rη αϵ (r)
rη αϵ (r)
⃗. curl(⃗
dx
=
f
ω
) dx
curl f⃗.⃗
ω
r2
r2
Z
rη αϵ (r)
rη αϵ (r)
rη αϵ (r)
+
f
∂
(ω
)
+
f
ω
) dx
= (−fr ∂z ωθ
r
r
θ
r
θ
r2
r2
r3
Z
fz
ωθ
fr
ωθ
1
= (− ∂z ( ) + ∂r ( ))rη αϵ (r) + fr ωθ 2 ∂r (rη α(r/ϵ)) dx
r
r
r
r
r
and
Z
∆⃗
ω .⃗
ω
rη αϵ (r)
dx =
r2
Z
ωθ
ωθ
1
ωθ
ωθ
( ∂z2 ( ) + ∂r ( ∂r ωθ ) − (∂r ( )2 )rη αϵ (r) dx
r
r
r
r
r
Z
ωθ 2
ωθ 2 η
= − ((∂r ( ) + (∂z ( ) ))r αϵ (r) dx
r
r
Z Z
ωθ
+ 2π
∂r ( ∂r ωθ )rη αϵ (r) dz dr
r
Z
ωθ 2
ωθ
= − ((∂r ( ) + (∂z ( )2 ))rη αϵ (r) dx
r
r
Z 2
1
dx
ωθ
∂r ( ∂r (rη αϵ (r)))
+
2
r
r
We find that:
Z
Z
Z Z ⃗2
|⃗
ω |2 η
|⃗
ω0 |2 η
1 t
|f | η
r αϵ (r)dx ≤
r αϵ (r)dx +
r αϵ (r) dx ds
r2
r2
ν 0
r2
Z tZ
ωθ
ωθ
−ν
((∂r ( )2 + (∂z ( )2 ))rη αϵ (r) dx ds
r
r
0
Z tZ
1
rη
−η
(ωθ2 ur + 2fr ωθ + ν(2 − η) ωθ2 ) 3 α(r/ϵ) dx ds
r
r
0
Z tZ
η
r
rη
+C
(ωθ2 |ur | + |fr ωθ |) 3 + ωθ2 4 dx ds
r
r
0
ϵ<r<2ϵ
Special Examples of Solutions
267
Letting ϵ go to 0, we find
Z
Z Z ⃗2
Z
|⃗
ω0 |2 η
1 t
|f | η
|⃗
ω |2 η
r
dx
≤
r
dx
+
r dx ds
r2
r2
ν 0
r2
Z tZ
ωθ
ωθ
((∂r ( )2 + (∂z ( )2 ))rη dx ds
−ν
r
r
0
Z tZ
1
rη
(ωθ2 ur + 2fr ωθ + ν(2 − η) ωθ2 ) 3 dx ds
−η
r
r
0
Z
Z tZ ⃗2
2
|⃗
ω0 | η
1
|f | η
≤
r dx + +
r dx ds
2
r
ν 0
r2
Z tZ
Z Z ⃗2
rη
η t
|f | η
ωθ2 ur 3 dx ds +
−η
r dx ds
r
ν
r2
0
0
Letting η go to 0, we find
Z
|⃗
ω |2
1
dx ≤ ∥⃗
ω0 ∥2Ḣ 1 +
r2
ν
Z
0
t
∥f⃗∥2Ḣ 1 ds
(10.27)
The end of the proof is now easy. We have
1/2
1/2
1/2
1/2
∥ur ∥∞ ≤ C∥⃗u∥Ḣ 1 ∥⃗u∥Ḣ 2 ≤ C ′ ∥⃗
ω ∥2 ∥⃗
ω ∥Ḣ 1
so that
∂t (∥⃗
ω ∥22 ) =2
Z
∂t ω
⃗ .⃗
ω dx
Z
⃗ u − ⃗u.∇⃗
⃗ ω ).⃗
(ν∆⃗
ω + curl f⃗ + ω
⃗ .∇⃗
ω dx
Z
Z
1
= − 2ν∥⃗
ω ∥2Ḣ 1 + 2 f⃗. curl ω
⃗ dx + 2 ur ωθ2 dx
r
ω
⃗
1
≤ − ν∥⃗
ω ∥2Ḣ 1 + ∥f⃗∥22 + ∥ur ∥∞ ∥⃗
ω ∥2 ∥ ∥2
ν
r
ω
⃗
1 ⃗ 2
C
4/3
ω ∥22 ∥ ∥2
≤ ∥f ∥2 + 1/3 ∥⃗
ν
r
ν
=2
We then conclude by Grönwall’s lemma and (10.27) that ∥⃗
ω ∥2 remains bounded.
Gallay and Šverák [203] considered another class of axisymmetric vector fields which lead
to global solutions. Their motivation was to investigate global existence in a space which
corresponds to the scale invariance of the Navier–Stokes equations (i.e. in a space E such
that, for ⃗u0 ∈ E, ∥λ⃗u0 (λx)∥E = ∥⃗u0 ∥E , such as the space Ḣ 1/2 considered by Abidi [1] or
the space L2 ( √ 21 2 dx) considered by Gallagher, Ibrahim and Majdoub [197]). The case
x1 +x2
1
x21 +x22
they considered concerns the vorticity and is ω
⃗ 0 ∈ L1 ( √
dx); their choice of measures
as initial vorticities aimed to help to understand the problem of vortex filaments.
Let us remark that, if ω
⃗ 0 ∈ L1 ( √ 21 2 dx) and is axisymmetric, then it belongs to the
x1 +x2
3
1
x21 +x22
Morrey space Ṁ 1, 2 : indeed, |⃗
ω0 |1/2 is axisymmetric and belongs to L2 ( √
saw that this implies that |⃗
ω0 |1/2 ∈ Ṁ 2,3 , and thus ω
⃗ 0 ∈ Ṁ
For ω
⃗ 0 = ωθ,0 (r, z)⃗eθ , the fact that ω
⃗ 0 ∈ L1 ( √ 21
dx), and we
1, 32
x1 +x22
.
dx) is equivalent to ωθ,0 ∈
L1 ((0, +∞) × R, dr dz). More generally, if ωθ,0 is a finite Borel measure f dµ on (0, +∞) × R
268
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ ∈ (D(R3 ))3
(with |f (r, z)| = 1 and µ a non-negative finite measure), then ω
⃗ 0 ∈ Ṁ 1,3/2 : if ψ
is supported in B(x0 , R) with x0 = (r0 cos θ0 , r0 sin θ0 , z0 ), then we may estimate
⃗ D′ ,D =
⟨⃗
ω (x) | ψ⟩
Z
Z
f (r, z)r(
2π
⃗ cos θ, r sin θ, z) · ⃗eθ dθ) dµ(r, z)
ψ(r
0
(0,+∞)×R
in the following way:
ˆ if r0 < 9R,
⃗ D′ ,D ≤(R + r0 )2π∥ψ∥
⃗ ∞ µ((0, +∞) × R)
⟨⃗
ω (x) | ψ⟩
⃗ ∞ µ((0, +∞) × R)
≤20πR∥ψ∥
ˆ if r0 > 9R and |x − x0 | < R, then
that
8r0
9
<r<
10r0
9 ,
|z − z0 | < R and |θ − θ0 | <
π 9R
2 8r0
so
⃗ D′ ,D ≤ 10 r0 9πR ∥ψ∥
⃗ ∞ µ((0, +∞) × R)
⟨⃗
ω (x) | ψ⟩
9 8r0
5π
⃗ ∞ µ((0, +∞) × R).
≤ R∥ψ∥
4
The Navier–Stokes equations in Morrey spaces have been studied for many years, with
contributions by Giga and Miyakawa [212], Kato [256] and Taylor [467]. In particular, we
have the following result (which is a variant of Theorem 8.3):
Proposition 10.3.
Let ω
⃗ 0 ∈ Ṁ 1,3/2 with div ω
⃗ 0 = 0.Let ⃗u0 be the solution of
−1
⃗ ∧ ⃗u0 = ω
∇
⃗ 0 , div ⃗u0 = 0, ⃗u0 ∈ Ḃ∞,∞
.
There exists a constant ϵν > 0 such that, if, for 0 < T ≤ +∞, we have
∥⃗
ω0 ∥3Ṁ 1,3/2 sup t∥Wνt ∗ ω
⃗ 0 ∥∞ < ϵν ,
0<t<T
then the Navier–Stokes equations
Z
⃗u = Wνt ∗ ⃗u0 −
t
Wν(t−s) ∗ P div(⃗u ⊗ ⃗u) ds
0
have a solution such that
√
• sup0<t<T t∥⃗u(t, .)∥∞ < +∞
⃗ ∧ ⃗u
⃗ =∇
• sup0<t<T ∥⃗
ω (t, .)∥Ṁ 1,3/2 < +∞, where ω
• sup0<t<T t∥⃗
ω (t, .)∥∞ < +∞.
Special Examples of Solutions
269
−2
Proof. First, we remark that Ṁ 1,3/2 is contained in Ḃ∞,∞
:
Z
Z
√
| Wt (x0 − y)f (y) dy| =| W (y)f (x − ty) dy|
≤∥f (x0 −
√
tx)∥M 1,3/2 (1 +
+∞
X
2n+1
sup
W (y))
2n <|y|≤2n+1
n=0
≤C∥f ∥Ṁ 1,3/2 t−1 .
If ω
⃗ 0 is a divergence-free vector field in Ṁ 1,3/2 , we may define
Z t
⃗ ∧ω
⃗u0 = lim
Wt ∗ ( ∇
⃗ 0 ) ds
t→+∞
since
0
+∞
Z
1
⃗ ∧ω
∥Wt ∗ (∇
⃗ 0 )∥∞ dt ≤ C∥⃗
ω0 ∥Ṁ 1,3/2
Z
+∞
1
dt
.
t3/2
We have
+∞
Z
⃗ ∧ (W(t+θ)/2 ∗ ω
∥W(t+θ)/2 ∗ ∇
⃗ 0 )∥∞
∥Wθ ∗ ⃗u0 ∥∞ ≤
0
+∞
Z
√
≤C
0
ω0 ∥Ṁ 1,3/2
∥⃗
ω0 ∥Ṁ 1,3/2
1 ∥⃗
√
dt = 2C
.
t+θ
t+θ
θ
−1
Thus, ⃗u0 ∈ Ḃ∞,∞
. We have div ⃗u0 = 0 and
⃗ ∧ ⃗u0 = −
∇
Z
+∞
Wt ∗ ∆⃗
ω0 dt = ω
⃗0
0
since
⃗ ∧ (∇⃗
⃗ ω0 ) = −∆⃗
⃗
∇
ω0 + ∇(div
ω
⃗ 0 ) = −∆⃗
ω0 .
We now solve the Navier–Stokes equations
∂t ⃗u = ν∆⃗u − P(⃗
ω ∧ ⃗u), ⃗u(0, .) = ⃗u0 , div ⃗u = 0
⃗ N (and Ω
⃗N = ∇
⃗ ∧U
⃗ N ) as
through Picard iterations: we define U
Z t
⃗ 0 = Wνt ∗ ⃗u0 and U
⃗ N +1 = U
⃗0 −
⃗N ∧U
⃗ N ) ds.
U
Wν(t−s) ∗ P(Ω
0
Taking the curl of
⃗ N +1 = ν∆U
⃗ N +1 − P(Ω
⃗N ∧U
⃗ N ),
∂t U
we find
⃗ N +1 = ν∆ΩN +1 + div(Ω
⃗N ⊗U
⃗N − U
⃗N ⊗ Ω
⃗ N)
∂t Ω
and thus
⃗ N +1 = Ω
⃗0 +
Ω
Z
t
⃗N ⊗U
⃗N − U
⃗N ⊗ Ω
⃗ N ) ds.
Wν(t−s) ∗ div(Ω
0
Thus, we study the bilinear operator
Z t
Z t
⃗ , Ω),
⃗ (V,
⃗ O))
⃗ = (− Wν(t−s) ∗ P(Ω
⃗ ∧V
⃗ ) ds, Wν(t−s) ∗ div(Ω
⃗ ⊗V
⃗ −U
⃗ ⊗ O)
⃗ ds).
B((U
0
0
270
The Navier–Stokes Problem in the 21st Century (2nd edition)
We will work with the norm
√
⃗ , Ω)∥
⃗ E = sup
∥(U
T
0<t<T
⃗ (t, .)∥∞ + t1/4 ∥Ω(t,
⃗ .)∥ 4/3,2 + t3/4 ∥Ω(t,
⃗ .)∥ 4,6 .
t∥U
Ṁ
Ṁ
We have
t
Z
⃗ ∧V
⃗ ) ds∥∞ ≤
Wν(t−s) ∗ P(Ω
∥
t
Z
0
⃗ ∧V
⃗ )∥∞ ds
∥Wν(t−s) ∗ P(Ω
0
Z
≤C
0
t
Z
≤C
0
1
ds
(ν(t − s))3/4 s3/4
t
1
⃗ ∧V
⃗ )∥ 4/3,2 ds
∥P(Ω
Ṁ
(ν(t − s))3/4
√
⃗ ∥∞ sup s1/4 ∥Ω∥
⃗ M 4/3,2
sup s∥V
0<s<t
′ −3/4
≤C ν
0<s<t
⃗ , Ω)∥
⃗ E ∥(V
⃗ , O)∥
⃗ E √1 .
∥(U
T
T
t
Similarly, we have
Z t
⃗ ⊗V
⃗ −U
⃗ ⊗ O)
⃗ ds∥ 4/3,2
∥
Wν(t−s) ∗ div(Ω
Ṁ
0
Z t
⃗ ⊗V
⃗ −U
⃗ ⊗ O)∥
⃗
≤
∥Wν(t−s) ∗ div(Ω
Ṁ 4/3,2 ds
0
Z t
1
⃗ ⊗V
⃗ ∥ 4/3,2 + ∥U
⃗ ⊗ O)∥
⃗
≤C
(∥Ω
Ṁ
Ṁ 4/3,2 ) ds
(ν(t
−
s))1/2
0
Z t
√
√
1
ds
⃗ ∥∞ s 14 ∥Ω∥
⃗
⃗ ∥∞ s 14 ∥O∥
⃗
4 ,2 +
4
≤C
sup ( s∥V
s∥U
)
1
3
3
M
M 3 ,2
2
4
0 (ν(t − s)) s 0<s<t
⃗ , Ω)∥
⃗ E ∥(V
⃗ , O)∥
⃗ E 1 .
≤C ′ ν −1/2 ∥(U
T
T 1/4
t
Finally, we have
Z t
⃗ ⊗V
⃗ −U
⃗ ⊗ O)
⃗ ds∥ 4,6
∥
Wν(t−s) ∗ div(Ω
Ṁ
0
Z t
⃗ ⊗V
⃗ −U
⃗ ⊗ O)∥
⃗
≤
∥Wν(t−s) ∗ div(Ω
Ṁ 4,6 ds
0
Z
≤C
t/2
1
⃗ ⊗V
⃗ ∥ 4/3,2 + ∥U
⃗ ⊗ O)∥
⃗
(∥Ω
Ṁ
Ṁ 4/3,2 ) ds
ν(t
− s)
0
Z t
1
⃗ ⊗V
⃗ ∥ 4,6 + ∥U
⃗ ⊗ O)∥
⃗
+C
(∥Ω
Ṁ
Ṁ 4,6 ) ds
(ν(t
−
s))1/2
t/2
t/2
√
√
1
ds
⃗ ∥∞ s 14 ∥Ω∥
⃗
⃗ ∥∞ s 14 ∥O∥
⃗
4 ,2 +
4
sup ( s∥V
s∥U
)
3
3
M
M 3 ,2
ν(t
−
s)
4
0<s<t
s
0
Z t
√
√
1
ds
⃗ ∥∞ s 34 ∥Ω∥
⃗ M 4,6 + s∥U
⃗ ∥∞ s 34 ∥O∥
⃗ M 4,6 )
+C
sup ( s∥V
1
5
2
4
t/2 (ν(t − s)) s 0<s<t
⃗ , Ω)∥
⃗ E ∥(V
⃗ , O)∥
⃗ E 1 .
≤C ′ (ν −1/2 + ν −1 )∥(U
T
T 3/4
t
Z
≤C
Thus, we have
⃗ , Ω),
⃗ (V
⃗ , O))∥
⃗ E ≤ Cν ∥(U
⃗ , Ω)∥
⃗ E ∥(V
⃗ , O)∥
⃗ E
∥B((U
T
T
T
Special Examples of Solutions
271
where Cν does not depend on T . Moreover, we have
3/4
∥Wνt ∗ ω
⃗ 0 ∥Ṁ 4/3,2 ≤∥Wνt ∗ ω
⃗ 0 ∥Ṁ 1,3/2 ∥Wνt ∗ ω
⃗ 0 ∥1/4
∞
1/4
− 14
3
≤t
∥⃗
ω0 ∥Ṁ 1,3/2 t∥Wνt ∗ ω
⃗ 0 ∥∞
,
∥Wνt ∗ ω
⃗ 0 ∥Ṁ 4,6 =∥Wνt/2 ∗ Wνt/2 ∗ ω
⃗ 0 ∥Ṁ 4,6
3/4
≤∥Wνt/2 ∗ ω
⃗ 0 ∥Ṁ 1,3/2 ∥Wνt ∗ ω
⃗ 0 ∥1/4
∞
1
≤C(νt)− 2 ∥Wνt/2 ∗ ω
⃗ 0 ∥Ṁ 4/3,2
1/4
3
1
ω0 ∥3Ṁ 1,3/2 t∥Wνt ∗ ω
⃗ 0 ∥∞
,
≤C √ t− 4 ∥⃗
ν
and
+∞
Z
⃗ ∧ (Wνt ∗ ω
Wτ ∗ ∇
⃗ 0 ) dτ ∥∞
∥Wνt ∗ ⃗u0 ∥∞ =∥
0
νt
Z
3
τ − 4 ∥Wνt ∗ ω
⃗ 0 ∥Ṁ 4,6 dτ
≤C
0
Z
+∞
νt
≤C
1
ν 1/4
5
τ − 4 ∥Wνt ∗ ω
⃗ 0 ∥Ṁ 4/3,2 dτ
+C
3
t− 4 ∥⃗
ω0 ∥3Ṁ 1,3/2 t∥Wνt ∗ ω
⃗ 0 ∥∞
1/4
.
Thus,
∥(Wνt ∗ ⃗u0 , Wνt ∗ ω
⃗ 0 )∥ET ≤ Cν sup
0<t<T
∥⃗
ω0 ∥3Ṁ 1,3/2 t∥Wνt ∗ ω
⃗ 0 ∥∞
1/4
and the Picard iterates converge if ∥⃗
ω0 ∥3Ṁ 1,3/2 sup0<t<T t∥Wνt ∗ ω
⃗ 0 ∥∞ is small enough.
It is easy to control the norm of the solution ω
⃗ in L∞ :
Z t
∥
Wν(t−s) ∗ div(⃗
ω ⊗ ⃗u − ⃗u ⊗ ω
⃗ ) ds∥∞
0
Z
≤C
≤C
t/2
1
∥⃗
ω ∥Ṁ 4/3,2 ∥⃗u)∥∞ ds
(ν(t − s))5/4
0
Z t
1
+C
∥⃗
ω ∥Ṁ 4,6 ∥⃗u∥∞ ds
3/4
t/2 (ν(t − s))
√
sup0<s<t s1/4 ∥⃗
ω ∥Ṁ 4/3,2
sup0<s<t s5/4 ∥⃗
ω ∥Ṁ 4,6
1
sup s∥⃗u(s, .)∥∞ (
+
).
t 0<s<t
ν 5/4
ν 3/4
For estimating the norm of ω
⃗ in Ṁ 1,3/2 , we define the following quantities: βN =
N
X
√
⃗N − Ω
⃗ N −1 ∥ 1,3/2 , BN =
⃗N − U
⃗ N −1 ∥∞ and η =
sup ∥Ω
βn , ϵN = sup
t∥U
Ṁ
0<t<T
0<t<T
n=0
√
⃗ N ∥∞ . We have
sup sup
t∥U
N ≥0 0<t<T
βN +1 ≤ Cν (ηβN + (BN + BN −1 )ϵN ).
This gives
B∞ ≤ B0 + Cν ηB∞ + 2Cν B∞
X
N ≥0
ϵN .
272
The Navier–Stokes Problem in the 21st Century (2nd edition)
If if ∥⃗
ω0 ∥3Ṁ 1,3/2 sup0<t<T t∥Wνt ∗ ω
⃗ 0 ∥∞ is small enough, then we have
Cν η ≤ Cν
X
ϵN ≤
N ≥0
1
4
and thus B∞ ≤ 41 B0 . Thus,
sup ∥⃗
ω (t, .)∥Ṁ 1,3/2 ≤ B∞ ≤ 4B0 ≤ 4∥⃗
ω0 ∥Ṁ 1,3/2 .
0<t<T
Corollary 10.2.
Let ω
⃗ 0 ∈ Ṁ 1,3/2 ∩ Ṁ 2,3 with div ω
⃗ 0 = 0.Let ⃗u0 be the solution of
−1
⃗ ∧ ⃗u0 = ω
∇
⃗ 0 , div ⃗u0 = 0, ⃗u0 ∈ Ḃ∞,∞
.
Then the Navier–Stokes equations
Z
⃗u = Wνt ∗ ⃗u0 −
t
Wν(t−s) ∗ P div(⃗u ⊗ ⃗u) ds
0
have a solution on (0, T ) × R3 with T ≥ C0 ∥⃗ω0 ∥3
1
Ṁ 1,3/2
∥⃗
ω0 ∥Ṁ 2,3
.
We may now state the result of Gallay and Šverák [203]:
Global existence of axisymmetrical solutions without swirl II
Theorem 10.5.
Let ⃗u0 with div ⃗u0 = 0 be an axisymmetric vector field without swirl, such that the vorticity ω
⃗ 0 = ωθ,0⃗eθ with ωθ,0 a finite Borel measure f dµ on (0, +∞)×R (with |f (r, z)| = 1
and µ a non-negative finite measure on (0, +∞) × R.
We assume that
lim ∥⃗
ω0 ∥3Ṁ 1,3/2 sup t∥Wνt ∗ ω
⃗ 0 ∥∞ < ϵν ,
T →0
0<t<T
where ϵν is the constant in Theorem 10.3. (This is the case when dµ is absolutely
continuous with respect to the Lebesgue measure).
Then the problem

⃗ u = ν∆⃗u − ∇p
⃗
∂t ⃗u + ⃗u.∇⃗
(10.28)
⃗u(0, .) = ⃗u0

div ⃗u = 0
has a global axisymmetric solution ⃗u on (0, +∞) × R3 with
ZZ
sup
|ωθ (t, r, z)| dr dz < +∞.
0<t
Proof.
Strategy of proof
Proposition 10.3 gives us a solution ⃗u on a small interval (0, T0 ) with sup0<t<T0 ∥⃗
ω (t, .)∥Ṁ 1,3/2 <
√
ω (t, .)∥∞ < +∞. In particular, sup0<t<T0 t∥⃗
+∞ and sup0<t<T0 t∥⃗
ω (t, .)∥Ṁ 2,3 < +∞. By
Special Examples of Solutions
273
Corollary 10.2, if T ∗ is the maximal time of existence and if T ∗ < +∞, we must have
lim supt→T ∗ ∥⃗
ω (t, .)∥Ṁ 1,3/2 = +∞ or lim supt→T ∗ ∥⃗
ω (t, .)∥Ṁ 2,3 = +∞. Thus, we shall prove
global existence by proving that, for every T < +∞, we have1
sup
T0 /2<t<inf(T,T ∗ )
∥⃗
ω (t, .)∥Ṁ 1,3/2 < ∞
(10.29)
∥⃗
ω (t, .)∥Ṁ 2,3 < ∞.
(10.30)
and
sup
T0 /2<t<inf(T,T ∗ )
Small times
First, we show that the local solution ⃗u given by Proposition 10.3 is axisymmetric
⃗ N are axisymmetric without swirl) and
without swirl (it is enough to check that the iterates U
∞
that ω
⃗ is given by ω
⃗ = ωθ (t, θ, r, z)⃗eθ with ωθ ∈ L ((0, T ), L1 ( dr dz)). As ω
⃗ is the pointwise
R
⃗ N , we shall prove that we have a uniform control of IN = |Ω
⃗ N|√ 1
dx. For
limit of Ω
2
2
x1 +x2
ωθ,0 = f dµ, we write
2π
Z
Z
|Wνt ∗ ω
⃗ 0| ≤
Wνt (x − (r cos θ, r sin θ, z)))|f (r, z)| r dθ dµ(r, z)
0
(0,+∞)×R
so that
Z 2πZ
Z
I0 (t) ≤C
0
≤C ′
Wνt (x − (r cos θ, r sin θ, z)) p
(
(0,+∞)×R
2π
Z
0
≤2πC
′
Z
√
(0,+∞)×R
dx
x21
+ x22
)|f (r, z)| r dθ dµ
1
|f (r, z)| r dθ dµ(r, z)
νt + r
Z
|f (r, z)| dµ(r, z).
(0,+∞)×R
Moreover, we have (writing again η = sup sup
√
⃗ N ∥∞ )
t∥U
N ≥0 0<t<T
Z tZ
⃗N ⊗U
⃗N − U
⃗N ⊗ Ω
⃗ N )| ds p dx
|Wν(t−s) ∗ div(Ω
x21 + x22
0
Z tZ
Z
ds
⃗ N (s, y)|( |∇W
⃗ ν(t−s) (x − y)| p dx
≤ I0 (t)+2η
|Ω
) dy √
2
2
s
x1 + x2
0
Z tZ
1
ds
⃗ N (s, y)| p 1
p
p
≤I0 (t) + Cη
|Ω
dy √
2
2
s
ν(t − s) ν(t − s) + y1 + y2
0
IN +1 (t) ≤I0 (t) +
C′
≤I0 (t) + √ η sup IN (s).
ν 0<s<t
For ϵν small enough, we have
C′
√
η
ν
< 12 , so that
Z
sup sup IN (t) ≤ 2 sup I0 (t) ≤ C
N ≥0 0<t<T
0<t<T
|f (r, z)| dµ(r, z).
(0,+∞)×R
1 Actually, Gallay and Šverák proved a global control with sup
ω (t, .)∥Ṁ 1,3/2
t>0 ∥⃗
√
supt>0 t∥⃗
ω (t, .)∥Ṁ 2,3 < +∞.
< +∞ and
274
The Navier–Stokes Problem in the 21st Century (2nd edition)
R
Decay of
|⃗
ω| √
1
x21 +x22
dx.
Now, we consider an interval of time (t0 , t1 ) with T0 /2 ≤ t0 ≤ t1 < T ∗ . ⃗u is bounded on
(t0 , t1 ) × R3 and
Z
1
sup
|⃗
ω| p 2
dx < +∞.
t0 <t<t1
x1 + x22
The
next step in Gallay and Šverák’s proof is to prove that the function t 7→
R
|⃗
ω (t, x)| √ 21 2 dx is non-increasing.
x1 +x2
We know that ⃗u is smooth on (t0 , t1 ) × R3 (and even analytic, see Theorem 9.12). We
consider a non-negative smooth function α which is compactly supported in (0, +∞) and
a non-negative smooth function β which is compactly supported in R, and we want to
estimate
Z
ZZ
1
Iα,β (t) = |⃗
ω (t, x)|α(r)β(z) dx =
|ωθ (t, r, z)|α(r)β(z) dr dz.
r
(0,+∞)×R
For t0 < τ0 < τ1 < t1 , we have
Iα,β (τ1 ) − Iα,β (τ0 )
Z p
p
1
( ωθ (τ1 , x)2 + ϵ − ωθ (τ0 , x)2 + ϵ)α(r)β(z) dx
r
ϵ→0+
Z Z τ1
p
1
= lim+
∂t ( ωθ (s, x)2 + ϵ)α(r)β(z) dx ds.
r
ϵ→0
τ0
= lim
We have
∂t ω
⃗ = (∂t ωθ )⃗eθ , hence ∂t ωθ = ∆ωθ −
1
ωθ − ∂r (ur ωθ ) − ∂z (uz ωθ )
r2
and thus
q
ωθ
∂t ( ωθ2 + ϵ) = p 2
∂ t ωθ
ωθ + ϵ
q
q
q
q
1
=∆( ωθ2 + ϵ) − ∂r (ur ωθ2 + ϵ) − ∂z (uz ωθ2 + ϵ) − 2 ωθ2 + ϵ
r
⃗ θ |2
√
√
1
1
1
|∇ω
+ ϵq
+ ϵq
(∂r ur + ∂z uz ) − ϵ
2
2
2
(ϵ + ωθ2 )3/2
ω r
ω
1 + ϵθ
1 + ϵθ
q
q
q
q
√ 1
1
≤ ∆( ωθ2 + ϵ) − 2 ωθ2 + ϵ − ∂r (ur ωθ2 + ϵ) − ∂z (uz ωθ2 + ϵ) + ϵ( 2 + |∂r ur + ∂z uz |).
r
r
We get
Z
p
1
∂t ( ωθ (s, x)2 + ϵ)α(r)β(z) dx
r
Z
ZZ
q
p
1
1
≤ ∆( ωθ2 + ϵ)α(r)β(z) dx − 2π
ωθ (s, r, z)2 + ϵ 2 α(r)β(z) dr dz
r
r
(0,+∞)×R
ZZ
p
+2π
ωθ (s, r, z)2 + ϵ(ur ∂r α(r)β(z) + α(r)uz ∂z β(z)) dr dz
(0,+∞)×R
ZZ
√
1
⃗ ⊗ ⃗u(s, .)∥∞ )
+2π 2ϵ(1 + ∥∇
(1 + 2 )α(r)β(z)) dr dz.
r
(0,+∞)×R
Special Examples of Solutions
275
Letting ϵ go to 0, we get
Iα,β (τ1 ) − Iα,β (τ0 )
Z τ1 Z
ZZ
1
1
≤
∆|ωθ | α(r)β(z) dx ds − 2π
|ωθ | 2 α(r)β(z) dr dz ds
r
r
τ0
(0,+∞)×R
Z τ1 Z Z
+2π
|ωθ |(ur ∂r α(r)β(z) + α(r)uz ∂z β(z)) dr dz ds
τ0
Z
(0,+∞)×R
τ1
ZZ
(∆|ωθ | − |ωθ |
=2π
Z
τ0
τ1 Z Z
(0,+∞)×R
1
)α(r)β(z) dr dz ds
r2
|ωθ |(ur ∂r α(r)β(z) + α(r)uz ∂z β(z)) dr dz ds
+2π
τ0
(0,+∞)×R
and thus
1
(Iα,β (τ1 ) − Iα,β (τ0 )) ≤
2π
τ1
ZZ
τ0
τ1
ZZ
Z
Z
1
1
(∂r2 + ∂z2 + ∂r − 2 )|ωθ | α(r)β(z) dr dz ds
r
r
(0,+∞)×R
|ωθ |(ur ∂r α(r)β(z) + α(r)uz ∂z β(z)) dr dz ds
+
τ0
(0,+∞)×R
τ1
ZZ
τ0
τ1
ZZ
τ0
τ1
ZZ
Z
=−
∂r |ωθ | ∂r α(r)β(z) dr dz ds
Z
(0,+∞)×R
−
Z
(0,+∞)×R
|ωθ | α(r)∂z2 β(z) dr dz ds
+
τ0
Z
τ1
1
|ωθ | ∂r α(r)β(z) dr dz ds
r
(0,+∞)×R
ZZ
|ωθ |ur ∂r α(r)β(z) dr dz ds
+
τ0
Z τ1
(0,+∞)×R
ZZ
|ωθ |α(r)uz ∂z β(z) dr dz ds
+
τ0
(0,+∞)×R
=Aα,β + Bα,β + Cα,β + Dα,β + Eα,β .
Let γ ∈ D(R) be an even function, radially non-increasing and equal to 1 on [−1, 1] and
to 0 on (2, +∞). For R > 1, we define the function αR (r) = γ( Rr ) − γ(Rr). We shall write
limα→1 meaning limR→+∞ . We have obviously
Z
1
lim Iα,β (τ ) = Iβ (τ ) = |⃗
ω (τ, x)|β(z) dx
α→1
r
Z τ1 Z Z
lim Cα,β = Cβ =
|ωθ | ∂z2 β(z) dr dz ds
α→1
τ0
Z
τ1
(0,+∞)×R
ZZ
|ωθ | uz ∂z β(z) dr dz ds
lim Eα,β = Eβ =
α→1
τ0
(0,+∞)×R
Next, we remark that γ ′ ≤ 0 so that ∂r α ≥
1 ′ r
Rγ (R)
≥ − R1 ∥γ ′ ∥∞ 1r>R while
2
1
|∂r α| ≤ ∥γ ′ ∥∞ ( 1r< R2 + 1r>R ).
r
R
This gives
lim sup Bα,β ≤ 0
α→1
276
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
lim sup Dα,β ≤ 2∥γ ′ ∥∞ lim sup
α→1
As
τ1
Z
α→1
ZZ
|ωθ |
2
(0, R
)×R
τ0
|ur |
β(z) dr dz ds.
r
|ur |
r
⃗ ⊗ ⃗u∥∞ , we find that lim supα→1 Dα,β ≤ 0.
≤ ∥∇
We now control Aα,β . We have
Aα,β = Fα,β + Gα,β
with
τ1
Z
ZZ
|ωθ |
Fα,β =
τ0
and
Z
(0,+∞)×R
τ1
ZZ
|ωθ | R2 γ ′′ (Rr)β(z) dr dz ds.
Gα,β = −
τ0
1 ′′ r
γ ( )β(z) dr dz ds
R2
R
(0,+∞)×R
We have
lim Fα,β = 0.
α→1
In order to estimate Gα,β , we introduce for η > 0
Z τ1 Z Z
p
Gη,α,β = −
|ωθ |2 + ηr2 R2 γ ′′ (Rr)β(z) dr dz ds.
τ0
(0,+∞)×R
We have
Z
τ1
√
ZZ
|Gα,β − Gη,α,β | ≤
τ0
and thus
lim sup Gα,β ≤ lim sup Gη,α,β +
α→1
ηr R2 |γ ′′ (Rr)|β(z) dr dz ds
(0,+∞)×R
√
Z
η(τ1 − τ0 )(
r|γ ′′ (r)| dr)∥β∥1 .
α→1
We write
Z
τ1
ZZ
p
∂r ( |ωθ |2 + ηr2 ) Rγ ′ (Rr)β(z) dr dz ds
Gη,α,β =
τ0
=−
1
2
Z
(0,+∞)×R
τ1 Z Z
τ0
(0,+∞)×R
r∂r |ωθ2 | + 2ηr2
p
R|γ ′ (Rr)|β(z) dr dz ds.
r |ωθ |2 + ηr2
We have
r∂r |ωθ2 | + 2ϵr2 =x1 ∂1 |⃗
ω |2 + x2 ∂2 |⃗
ω |2 + 2ϵ(x21 + x22 ).
The function ω
⃗ is smooth on (τ0 , τ1 ) × R3 and its derivatives are bounded. Moreover, it
⃗ ⊗ω
vanishes for x1 = x2 = 0, as |⃗
ω | ≤ r|∇
⃗ |. Thus, |⃗
ω |2 has a minimum at x when
2
x1 = x2 = 0, so that the derivatives ∂1 |ωθ | (τ, 0, 0, x3 ) and ∂2 |ωθ |2 (τ, 0, 0, x3 ) are equal to
0 and the quadratic form on R2
Q(u, v) =u1 v1 ∂12 |ωθ |2 (τ, 0, 0, x3 ) + u2 v2 ∂22 |ωθ |2 (τ, 0, 0, x3 )
+ (u1 v2 + u2 v1 )∂1 ∂2 |ωθ |2 (τ, 0, 0, x3 )
Special Examples of Solutions
277
is non-negative: Q(u, u) ≥ 0. We then write
x1 ∂1 |⃗
ω |2 + x2 ∂2 |⃗
ω |2
=x21 ∂12 |ωθ |2 (τ, 0, 0, x3 ) + x22 ∂22 |ωθ |2 (τ, 0, 0, x3 ) + 2x1 x2 ∂1 ∂2 |ωθ |2 (τ, 0, 0, x3 )
X Z 1
+ x1
∂1 ∂i ∂j |ωθ |2 (τ, λx1 , λx2 , x3 )(1 − λ)xi xj dλ
0
1≤i,j≤2
1
X Z
+ x2
∂2 ∂i ∂j |ωθ |2 (τ, λx1 , λx2 , x3 )(1 − λ)xi xj dλ
0
1≤i,j≤2
≥ − 4∥⃗
ω ∥W 3,∞ r3 .
For R > η4 supτ0 <t<τ1 ∥⃗
ω ∥W 3,∞ , we find Gη,α,β ≤ 0 and thus lim supα→1 Gη,α,β ≤ 0.
We then conclude by letting β go to 1 (β = γ(z/R) and R → +∞).
Thus, we proved that
ZZ
ZZ
sup
|⃗
ω (t, r, z)| dr dz ≤
|⃗
ω (t0 , r, z)| dr dz.
T0 /2<t<T ∗
As ∥⃗
ω ∥Ṁ 1,3/2 ≤ C
Control of
Let η =
we have
RR
|⃗
ω (t, r, z)| dr dz, we proved inequality (10.29).
ωθ
r .
ωθ
r .
We have just proved that, on the maximal time interval of existence (0, T ∗ )
sup ∥η(t, .)∥L1 (R3 )) = lim inf ∥η(T, .)∥L1 (R3 )) ;
0<t<T ∗
t→0
if ωθ,0 ∈ L1 ((0, +∞) × R, dr dz), we have
lim inf ∥η(t, .)∥L1 (R3 ),dx) = ∥η0 ∥L1 (R3 ),dx) = ∥ωθ,0 ∥L1 (dr dz) .
t→0
Recall that the proof of global existence for axisymmetric vector fields in H 1 relied on
inequality (10.27), i.e. on the control of η(t, .) in L2 (dx). Here, we shall prove a similar
control on ∥η(t, .)∥L2 (dx) , when T0 /2 < t < T ∗ .
We want to estimate, for T0 /2 ≤ τ0 ≤ t ≤ τ1 < T ∗ ,
Z
1
I(t) = |⃗
ω (t, x)|2 2 dx.
r
⃗ ⊗ω
A first remark is that |η(t, x)| ≤ |∇
⃗ (t, x)|, so that η is bounded on [τ0 , τ1 ] × R3 , and,
1
since we control the L norm of η, I(t) < +∞. A second remark is that |⃗
ω |2 is smooth and
axisymmetrix; in particular, looking at the Taylor polynomial of order 4 for (x, y) close to
(0, 0) and writing
x+y x−y
|⃗
ω (x, y, z)|2 = |⃗
ω (x, −y, z)|2 = |⃗
ω (−x, y, z)|2 = |⃗
ω (y, x, z)|2 = |⃗
ω ( √ , √ , z)|2 ,
2
2
one gets (since ω
⃗ (t, 0, 0, z) = 0)
|⃗
ω (t, x, y, z)|2 =
r2 2
r4
∂1 (|⃗
ω |2 )(t, 0, 0, z) + ∂14 (|⃗
ω |2 )(t, 0, 0, z) + r6 ϵ(t, x, y, z)
2!
4!
where ϵ(t, x, y, z) is bounded on [τ0 , τ1 ] × R3 .
278
The Navier–Stokes Problem in the 21st Century (2nd edition)
A third remark is that we have the identity
1
1
1
⃗ω−ω
⃗ u) · ω
ω
⃗ · ∂t ω
⃗ = 2 ν⃗
ω · ∆⃗
ω − 2 (⃗u · ∇⃗
⃗ · ∇⃗
⃗
r2
r
r
2
1
1
1
⃗ ωθ ) + ur ωθ2
= 2 νωθ ∆ωθ − 4 νωθ2 − 2 ⃗u · ∇(
r
r
r
2
r3
2
ωθ
ωθ
ωθ
ωθ
⃗ ωθ )
=ν ∆( ) + 2ν 2 ∂r ( ) − ⃗u · ∇(
r
r
r
r
2r2
or equivalently
ν
⃗ 2 ).
∂t (η 2 ) = 2νη∆η + 2 ∂r (η 2 ) − ⃗u · ∇(η
r
(10.31)
As η(t, .) ∈ L1 ∩ L∞ , we have, for ϕR = ϕ(x/R), where ϕ ∈ D is radial, non-negative and is
equal to 1 on the ball B(0, 1),
Z
Z 2
Z
η
⃗ 2 dx;
η ∆η ϕR dx =
∆(ϕR ) dx − |∇η|
2
the first term is O( R12 ) and the second one is non-positive. Moreover, we have
Z
ZZ
1
2
∂r (η )ϕR dx =
∂r (η 2 )ϕR (r, 0, z) dr dz
r
(0,+∞)×R
Z
Z
= − η 2 (t, 0, 0, z)ϕR (0, 0, z) dz − η 2 r∂r ϕR dx.
R
The first term is non-positive and the second one is O( |x|>R |η| dx) = o(1). On the other
hand, as ⃗u is bounded on [τ0 , τ1 ] × R3 and divergence-free, we find that
Z
Z
⃗ 2 ) dx = − η 2 ⃗u · ∇ϕ
⃗ R dx = O( 1 ).
ϕR ⃗u · ∇(η
R
R
d
Thus, we easily check that I is non-increasing, as dt
( η 2 (t, x)ϕR (x) dx) ≤ o(1). We finally
get
sup
∥η(t, .)∥2 ≤ ∥η(T0 , .)∥2 .
T0 /2≤t<T ∗
We control ∥η∥4 in a similar way. We write
Z
Z
d
η(t, x)4 ϕR dx = 2 η 2 ∂t (η 2 ) ϕR dx.
dt
Using identity (10.31), we get
ν
⃗ 2)
η 2 ∂t (η 2 ) =2νη 3 ∆η + 2 η 2 ∂r (η 2 ) − η 2 ⃗u · ∇(η
r
⃗ 4 ).
⃗ 2 + ν ∂r (η 4 ) − 1 ⃗u · ∇(η
=νη 2 ∆(η 2 ) − 2νη 2 |∇η|
r
2
R
d
this gives that dt
( η 4 (t, x)ϕR (x) dx) ≤ o(1), and thus
sup
T0 /2≤t<T ∗
∥η(t, .)∥4 ≤ ∥η(T0 , .)∥4 .
Special Examples of Solutions
279
Control of ur (t, x).
Recall that
⃗ ∧ω
⃗u = G ∗ ∇
⃗ with G =
1
.
4π|x|
We have
1
⃗ ∧ω
∇
⃗ = −∂z ωθ ⃗er + (∂r ωθ + ωθ )⃗ez .
r
Thus, we have, for y = (ρ cos σ, ρ sin σ, w) and x = (r cos θ, r sin θ, z),
Z
ur (t, r, z) =⃗u · ⃗er = ∂z G(r⃗er − y, z − w)ωθ (t, ρ, w) ⃗er · ⃗eρ dy
We split the domain of integration in (ρ, w) ∈ ∆1 and (ρ, w) ∈ ∆2 , where
∆1 = {(ρ, w) ∈ (0, +∞) × R/ ρ ≤ 2r}.
We thus have
ur = A(t, x) + B(t, x)
with
(w − z)
ωθ (t, y) dy
|y − x|3
Z
1
A(t, x) =
4π
(ρ,w)∈∆1
and
ZZ
B(t, x) =
(ρ,w)∈∆2
w−z
4π
2π
Z
0
⃗er · ⃗eρ
dσ ωθ (ρ, w)ρ dρ dw
(|r⃗er − ρ⃗eρ |2 + (z − w)2 )3/2
ZZ
=
K(r, z, ρ, w)ωθ (ρ, w) dρ dw
(ρ,w)∈∆2
with
K(r, z, ρ, w) =
ρ(w − z)
4π
Z
2
=
2π
(r2
0
3rρ (w − z)
4π
Z
0
+
ρ2
cos γ
dγ
− 2rρ cos γ + (z − w)2 )3/2
2π
(r2
+
ρ2
sin2 γ
dγ.
− 2rρ cos γ + (z − w)2 )5/2
We have
|A(t, x)| ≤
1
1
∥
∥ 3/2,∞ ∥1(ρ,w)∈∆1 ωθ ∥L3,1
4π |y|2 L
1/9
8/9
≤C∥1(ρ,w)∈∆1 ωθ ∥L1 (R3 ) ∥1(ρ,w)∈∆1 ωθ ∥L4 (R3 )
1/9
8/9
≤2Cr∥η(t, .)∥1 ∥η(t, .)∥4 .
On the other hand, we have, if (ρ, w) ∈ ∆2 , |r⃗er − ρ⃗eρ | ≥ ρ/2 and thus
K(r, z, ρ, w) ≤ 24 r
1
ρ2 + (z − w)2
and
Z
|B(t, x)| ≤24 r
(ρ,w)∈∆2
ρ2
≤24 r∥1∆2 (ρ, w)
1
|ωθ (ρ, w)| dρ dw
+ (z − w)2
ρ1/3
∥L3 (dρ dw) ∥ρ−1/3 ωθ (t, ρ, w)∥L3/2 (dρ dw)
ρ2 + (z − w)2
=C∥η(t, .)∥L3/2 (dx)
We thus get
|ur (t, x)| ≤ C(∥η(t, .)∥1 + ∥η(t, .)∥4 )(r + 1).
280
The Navier–Stokes Problem in the 21st Century (2nd edition)
Control of ∥⃗
ω (t, .)∥Ṁ 2,3
R
We know that ∥⃗
ω (t, .)∥Ṁ 2,3 ≤ C∥ω(t, .)∥L2 ( dx ) , thus we will try and control J(t) =
r
|⃗
ω |2 dx
.
For
T
/2
≤
τ0 ≤ τ1 < T ∗ , ω
⃗ is bounded on (τ0 , τ1 ) × R3 so that
0
r
J(t) ≤ ∥⃗
ω (t, .)∥∞ ∥η(t, .)∥1 .
We have
d
dt
Z
|⃗
ω | 2 ϕR
dx
=
r
Z
∂t (η(t, x)2 )ϕR r dx.
with
ν
⃗ 2 ))
r∂t (η(t, x)2 ) =r(2νη∆η + 2 ∂r (η 2 ) − ⃗u · ∇(η
r
2
⃗ 2 + 2ν∂r (η 2 ) − ⃗u · ∇(rη
⃗
=νr∆(η 2 ) − 2νr|∇η|
) + ur η 2
2
⃗ 2 + 2ν∂r (η 2 ) − ⃗u · ∇(rη
⃗
= νr∂r2 (η 2 ) + νr∂z2 (η 2 ) + 3ν∂r (η 2 ) − 2νr|∇η|
) + ur η 2
2ν 2
ν
2
⃗ 2 − ⃗u · ∇(rη
⃗
η − 2νr|∇η|
) + ur η 2
= (∂r2 (ωθ2 ) + ∂z2 (ωθ )2 ) − ν∂r (η 2 ) −
r
r
2
2
ν
ν
ω2
ω2
⃗ ωθ )|2 − ⃗u · ∇(
⃗ ωθ ) + ur ωθ .
= (∂r2 (ωθ2 ) + ∂z2 (ωθ )2 ) − ∂r ( θ ) − ν 3θ − 2νr|∇(
2
r
r
r
r
r
r
r
This gives
Z
ZZ
1
d
dx
|⃗
ω | 2 ϕR
=2πν
ωθ2 (∂r2 ϕR + ∂r2 ϕR + ∂r ϕR ) dr dz
dt
r
r
r>0
Z
Z 2
Z
ωθ2
ω
dx
ω
ω2
θ
⃗ θ )|2 )
− ν( 2 + 2r2 |∇(
+
ur ∂r ϕR dx + ur 2θ ϕR dx.
r
r
r
r
r
As |ur | ≤ C0 (1 + r) on (T0 /2, T ∗ ), we obtain
Z
t
Z
J(t) ≤J(T0 /2) +
ωθ (s, x)2
dx ds
r2
Z t
J(s) ds + C0
∥η(s, .)∥22 ds.
ur
T0 /2
Z
t
≤J(T0 /2) + C0
T0 /2
T0 /2
We then conclude with the Grönwall lemma.
The case of axisymmetric flows with swirls has been studied by many authors. Regularity
criteria have been given by Chen, Fang and T. Zhang in 2017 [114]; those criteria were used
by Lei and Q. Zhang [307] to prove existence when the swirl component is small enough:
Theorem 10.6.
Let ⃗u0 ∈ H 1/2 with div ⃗u0 = 0 be an axisymmetric vector field, with vorticity ω
⃗0 =
u2θ,0
ω
θ,0
2
2
2
∞
⃗ ∧ ⃗u0 , such that
∇
r ∈ L , r ∈ L and ruθ,0 ∈ L ∩ L . Then there is a constant
ϵν (which does not depend on ⃗u0 ) such that, if
(∥
u2θ,0
ωθ,0
∥2 + ∥
∥2 )∥ruθ,0 ∥2 ∥ruθ,0 ∥∞ < ϵ0 ,
r
r
Special Examples of Solutions
281
then the problem

⃗ u = ν∆⃗u − ∇p
⃗
∂t ⃗u + ⃗u.∇⃗
⃗u(0, .) = ⃗u0

div ⃗u = 0
(10.32)
has a global regular axisymmetric solution ⃗u on (0, +∞) × R3 .
10.4
Helical Solutions
In this section, we consider the Navier–Stokes problem with the following symmetry property: ⃗u is invariant under the action of a one-parameter group of screw motions
Rθ (x1 , x2 , x3 ) = (x1 cos θ − x2 sin θ, x1 sin θ + x2 cos θ, x3 + αθ) (where α ̸= 0 is fixed): this
is the case of helical symmetry.
In cylindrical cordinates, we find that ⃗u = ur ⃗er + uθ ⃗eθ + uz ⃗ez , where ur , uθ and uz
depend only on r and η = θ cos γ + z sin γ with tan γ = −1/α. The case γ = π/2 would
correspond to axisymmetrical solutions, the case γ = 0 to two-and-a-half dimensional flows.
For helical symmetry, we consider γ ∈ (−π/2, 0) ∪ (0, π/2).
It will be more convenient to define ξ = θ − z/α = η/ cos γ. A scalar function A(x) with
helical symmetry may be written as A(x) = B(r, ξ), where B is 2π-periodical in ξ. If Γ is a
cylindrical domain of R3 of the form Γ = {x ∈ R3 / z ∈ I, r ∈ J} for an interval I ⊂ R and
an interval J ⊂ (0, +∞), we find
Z
ZZ
Z Z 2π
2
2
|A(x)| dx = |I|
|A(x1 , x2 , 0)| dx1 dx2 = |I|
|B(r, θ)|2 r dr dθ.
Γ
r∈J
J
0
Thus, we can see that we have the equivalence for a flow ⃗u0 with helical symmetry:
⃗u0 ∈ Ṁ 2,3 ⇔ ⃗u0 (x1 , x2 , 0) ∈ L2 (R2 )
and the condition limt→0 t1/2 ∥Wνt ∗ ⃗u0 ∥∞ = 0 is automatically fulfilled, as L∞ ∩ L2 is dense
in L2 . This situation is very similar to the case of two-and-a-half dimensional flows.
Helical flows have been studied by Mahalov, Titi, and Leibovich [347]. We have the
following result:
Global existence of helical symmetrical solutions
Theorem 10.7.
Let ⃗u0 ∈ Ṁ 2,3 with div ⃗u0 = 0 be a vector field with helical symmetry, and let f⃗ ∈
L1 ((0, T ), , Ṁ 2,3 ) with helical symmetry. Then the problem

⃗ u = ν∆⃗u + f⃗ − ∇p
⃗
∂t ⃗u + ⃗u.∇⃗
(10.33)
⃗u(0, .) = ⃗u0

div ⃗u = 0
2,3
⃗ ⊗ ⃗u ∈
has a unique global helical solution ⃗u on (0, T ) × R3 with ⃗u ∈ L∞
and ∇
t Ṁ
2
2,3
Lt Ṁ .
282
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proof. First, we consider ⃗u1 = Wνt ∗ ⃗u0 . If ⃗u0 is helical, so is ⃗u1 . If ⃗u0 belongs to Ṁ 2,3 , then
2,3
⃗u1 belongs to L∞
. Moreover, ⃗u0 ∈ L2x1 ,x2 L2per,x3 and thus ⃗u1 will satisfy
t Ṁ
Z
Z
Z
2
⃗ ⊗ ⃗u1 |2 dx
∂t
|⃗u1 (t, x)| dx = 2ν
⃗u1 .∆⃗u1 dx = −2ν
|∇
0<x3 <2πα
0<x3 <2πα
0<x3 <2πα
2
⃗ ⊗ ⃗u1 belongs to L2 L2
⃗ u1 | is helical, we find that ∇
⃗ ⊗ ⃗u1 belongs
Thus, ∇
x1 ,x2 Lper,x3 ; as |∇ ⊗ ⃗
to L2t Ṁ 2,3 .
Rt
We consider now ⃗u2 = 0 Wν(t−s) ∗Pf⃗(s, .) ds. As f⃗ is helical, Pf⃗ is helical. Thus, writing
Z t
∥⃗u2 (t, .)∥Ṁ 2,3 ≤
∥Wν(t−s) ∗ Pf⃗(s, .)∥Ṁ 2,3 ds,
0
2,3
we find that ⃗u2 belongs to L∞
. Moreover,
t Ṁ
Z T
⃗
∥∇ ⊗ ⃗u2 ∥L2 Ṁ 2,3 ≤
∥1t>s Wν(t−s) ∗ Pf⃗(s, .)∥L2 Ṁ 2,3 ds
0
⃗ ⊗ ⃗u2 ∈ L2 Ṁ 2,3 .
so that ∇
⃗ ∈ Ṁ 2,3 , we find that v belongs to
Now, if v is a function such that v ∈ Ṁ 2,3 and ∇v
Ṁ 4,6 : this is easily seen with Hedberg’s inequality. Indeed, we write
Z +∞
Z ∞
M∇v(x)
⃗
∥v∥Ṁ 2,3
|v(x)| = |
∆Wt∆ ∗ v(x) dt| ≤C
min( 1/2 ,
) dt
t
t3/2
0
0
q
q
=C ′ ∥v∥Ṁ 2,3 M∇v(x)
.
⃗
⃗ ⊗ ⃗u and ∇
⃗ ⊗ ⃗v in L2 Ṁ 2,3 that
In particular, we obtain, for ⃗u and ⃗v in L∞ Ṁ 2,3 with ∇
1
4/3
4/3,2
⃗
⃗v =
⃗u.∇⃗v belongs to L Ṁ
. Moreover, if ⃗u is divergence free, we have that √−∆
⃗u · ∇⃗
√1
−∆
div(⃗u ⊗ ⃗v ) belongs to L2 Ṁ 2,3 .
Rt
The next step is to consider ⃗u3 = 0 Wν(t−s) ∗ P⃗g (s, .) ds for a helical ⃗g ∈ L4/3 Ṁ 4/3,2
1
such that √−∆
⃗g belongs to L2 Ṁ 2,3 . We have
⃗ ⊗ ⃗u3 ∥ 2,3 ≤
∥∇
Ṁ
t
Z
⃗ ⊗ Wν(t−s) ∗ P⃗g (s, .)∥ 2,3 ds
∥∇
Ṁ
0
Z
≤C
0
t
1
∥⃗g (s, .)∥Ṁ 4/3,2 ds.
(t − s)3/4
⃗ ⊗ ⃗u3 is controlled in L2 Ṁ 2,3 :
so that ∇
⃗ ⊗ ⃗u3 ∥ 2 2,3 ≤ C∥⃗g ∥ 4/3 4/3,2 .
∥∇
L Ṁ
L
Ṁ
In order to estimate ⃗u3 in Ṁ 2,3 , we shall use the helicity of ⃗u3 , and thus just try and
estimate
⃗u3 (x1 , x2 , 0) in (L2 (R2 ))3 . Thus, we consider ⃗v0 in (L2 (R2 ))3 and compute I =
RR
⃗u3 (t, x1 , x2 , 0)).⃗v0 (x1 , x2 ) dx1 dx2 . We write ⃗v0 = vr′ (r, θ)⃗er + vθ (r, θ)⃗eθ + vz (r, θ)⃗ez , and
consider the helical extension of ⃗v0 : ⃗v (r, θ, z) = vr′ (r, θ − z/α)⃗er + vθ (r, θ − z/α)⃗eθ + vz (r, θ −
z/α)⃗ez . We have
Z
1
I=
⃗u3 · ⃗v dx
2πα 0<x3 <2πα
Z tZ
√
1
1
=
( −∆Wν(t−s) ∗ ⃗v ) · √
P⃗g (s, .) dx ds
2πα 0 0<x3 <2πα
−∆
Special Examples of Solutions
283
so that
⃗ ⊗ Wν(t−s) ∗ ⃗v ∥ 2 2,3 ∥ √ 1 ⃗g ∥ 2 2,3
|I| ≤C∥∇
L Ṁ
−∆ L Ṁ
1
≤C ′ ∥⃗v ∥Ṁ 2,3 ∥ √
⃗g ∥ 2 2,3
−∆ L Ṁ
1
=C ′ ∥⃗v0 ∥L2 (R2 ) ∥ √
⃗g ∥ 2 2,3 .
−∆ L Ṁ
and we get
1
⃗g ∥ 2 2,3 .
−∆ L Ṁ
Thus, if we look at the Picard iterates in the space
∥⃗u3 ∥L∞ Ṁ 2,3 ≤ C∥ √
⃗ ⊗ ⃗u ∈ L2 ((0, T0 ), Ṁ 2,3 )}
YT0 = {⃗u is helical / div ⃗u = 0, ⃗u ∈ L4 ((0, T0 ), Ṁ 4,6 ), ∇
we shall find convergence
to a mild solution as soon as T0 is small enough to grant that
R
⃗ 0 = Wνt ⃗u0 + t Wν(t−s) ∗ Pf⃗(s, .) ds satisfies for some constant ϵν (which depends only on
U
0
ν):
⃗ 0∥ 4
⃗
⃗
∥U
L ((0,T0 ),Ṁ 4,6 ) + ∥∇ ⊗ U0 ∥L2 ((0,T0 ),Ṁ 2,3 ) < ϵν .
Let T ∗ be the maximal time of existence of our solution. If ⃗u is bounded in
⃗ ⊗ ⃗u is bounded in L2 ((0, T ∗ ), Ṁ 2,3 ), we find that ⃗u ∈
L ((0, T ∗ ), Ṁ 4,6 ) and ∇
∞
∗
2,3
L ((0, T ), Ṁ ). But we have a more precise statement: due to helicity, the norm of
⃗u in the non-separable space L4 ((0, T ∗ ), Ṁ 4,6 ) is equal to the norm of ⃗u(t, x1 , x2 , 0) in the
⃗ ⊗ ⃗u in the non-separable
separable space L4 ((0, T ∗ ), L4 (R2 )), and similarly the norm of ∇
2
∗
2,3
⃗
space L ((0, T ), Ṁ ) is equal to the norm of (∇ ⊗ ⃗u)(t, x1 , x2 , 0) in the separable space
L2 ((0, T ∗ ), L2 (R2 )), Thus, we may approximate ⃗u by smooth functions and we get in return that ⃗u belongs actually to C([0, T ∗ ], Ṁ 2,3 ). This would give that ⃗u(T ∗ , .) ∈ Ṁ 2,3 and
if T ∗ < T we might reiterate the construction of the solution by considering the Cauchy
problem with initial time t = T ∗ .
Thus, in order to prove that the solution is global, we just need to control the size of ⃗u
⃗ ⊗ ⃗u. We have proven enough regularity on ⃗u to be allowed to write:
and of ∇
Z
1
2
∂t ∥⃗u(t, .)∥Ṁ 2,3 =
∂t
|⃗u|2 dx
2πα
0<x3 <2απ
Z
2
=
⃗u.∂t ⃗u dx
2πα 0<x3 <2απ
Z
Z
2ν
⃗ ⊗ ⃗u|2 dx + 2
=−
|∇
⃗u.f⃗ dx
2πα 0<x3 <2απ
2πα 0<x3 <2απ
ZZ
⃗ ⊗ ⃗u(t, x1 , x2 , 0)|2 dx1 dx2
= − 2ν
|∇
ZZ
+2
⃗u(t, x1 , x2 , 0).f⃗(t, x1 , x2 , 0) dx1 dx2
4
⃗ ⊗ ⃗u(t, x1 , x2 , 0)|∥2 2 2
≤ − 2ν∥|∇
L (R )
+ 2∥⃗u(t, x1 , x2 , 0)∥L2 (R2 ) ∥f⃗(t, x1 , x2 , 0)∥L2 (R2 )
⃗ ⊗ ⃗u∥2 2,3 + 2∥⃗u∥M 2,3 ∥f⃗∥M 2,3 .
= − 2ν∥∇
M
This gives ∂t ∥⃗u∥M 2,3 ≤ ∥f⃗∥M 2,3 , so that
∥⃗u∥Ṁ 2,3 ≤ ∥⃗u0 ∥Ṁ 2,3 + ∥f⃗∥L1 Ṁ 2,3 .
284
The Navier–Stokes Problem in the 21st Century (2nd edition)
Moreover, we have
⃗ ⊗ ⃗u∥2 2 2,3 ≤ ∥⃗u0 ∥ 2,3 +
2ν∥∥∇
L M
Ṁ
Z
0
T∗
∥⃗u∥Ṁ 2,3 ∥f⃗∥Ṁ 2,3 dt ≤ (∥⃗u0 ∥Ṁ 2,3 + ∥f⃗∥L1 Ṁ 2,3 )2
We thus have global existence.
10.5
Brandolese’s Symmetrical Solutions
In this section, we consider the Navier–Stokes problem with the following symmetry
property: ⃗u is invariant under the action of the discrete group generated by the isometries
W1 : (x1 , x2 , x3 ) 7→ (x2 , x3 , x1 ) and W2 : (x1 , x2 , x3 ) 7→ (−x1 , x2 , x3 ). In that case, we get
the symmetrical solutions of Brandolese, that satisfy u1 (t, x1 , x2 , x3 ) = u2 (t, x3 , x1 , x2 ) =
u3 (t, x2 , x3 , x1 ) and u1 (t, −x1 , x2 , x3 ) = −u1 (t, x1 , x2 , x3 ) [59, 62].
Thus, if we consider the solutions for the Cauchy problem described in Theorem 4.10,
and if we start with a data ⃗u0 and a force f⃗ which are invariant under the action of the
isometries W1 and W2 , we obtain a solution that is still invariant. But such a solution and
such a force clearly satisfy the Dobrokhotov and Shafarevich conditions
R tR

for 1 ≤ i ≤ 3, 0 fi dx ds = 0





R tR
(10.34)
for 1 ≤ i < j ≤ 3, 0 2ui uj + xi fj + xj fi dx ds = 0




R tR

for 1 ≤ i < j ≤ 3, 0 u2i − u2j + xi fi − xj fj dx ds = 0
Thus, we get a better decay at infinity (⃗u = o(|x|−4 ) than for the generic solutions of the
Navier–Stokes equations.
As a final remark, let us recall that Brandolese studied more generally the finite groups
of isometry of R3 and the solutions that are invariant under the action of such a group, in
order to determine which decay estimate was obtainable in that case [60].
10.6
Self-similar Solutions
In this section, we consider the Navier–Stokes problem with the following symmetry
property: ⃗u is invariant under the action of time-space rescalings, i.e., we consider self–
similar solutions:
for every λ > 0, λ⃗u(λ2 t, λx) = ⃗u(t, x).
(10.35)
Those solutions are associated to homogeneous initial values
λ⃗u0 (λx) = ⃗u0 (x)
(10.36)
λ3 f⃗(λ2 t, λx) = f⃗(t, x).
(10.37)
and self-similar forcing terms
Special Examples of Solutions
285
It is easy to check that the only homogeneous ⃗u0 (satisfying (10.36)) belonging to a Lebesgue
space Lp or a Sobolev space H s or Ḣ s is the null function ⃗u0 = 0. Thus, the study of selfsimilar solutions was an argument for the study of mild solutions in more general spaces as
Morrey spaces (Giga and Miyakawa [212]), homogeneous Besov spaces (Cannone [81]) or
Lorentz spaces (Barraza [22]).
Recall that the homogeneous distributions T (of homogeneity degree −1) may be written
as T (x) =
x
ω( |x|
|x|
, where ω is a distribution on the sphere S 2 , in the sense that
Z
⟨T |φ⟩S ′ ,S = ⟨ω(σ)|
+∞
φ(rσ) dr⟩D′ (S 2 ),D(S 2 )
0
(see Lemarié-Rieusset [313], chapter 23, for instance). In particular, we have T ∈ L3,∞ if
and only if ω ∈ L3 (S 2 ), and T ∈ Ṁ p,3 with 2 ≤ p < 3 if and only if ω ∈ Lp (S 2 ).
Existence of self-similar solutions is then a direct consequence of the theory of mild
solutions for small data in Besov spaces [81, 313].
Self-similar mild solutions
Theorem 10.8.
Let X be a Banach space such that
• for λ > 0, ∥λα u(λx)∥X = ∥u∥X , where α is a positive constant
• ∥φ ∗ u∥X ≤ ∥φ∥1 ∥u∥X .
Assume moreover that, for some β ∈ (max(0, α − 1), α), pointwise multiplication maps
β
β−α
X × ḂX,1
to ḂX,∞
:
∥uv∥Ḃ β−α ≤ C∥u∥X ∥v∥Ḃ β .
X,∞
X,1
(This is the case for instance if we have the inequality for Riesz potentials
∥Iα−β (u Iβ v)∥X ≤ C∥u∥X ∥u∥X .)
Finally, let γ such that max(0, 1 − α) < γ < 2 − β. Then, there exists an ϵ0 > 0
(depending on X, on γ and on ν) such that, if ⃗u0 and f⃗ satisfy
• ⃗u0 ∈ Y, div ⃗u0 = 0 and ⃗u0 is homogeneous of degree −1 (λ⃗u0 (λx) = ⃗u0 (x)),
−1+α
where Y = X if α = 1, Y = ḂX,∞
if α < 1, Y = {0} if α > 1
• f⃗(t, x) =
1 ⃗ √
F ( xt )
t3/2
−γ
with F⃗ ∈ ḂX,∞
• ∥⃗u0 ∥Y + ∥F⃗ ∥Ḃ −γ < ϵ0
X,∞
then the Navier–Stokes problem

⃗
∂t ⃗u + div(⃗u ⊗ ⃗u) = ν∆⃗u + f⃗ − ∇p
div ⃗u = 0

⃗u(0, .) = ⃗u0
has a self-similar solution ⃗u(t, x) =
1 ⃗ √
√
U ( xt ),
t
⃗ ∈ X ∩ Ḃ β .
with U
X,1
286
The Navier–Stokes Problem in the 21st Century (2nd edition)
R
⃗ 0 = Wν ∗ ⃗u0 , U
⃗ 1 = 1 Wν(1−s) ∗ Pf⃗(s, .) ds and V
⃗0 = Wνt ∗ ⃗u0 +
Proof. Let us write U
0
Rt
⃗
Wν(t−s) ∗ Pf (s, .) ds. Then we have
0
x
x
⃗ 0( √
⃗ 1( √
⃗0 (t, x) = √1 (U
)+U
)).
V
t
t
t
Moreover, we have
⃗ 0 ∥X ≤ ∥⃗u0 ∥X
∥U
and, for all β > 0,
⃗ 0 ∥ β ≤ Cν,β ∥⃗u0 ∥X .
∥U
Ḃ
X,1
If α < 1, we write for all positive δ,
⃗ 0 ∥ −1+α+δ ≤ Cν,α,δ ∥⃗u0 ∥ −1+α ;
∥U
Ḃ
Ḃ
X,∞
X,1
0
we shall use it for δ = 1 − α + β and for δ = 1 − α (since we have ḂX,1
⊂ X). Similarly, we
have, for all δ > 0,
∥Wν(1−s) ∗ f⃗(s, .)∥Ḃ −γ+δ ≤ C∥F⃗ ∥Ḃ −γ s−
(3−α−γ)
2
(1 − s)−δ/2 ;
X,∞
X,1
⃗ 1 in the Ḃ −γ+δ norm, provided that α + γ > 1 and δ < 2.
This will give a control on U
X,1
⃗ 1 in X ∩ Ḃ β (provided γ < 2 − β).
Taking δ = γ, then δ = β + γ gives the control on U
X,1
⃗ belongs to X and V
⃗ belongs to Ḃ β and if W
⃗ = div(U
⃗ ⊗V
⃗ ), we know that
Now, if U
X,1
β−α−1
β
⃗ belongs to Ḃ
⃗ and V
⃗ belong to Ḃ , we may use the
W
. On the other hand, when U
X,∞
X,1
⃗ as
paradifferrential calculus and write the Littlewood–Paley decomposition [313] of W
X
⃗ =∆j (
∆j W
⃗ ⊗ ∆k V
⃗ + ∆k U
⃗ ⊗ Sk V
⃗)
(div(Sk U
|k−j|≤2
X
+ ∆j (
X
⃗ ⊗ ∆l V
⃗ ))
div(∆k U
k≥j−3 |k−l|≤1
and thus
⃗ ∥X ≤C2j (
2j(β−α) ∥∆j W
X
⃗ ∥ β ∥∆k V
⃗ ∥X + ∥Sk V
⃗ ∥ β ∥∆k U
⃗ ∥X )
∥Sk U
Ḃ
Ḃ
X,1
|k−j|≤2
+ C2j
X
X
X,1
⃗ ∥ β ∥∆l V
⃗ ∥X
∥∆k U
Ḃ
X,1
k≥j−3 |k−l|≤1
⃗ ∥ β ∥V
⃗∥ β
≤ C2j(1−β) ∥U
Ḃ
Ḃ
X,1
X,1
⃗ belongs to Ḃ 2β−α−1 .
so that W
X,∞
⃗ ∈ Ḃ δ .
Thus, we may find δ ∈ (max(0, 1 − α), 1 + α − β) ∩ (1 + α − 2β, 2 − β) with W
X,∞
This gives
Z 1
1 ⃗ .
1 ⃗ .
⃗∥
⃗
(√ ) ⊗ √ V
( √ )) ds∥X∩Ḃ β ≤ C∥U
∥
Wν(t−s) ∗ P div( √ U
β ∥V ∥
β .
X∩ḂX,1
X∩ḂX,1
X,1
s
s
s
s
0
This gives the existence of self-similar solutions for small data.
Special Examples of Solutions
287
Theorem 10.8 may be applied to a lot of spaces X. For instance, in the case X = Lp ,
⃗ ( √x ) with a profile U
⃗ ∈ Lp ∩ Lq , where
p > 3, we find self-similar solutions ⃗u = √1t U
t
−1+ 3
p < q < ∞, for a small enough homogeneous initial value ⃗u ∈ Ḃp,∞ p and for a small
1 ⃗ √
enough forcing term f⃗ = t3/2
F ( xt ) with F⃗ ∈ Lr for some r ∈ (r∗ , p), where r1∗ = 1q + 23
3
−3
3
−3
p
q
p
r
(just use the embeddings Ḃp,1
⊂ Lq and Lr ⊂ Ḃp,∞
).
−γ
⃗
In Theorem 10.8, the force F is quite regular: it belongs to ḂX,∞
with γ < 2. In some
−2
⃗
cases, one may even consider a force F in Ḃ
; of course, we shall not have the extra-
X,∞
⃗ ∈ Ḃ β , and may only hope that U
⃗ ∈ X. We give here an easy lemma that allows
regularity U
X,1
as well to deal with discretely self-similar solutions (or DSS solutions), a class of solutions
which has been considered by Tsai, when the initial data is not homogeneous, but only
discretely homogeneous: the equality λ⃗u0 (λx) = ⃗u0 (x) holds only for λ ∈ {λ = λk0 / k ∈ Z}
(with λ0 > 1), a discrete subgroup of R∗+ .
Lemma 10.2.
Let X be a Banach space such that
• for λ > 0, ∥λu(λx)∥X = ∥u∥X
• ∥φ ∗ u∥X ≤ ∥φ∥1 ∥u∥X
Assume moreover that the pointwise product maps X × X to a shift-invariant space Y such
that
• for λ > 0, ∥λ2 u(λx)∥Y = ∥u∥Y
• ∥φ ∗ u∥Y ≤ ∥φ∥1 ∥u∥Y
−1
2
• [Y, ḂY,1
]1/2,∞ ⊂ X (so that in particular we have Y ⊂ ḂX,∞
).
R
t
If F ∈ L∞ ((0, +∞), Y), then ⃗v = 0 Wν(t−s) ∗ P div F ds belongs to L∞ ((0, +∞), X) and
sup ∥⃗v (t, .)∥X ≤ C0
t>0
1
sup ∥F(t, .)∥Y
ν t>0
where the constant C0 does not depend on ν.
Proof. Recall that the norm of f in [Y, L∞ ]1/2,∞ is equivalent to
sup inf A∥g∥Y + A−1 ∥h∥∞ .
A>0 f =g+h
We write
∥Wν(t−s) ∗ div F∥Y ≤ C p
and
∥Wν(t−s) ∗ div F∥∞ ≤ C
so that, for A < t,
Z
∥
1
ν(t − s)
1
∥F∥L3/2,∞ ,
(ν(t − s))3/2
max(A,0)
Wν(t−s) ∗ div F ds∥∞ ≤ 2C
0
and
Z
1
√
∥F∥L∞ Y
ν 3/2 t − A
√
t
∥
Wν(t−s) ∗ div F ds∥Y ≤ 2C
max(A,0)
∥F∥Y
t−A
√
∥F∥L∞ Y .
ν
This gives the control of ⃗v (t, .) in [Y, L∞ ]1/2,∞ , hence in X.
288
The Navier–Stokes Problem in the 21st Century (2nd edition)
Examples of Banach spaces X that satisfy assumptions of Lemma 10.2 are the Lorentz
−1+ 3
−2+ 3
space X = L3,∞ (with Y = L3/2,∞ ), the Besov spaces X = Ḃp,∞ p (with Y = Ḃp,∞ p ), where
2
1
1 ≤ p < 3 or the Besov space based on pseudo-measures X = ḂPM,∞
(with Y = ḂPM,∞
).
Theorem 10.9.
Let X be a Banach space such that
• for λ > 0, ∥λu(λx)∥X = ∥u∥X
• ∥φ ∗ u∥X ≤ ∥φ∥1 ∥u∥X
Assume moreover that the pointwise product maps X × X to a shift-invariant space Y
such that
• for λ > 0, ∥λ2 u(λx)∥Y = ∥u∥Y
• ∥φ ∗ u∥Y ≤ ∥φ∥1 ∥u∥Y
−1
2
• [Y, ḂY,1
]1/2,∞ ⊂ X (so that in particular we have Y ⊂ ḂX,∞
).
Then, there exists an ϵ0 > 0 (depending on X) and a constant C1 > 0 such that, if ⃗u0
and f⃗ satisfy
• ⃗u0 ∈ X, div ⃗u0 = 0
• F ∈ L∞ ((0, +∞), Y)
•
∥⃗
u0 ∥X
ν
+
∥F∥L∞ Y
ν2
< ϵ0
then the Navier–Stokes problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
div ⃗u = 0

⃗u(0, .) = ⃗u0
has a unique solution ⃗u(t, x) such that supt>0 ∥⃗u(t, .)∥X ≤ C1 .
If there exists λ0 > 1 such that λ0 ⃗u0 (λ0 x) = ⃗u0 (x) and λ20 F(λ20 t, λ0 x) = F(t, x), then
this solution ⃗u is discretely self-similar:
λ0 ⃗u(λ20 t, λ0 x) = ⃗u(t, x).
Proof. As usual, the proof is performed by using
R t Picard’s iterates. By Lemma 10.2, we
know that the bilinear operator B(⃗u, ⃗v ) = 0 Wν(t−s) ∗ P div(⃗u ⊗ ⃗v ) ds is bounded on
L∞ ((0, +∞), X, with an operator norm which is bounded by C0 ν1 . Thus the Picard iterates
defined by
Z
t
⃗ 0 = Wνt ∗ ⃗u0 +
U
Wν(t−s) ∗ P div F ds
0
and
⃗ N +1 = U
⃗0 −
U
Z
0
t
⃗N ⊗ U
⃗ N ) ds
Wν(t−s) ∗ P div(U
Special Examples of Solutions
⃗ 0 ∥L∞ X <
will converge to a solution ⃗u ∈ L∞ X if we have ∥U
estimates ∥Wνt ∗ ⃗u0 ∥L∞ X ≤ ∥⃗u0 ∥X and, again by Lemma 10.2,
Z
0
t
289
ν
4C0 .
We conclude by the
1
Wν(t−s) ∗ P div F ds∥L∞ X ≤ C0 ∥F∥L∞ Y .
ν
Self-similarity is then obvious, due to the symmetry of the equations under scaling and
the invariance of the norms and due to uniqueness of the solution in the ball.
10.7
Stationary Solutions
In this section, we consider the Navier-Stokes problem with the following symmetry
property: ⃗u is invariant under the action of time translations, i.e., we consider steady solutions.
When the force f⃗ is stationary (i.e., does not depend on time), one may ask whether
one might find a steady-state solution ⃗u of the Navier–Stokes equations. The problem to be
solved is then the following one:
Stationary Navier–Stokes equations
⃗
Given the force f (x), find the velocity ⃗u(x) and the pressure p(x) such that
⃗ u − ∇p
⃗ + f⃗,
−ν∆⃗u = −⃗u.∇⃗
div ⃗u = 0
(10.38)
This is an old and difficult problem, especially in bounded domains (Galdi’s Introduction
[194] to the steady state problem runs over more than 1000 pages). In the whole space,
however, the problem is quite easy.
We are going to consider a special formulation of the problem, turning the differential
equation into an integral one and considering only small data.
Stationary Navier–Stokes equations
Given the force f⃗(x), find the velocity ⃗u(x) such that
⃗u = −
⃗
1 ⃗ 1 ∇
Pf + P .(⃗u ⊗ ⃗u)
ν∆
ν ∆
(10.39)
This problem has been studied for solutions in the Lorentz space L3,∞ or in the Morrey
space Ṁ p,3 with 2 < p < 3 (remark: L3,∞ ⊂ Ṁ p,3 when p < 3) by Kozono ad Yamazaki
[280]. In that case, one starts from f⃗ = ∆F⃗ , with F⃗ small enough in L3,∞ or in Ṁ p,3 . The
case of F⃗ ∈ L3,∞ ∩ Lq with F⃗ small in L3,∞ (3/2 < q < ∞) has been discussed by Bjorland,
Brandolese, Iftimie, and Schonbek [44].
290
The Navier–Stokes Problem in the 21st Century (2nd edition)
Recently, Phan and Phuc [395] discussed the problem in the largest space where one can
look for steady solutions: as in Section 5.3, one looks at a dominating quadratic equation
with non-negative kernel
1
1
U 2,
U = F + C0 √
ν
ν −∆
for which the space where to look for solutions is the multiplier space V 1 (R3 ) = M(Ḣ 1 7→
L2 ).
Existence of steady solutions
Theorem 10.10.
Let X be a space such that
• the Riesz transforms are bounded on X
• the bilinear operator (u, v) 7→
√ 1 (uv)
−∆
is bounded on X
Then, there exists ϵ0 > 0 and C0 > 0 (depending on X) such that, if f⃗(x) (independent
from t) satisfies
• f⃗ = ∆F⃗ with F⃗ ∈ X
• ∥F⃗ ∥X < ϵ0 ν 2
then there is a unique solution ⃗u ∈ X of the equation
⃗u = −
⃗
1 ⃗ 1 ∇
Pf + P .(⃗u ⊗ ⃗u)
ν∆
ν ∆
(10.40)
with ∥⃗u∥X < C0 ν.
Proof. This is obvious by Picard’s fixed-point theorem: we have
⃗
1 ∇
1
∥ P .(⃗u ⊗ ⃗v )∥X ≤ C1 ∥⃗u∥X ∥⃗v ∥X
ν ∆
ν
and we find that Equation (10.40) has a unique solution ⃗u with ∥⃗u∥X <
1
ν
∥ ν∆
Pf⃗∥X < 4C
.
1
ν
2C1
as soon as
Classical examples of such spaces X are the homogeneous Sobolev space Ḣ 1/2 , the
Lebesgue space L3 , the Lorentz space L3,∞ , the Morrey spaces Ṁ p,3 with 2 < p ≤ 3
and the mulitiplier space V 1 = M(Ḣ 1 7→ L2 ).
Notice that those examples, except Ḣ 1/2 are lattice spaces of Lebesque measurable
functions: if u ∈ X and if v is a Lebesque measurable function such that |v| ≤ |u|, then
v ∈ X and ∥v∥X ≤ ∥u∥X .
1
1
Moreover, we have that the bilinear operator (u, v) 7→ (−∆)
1/4 (u (−∆)1/4 v) is bounded
on X. For L3 , L3,∞ or Ṁ p,3 , this is obvious from Hedberg’s inequality:
|
1
1
1/3
2/3
(u
v)| ≤ C(Mu√Mv (x))2/3 ∥u∥X ∥v∥X
(−∆)1/4 (−∆)1/4
Special Examples of Solutions
291
For X = V 1 , we may use the results of Gala and Lemarié-Rieusset [191]: if v ∈ V 1 =
M(Ḣ 1 7→ L2 ), then
1
v ∈ M(Ḣ 1 7→ Ḣ 1/2 ) = M(Ḣ −1/2 7→ Ḣ −1 ) ⊂ M(L2 7→ H −1/2 );
(−∆)1/4
1
v
(−∆)1/4
−1/2
thus, if we first multiply with u, we map Ḣ 1 to L2 , then, multiplying the result by
1
1
to Ḣ
(that maps L2 to H −1/2 ); we find that the product u (−∆)
1/4 v maps Ḣ
1
1
(u (−∆)
1/4 v)
(−∆)1/4
1
, and
2
finally (using again [191]) we get
∈ M(Ḣ 7→ L ) = X.
Following Kozono and Yamazaki [280], and Phan and Phuc [395], we may then give a
simple proof of the stability of steady solutions under small perturbations:
Stability of steady solutions
Theorem 10.11.
Let X be a Banach space of Lebesque measurable functions such that
• X is a lattice
• the Hardy–Littlewood maximal function is a bounded operator on X
• the Riesz transforms are bounded on X
• the bilinear operator (u, v) 7→
√ 1 (uv)
−∆
• the bilinear operator (u, v) 7→
1
1
(u (−∆)
1/4 v)
(−∆)1/4
is bounded on X
is bounded on X
Then, there exists ϵ0 > 0 and C0 > 0 (depending on X) such that, if ⃗u0 and f⃗(x)
(independent from t) satisfy
• ⃗u0 ∈ X and div ⃗u0 = 0
• f⃗ = ∆F⃗ with F⃗ ∈ X
• ∥⃗u0 ∥X < ϵ0 ν and ∥F⃗ ∥X < ϵ0 ν 2
then
⃗ ∈ X of the equation
• there is a unique solution U
⃗
⃗ = − 1 Pf⃗ + 1 P ∇ .(U
⃗ ⊗U
⃗)
U
ν∆
ν ∆
⃗ ∥X < C0 ν
with ∥U
• there exists a unique solution ⃗u on (0, +∞) × R3 of the problem

⃗
∂t ⃗u − div(⃗u ⊗ ⃗u) = ν∆⃗u + f⃗ − ∇p
div ⃗u = 0

⃗u(0, .) = ⃗u0
(10.41)
such that supt>0 |⃗u(t, .)| ∈ X and ∥ supt>0 |⃗u(t, .)|∥X < C0 ν.
• moreover, we have
⃗ (x))∥X ≤ C0 ν 3/4
sup t1/4 ∥(−∆)1/4 (⃗u(t, x) − U
t>0
⃗ in S ′ (and in L2 ) as t goes to +∞.
so that ⃗u(t, .) converges to U
loc
(10.42)
292
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ (Theorem 10.10). For
Proof. We have already proved the existence of the steady solution U
⃗
the existence of ⃗u, we define w
⃗ = ⃗u − U . w
⃗ is a solution of the problem

⃗ ⊗w
⃗ ) = ν∆w
⃗
⃗ − div(w
⃗ ⊗w
⃗ +U
⃗ +w
⃗ ⊗U
⃗ − ∇q
∂t w
div w
⃗ =0

⃗
w(0,
⃗ .) = ⃗u0 − U
We rewrite the problem as
⃗ , w)
⃗)
w
⃗ = Wνt ∗ w
⃗ 0 − B(w,
⃗ w)
⃗ − B(U
⃗ − B(w,
⃗ U
with
Z
t
Wν(t−s) ∗ P div(⃗v ⊗ w)
⃗ ds.
B(⃗v , w)
⃗ =
0
We know that
|Wνt ∗ w
⃗ 0 (x)| ≤ M|w⃗ 0 |
so that
sup |Wνt ∗ w
⃗ 0 (x)| ∈ X.
t>0
Moreover, if |⃗v (t, x)| ≤ V (x) and |w(t,
⃗ x)| ≤ W (x), we find that
Z tZ
1
C′
|B(⃗v , w)(t,
⃗
x)| ≤ C
W
(y)
V
(y)
dy
ds
≤
I1 (V W )(x)
(ν(t − s))2 + |x − y|4
ν
0
1
(where I1 = √−∆
). This grants the existence of w
⃗ (hence of ⃗u), if ⃗u0 and f⃗ are small
enough.
Moreover, we have
|(νt)1/4 (−∆)1/4 Wνt ∗ w
⃗ 0 (x)| ≤ M|w⃗ 0 |
and thus
|Wνt ∗ w
⃗ 0 (x)| ≤ C
1
I1/2 M|w⃗ 0 | (x)
(νt)1/4
1
).
(−∆)1/4
If|⃗v (t, x)| ≤ V (x) and t1/4 |w(t,
⃗ x)| ≤ I1/2 W (x), we find that
Z tZ
(t − s)1/4 + s1/4
ds
t1/4 |(−∆)1/4 B(⃗v , w)(t,
⃗
x)| ≤C
V (y)I1/2 W (y) dy 1/4
2 )9/4
(ν(t
−
s)
+
|x
−
y|
s
0
′1
≤C I1/2 (V I1/2 W )(x)
ν
(where I1/2 =
and a similar estimate holds for |(−∆)1/4 B(w,
⃗ ⃗v )(t, x)|. This will give the regularity estimate
for our solution w.
⃗
Remark:
• An interesting point is that, while the perturbation w
⃗ is regular ((−∆)1/4 w
⃗ is a locally
square integrable function), the steady solution may be irregular (this proves that in
presence of a singular forcing term, the mild solutions of the Navier–Stokes equations
may be singular): for example, let θ ∈ L3 (R) be a compactly supported function with
R
Rs
θ(s) ds = 0 and let Θ(s) = −∞ θ(σ) dσ; if F⃗ ∈ L3 is defined by
F⃗ = (Θ(x1 )θ(x2 )θ(x3 ), θ(x1 )Θ(x2 )θ(x3 ), −2θ(x1 )θ(x2 )Θ(x3 )),
⃗
⃗ of U
⃗ = − 1 PF⃗ + 1 P ∇
⃗
⃗
then the steady solution U
ν
ν ∆ .(U ⊗ U ) is the sum of a term
⃗
1
∇
1/2
⃗ 1 = P .(U
⃗ ⊗U
⃗ ) that satisfies U
⃗ 1 ∈ Ḣ
⃗ 2 = − 1 F⃗ that is very
U
and of a term U
ν ∆
ν
irregular if θ is irregular.
Special Examples of Solutions
293
⃗ is the steady solution
• One may consider more singular initial values. For instance, if U
in V 1 associated to the small force f⃗ = ∆F⃗ , and if the initial value ⃗u0 for the Cauchy
problem (10.41) is small enough in Ṁ 2,3 , then the Cauchy problem has a solution ⃗u
such that:
sup |⃗u(t, .)| ∈ Ṁ 2,3
t>0
and
⃗ )| ∈ Ṁ 2,3 .
sup t1/4 |(−∆)1/4 (⃗u(t, .) − U
t>0
10.8
Landau’s Solutions of the Navier–Stokes Equations
1 ⃗
Let f⃗ ∈ L1 (R3 ). Then ∆
f ∈ L3,∞ ; thus, we may apply Theorem 10.11 if ∥f⃗∥1 is small
enough and find a steady solution ⃗u ∈ L3,∞ of the equation
⃗u = −
⃗
1 ⃗ 1 ∇
Pf + P .(⃗u ⊗ ⃗u)
ν∆
ν ∆
−1/2
1/2
This solution is quite regular: as L3/2,∞ ⊂ Ḃ2,∞ , we find that ⃗u ∈ Ḃ2,∞ .
Theorem 10.12.
There exist ϵ0 > 0 and C0 > 0 such that, if f⃗(x) (independent from t) satisfies
∥f⃗∥1 < ϵ0 ν 2
then
• there is a unique solution ⃗u ∈ L3,∞ of the equation
⃗u = −
⃗
1 ⃗ 1 ∇
Pf + P .(⃗u ⊗ ⃗u)
ν∆
ν ∆
with ∥⃗u∥L3,∞ < C0 ν
• the functions λ⃗u(λx) is *-weakly convergent in L3,∞ (as λ → +∞) to the unique
solution ⃗u∞ on (0, +∞) × R3 of the problem
⃗u∞ = −
with f⃗∞ = δ(x − 0)
R
⃗
1 ⃗
1 ∇
Pf∞ + P .(⃗u∞ ⊗ ⃗u∞ )
ν∆
ν ∆
f⃗ dy and ∥⃗u∞ ∥L3,∞ < C0 ν.
Proof. The existence of ⃗u and of ⃗u∞ are proved by Theorem 10.11. Moreover, the functions
λ⃗u(λx) are all contained in the closed ball B = {⃗v ∈ L3,∞ / ∥⃗v ∥L3,∞ ≤ ∥⃗u∥L3,∞ }, which is a
compact metrizable space for the *-weak convergence. Thus, in order to prove the theorem,
we have only to prove that ⃗u∞ is the only limit point of the family λ⃗u(λx) when λ → +∞.
Let us consider a limit point ⃗v of λ⃗u(λx). To ⃗u, one may associate p ∈ L3/2,∞ such that
⃗ + f⃗
∂t ⃗u = ν∆⃗u − div(⃗u ⊗ ⃗u) − ∇p
We have ⃗v = ∗ − lim λk ⃗u(λk x); we may assume that λ2k p(λk x) is *-weakly convergent as
well (to some q ∈ L3/2,∞ ). The, we get in the distribution sense that
Z
⃗
lim div(λk ⃗u(λk x) ⊗ λk ⃗u(λk x)) = ν∆⃗v + ( f⃗ dx)δ(x − 0) − ∇q.
294
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, the problem is just to study the convergence of the non-linear term. But this conver1/2
gence is easy, due to the Rellich lemma: the functions λk ⃗u(λk x) are bounded in Ḃ2,∞ , hence
s
in Hloc
for 0 < s < 1/2. Thus, applying Rellich theorem, we find that a subsequence will
converge strongly in L2loc to ⃗v and the convergence of the non-linear term is obtained.
Thus, we can see that the steady solutions associated to a Dirac mass
f⃗ = βδ(x − 0)⃗e
(β ∈ R, ⃗e unit vector) play a special role into the asymptotic behavior of steady solutions
when x goes to ∞. Up to a rotation, we may assume that ⃗e = ⃗e3 (= ⃗ez in cylindrical
coordinates); the solution will then be axisymmetrical with no swirl. Those solutions are
known as Landau’s (self-similar) solutions. Surprisingly enough, those solutions exist for all
β (even large ones) and have been described first (quite implicitly) by Slezkin [439]2 , then
independently by Landau [301] and Squire [447]; recently, Tian and Xin provided another
derivation of those solutions [473]. The role of Landau solutions in asymptotics of steady
solution has been discussed by Šverák [492].
Landau solutions
Theorem 10.13.
For β ∈ R, there exists one and only one solution of the problem
⃗ + βδ(x − 0)⃗e3 ,
ν∆⃗u − div(⃗u ⊗ ⃗u) − ∇p
div ⃗u = 0
(10.43)
such that ⃗u is axisymmetric with no swirl, homogeneous of homogeneity degree −1 and
C 2 on |x| ̸= 0.
For β =
̸ 0, this solution is given by the formula

x1 (Ax3 −|x|)

2ν |x|(A|x|−x
2

3)
u1 =




x2 (Ax3 −|x|)
u2 =
2ν |x|(A|x|−x
(10.44)
2
3)






u3 = 2ν A|x|2 +Ax23 −2x32 |x|
|x|(A|x|−x3 )
where A = A(β) is a constant with |A| > 1. The pressure p is given by
p = 4ν 2
Ax3 − |x|
|x|(A|x| − x3 )2
Remark: If ⃗u is a steady solution of the Navier-Stokes equations on R3 \ {0} (with null
forcing term) which is regular and homogeneous of degree −1, then it is a Landau solution
(see Šverák [492]).
Proof. In order to describe axisymmetric flows, recall that we found it convenient to use
cylindrical coordinates: x1 = r cos θ, x2 = r sin θ and x3 = z. Another interesting system
2 Galaktionov
[192] provides an English translation of Slezkin’s paper.
Special Examples of Solutions
295
of coordinates is the system of spherical coordinates (which amounts to write (r, z) =
ρ(sin φ, cos φ)).
We thus write our velocity as ⃗u = ur (ρ, φ)⃗er + uz (ρ, φ)⃗ez and the vorticity as ω
⃗ = ωθ ⃗eθ ,
with ωθ = ∂z ur − ∂r uz .
The equation div ⃗u = 0 gives
1
(∂r (rur ) + ∂z (ruz )) = 0.
r
Thus, we have (on the open set r > 0), rur = −∂z γ and ruz = ∂r γ for some function
γ(r, z), or equivalently
⃗ ∧ (ψθ ⃗eθ ) with ψθ = 1 γ.
⃗u = ∇
r
We then obtain an equation on γ: as div(ψθ ⃗eθ ) = 0, we have ω
⃗ = −∆(ψθ ⃗eθ ); let D be
the differential operator
1
1
Dh = ∂r2 h + ∂r h + ∂z2 h − 2 h;
r
r
we have ∆(ψθ ⃗eθ ) = (Dψθ )⃗eθ and ∆⃗
ω = (Dωθ )⃗eθ , so that, taking the curl of the Navier–
Stokes equations, we have the equation on ω
⃗:
⃗ u − (⃗u.∇)⃗
⃗ ω + curl f⃗ = 0
ν∆⃗
ω + (⃗
ω .∇)⃗
and thus (on r > 0)
1
νD2 ψθ = − ur Dψθ + ur ∂r Dψθ + uz ∂z Dψθ
r
1
1
= 2 ∂z ψθ Dψθ − ∂z ψθ ∂r Dψθ + ∂r (rψθ ) ∂z Dψθ
r
r
Let
We have
1
D0 h = ∂r2 h − ∂r h + ∂z2 h.
r
h
D0 h
h
D2 h
D( ) =
and D2 ( ) = 0 .
r
r
r
r
This gives
D0 γ
1
D0 γ
1
D0 γ
νD02 γ =r ∂z γ 3 − ∂z γ ∂r (
) + ∂r γ∂z (
)
r
r
r
r
r
D0 γ
1
1
=2∂z γ 2 − ∂z γ ∂r D0 γ + ∂r γ ∂z D0 γ
r
r
r
As rur and ruz are homogeneous of degree 0, Sleznik’s idea was to look for an axisymmetric function γ that would be homogeneous of degree 1, thus to write γ as
γ = ρ G(cos φ).
Indeed, we have ∂ρ γ = cos φ∂z γ + sin φ∂r γ = − cos φ rur + sin φ ruz ; thus, ∂ρ γ is homogeneous of order 0: ∂ρ γ = G(cos φ), and thus γ = ρG(cos φ) + H(cos φ); moreover,
⃗ ∧ ( γ ⃗eθ ) = ⃗u is homogeneous of degree −1: by homogeneity, we must have ∇(
⃗ H ⃗eθ ) = 0;
∇
r
r
H
1
⃗
since ∇( r ⃗eθ ) = − r ∂z H⃗er , we find ∂z H = 0, so that H is constant (and may be taken
equal to 0).
296
The Navier–Stokes Problem in the 21st Century (2nd edition)
A further change of variable τ = cos φ then gives:
p
r = ρ sin(φ) = ρ 1 − τ 2 and z = ρ cos(φ) = ρτ
so that
∂z = cos(φ)∂ρ −
sin(φ)
1 − τ2
∂φ = τ ∂ρ +
∂τ
ρ
ρ
and
∂r = sin(φ)∂ρ +
cos(φ)
r
rτ
∂φ = ∂ρ − 2 ∂τ
ρ
ρ
ρ
This gives
−
1
1
1
∂z γ ∂r D0 γ + ∂r γ ∂z D0 γ = 2 (∂ρ γ ∂τ D0 γ − ∂τ γ ∂ρ D0 γ)
r
r
ρ
and
2∂z γ
D0 γ
2
τ
1
= 2 D0 γ (
∂ρ γ + ∂τ γ).
r2
ρ
1 − τ2
ρ
We then write
1
1
1 − τ2 2
1
∂τ .
D0 = ∂ρ2 + ∂ρ + 2 ∂ϕ2 − ∂r = ∂ρ2 +
ρ
ρ
r
ρ2
We have
d
• ∂ρ γ = G(τ ) and ∂τ γ = ρ dτ
G(τ )
• D0 γ =
1−τ 2 d2
ρ dτ 2 G(τ )
• ∂ρ D0 γ = − 1−τ
ρ2
4
d2
dτ 2 G(τ )
and ∂τ D0 γ =
2
d2
1−τ 2 d2
dτ 2 G(τ ) + ρ3 dτ 2
d3
− 4τ dτ
3 G(τ ))
• D02 γ = 2 1−τ
ρ3
d
τ 2 ) dτ
4 G(τ )
2
1 d
ρ dτ
d2
(1 − τ )2 dτ
2 G(τ )
d2
(1 − τ 2 ) dτ
or equivalently D02 γ =
2 G(τ )
1−τ 2
ρ3 ((1 −
We get an equation on G:
ν((1 − τ 2 )
d2
d4
d3
d3
d
G(τ ) − 4τ 3 G(τ )) = G(τ ) 3 G(τ ) + 3 G(τ ) 2 G(τ )
4
dτ
dτ
dτ
dτ
dτ
which can be rewritten as
ν
d3
d
1 d3
((1 − τ 2 ) G(τ ) + 2τ G(τ )) =
(G(τ )2 )
3
dτ
dτ
2 dτ 3
and finally, for three constants of integration, we obtain Slezkin’s equation
ν((1 − τ 2 )
d
1
G(τ ) + 2τ G(τ )) = G(τ )2 + C2 τ 2 + C1 τ + C0 .
dτ
2
(10.45)
General solutions of Slezkin’s equation have been discussed by many authors (as Sedov [427]
or Vyskrebtsov [493]). Landau’s solutions correspond to the simple case
ν((1 − τ 2 )
d
1
G(τ ) + 2τ G(τ )) = G(τ )2 .
dτ
2
(10.46)
Indeed, we have
∂τ G(τ ) =
p
1
τ
∂r γ + ∂z γ = −ρτ uz − ρ 1 − τ 2 ur = −zuz − rur
∂τ γ = − √
2
ρ
1−τ
Special Examples of Solutions
297
thus, as ⃗u is continuous on |x| ̸= 0, ∂τ G is continuous on [−1, 1], and G is C 1 on [−1, 1].
z
Moreover, ⃗ur = − 1r ∂z γ = − ρr
G(τ ) − ρr2 ∂τ G(τ ); as ρur is bounded, we find that G(1) =
2
ρ G
G
G
G(−1) = 0 and 1−τ
2 =
r 2 is bounded. If we define H(τ ) = 1−τ 2 , we have that H is
continuous on [−1, 1], with H(1) = − 21 ∂τ G(1) and H(−1) = 12 ∂τ G(−1); this gives that,
near 1 and −1, we have
C2 τ 2 + C1 τ + C0 = ν((1 − τ 2 )
d
1
G(τ ) + 2τ G(τ )) − G(τ )2 = o(1 − τ 2 )
dτ
2
which is possible only if C0 = C1 = C2 = 0.
In the case C0 = C1 = C2 = 0, we have
1
G(τ )
)=
ν∂τ (
2
1−τ
2
G(τ )
1 − τ2
2
hence, if G is not the null function, for some constant A
G(τ ) =
2ν(1 − τ 2 )
.
A−τ
(10.47)
For G to be C 1 with G(1) = G(−1) = 0, we must have |A| > 1.
2
⃗ ∧ ( ρG(τ ) ⃗eθ ), with G(τ ) = 2ν(1−τ ) , we know that ω
⃗ ∧ ⃗u satisfies on
Now, if ⃗u = ∇
⃗ =∇
r
A−τ
|x| ̸= 0
⃗ u − (⃗u.∇)⃗
⃗ ω=0
ν∆⃗
ω + (⃗
ω .∇)⃗
so that ⃗u satisfies
⃗ ∧ (ν∆⃗u − ω
⃗ ∧ (ν∆⃗u − ⃗u.∇⃗
⃗ u) = 0.
∇
⃗ ∧ ⃗u) = ∇
⃗ ∧ (ν∆⃗u − div(⃗u ⊗ ⃗u)) is reduced to {0}, and w
Thus, the support of w
⃗ = −∇
⃗ is a sum
of derivatives of Dirac masses. But w
⃗ is homogeneous of homogeneous degree −4, so the
⃗ 1 + ∂2 δ E
⃗ 2 + ∂3 δ E
⃗ 3 for three constant vectors
derivatives are derivatives of order 1: w
⃗ = ∂1 δ E
⃗ 1, E
⃗ 2, E
⃗ 3 . As w
E
⃗ is divergence free, we find that






−∂y δ
∂z δ
0
w
⃗ = α −∂z δ  + β  ∂x δ  + γ  0 
0
−∂x δ
∂y δ
for three constants α, β, γ. Moreover, w
⃗ is axisymmetrical; rotating the axes inx1 and
 x2
−∂y δ
should let the component on ⃗ez invariant: this gives α = γ = 0. Thus, w
⃗ = β  ∂x δ  =
0
3
⃗
∇ ∧ (βδ⃗e3 ). We find that, on R we have
⃗ =0
ν∆⃗u − div(⃗u ⊗ ⃗u) + βδ⃗e3 − ∇p
for some distribution p. Thus, ⃗u satisfies the Navier–Stokes equations with forcing term
βδ⃗e3 .
It remains to state the exact range where β can be taken in. First, we have to compute
β as a function of the constant A in Equation (10.47). This value of β is given in Batchelor’s
book [25] and in the paper of Cannone and Karch [82]:
β = ν2
8πA
A+1
(2 + 6A2 − 3A(A2 − 1) ln(
))
3(A2 − 1)
A−1
(10.48)
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The Navier–Stokes Problem in the 21st Century (2nd edition)
We have that β is an odd function and satisfies
4
A+1
d
6
4
2
+
+ 6A ln(
β = −ν
+
) <0
dA
A2 − 1 (A − 1)2
(A + 1)2
A−1
with β(1) = +∞ and β(+∞) = 0. Thus, the mapping A ∈ (1, +∞) 7→ β ∈ (0, +∞) is a
bijection.
10.9
Time-Periodic Solutions
In this section, we consider the Navier–Stokes problem with the following symmetry
property: ⃗u is invariant under the action of a discrete group of time translations, i.e., ⃗u is
time-periodic.
When the force f⃗ is time-periodic (f⃗(t+T, x) = f⃗(t, x)), one may ask whether one might
find a time-periodic solution ⃗u of the Navier–Stokes equations. The problem to be solved is
then the following one:
Time-periodic Navier–Stokes equations
Given a time-periodic force f⃗(t, x), find a time-periodic velocity ⃗u(t, x) and a timeperiodic pressure p(t, x) such that
⃗ u − ∇p
⃗ + f⃗,
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0
(10.49)
The study of the time-periodic Navier–Stokes problem is now ancient. In the fifties,
Serrin studied the problem on bounded domains [433]. Then, in the nineties, there were
⃗ per (t, .) =
several works on the whole space. Maremonti [349] constructed periodic solutions U
⃗
Uper (t + N T, .) as the asymptotic limit of ⃗u(t + N T, .) of the Cauchy initial value problem
for a small arbitrary initial value ⃗u0 . Then, Yamazaki [509], generalizing a previous work of
Kozono and Nakao [272], proved that the formalism of mild solutions developed by Fujita
3,∞
.
and Kato [185] could be adapted to find solutions in the Lorentz space L∞
t L
In order to solve the time-periodic problem, we begin by showing a simple inequality:
Lemma 10.3.
Rt
−1
−3
The operator f 7→ −∞ Wν(t−s) ∗f ds is bounded from L1 ((−∞, +∞), Ḃ∞,∞
(R3 )), L∞ B∞,∞
−3+ 2
−1
or Lp,∞ Ḃ∞,∞p (1 < p < ∞) to L∞ Ḃ∞,∞
.
Rt
Proof. Let u = −∞ Wν(t−s) ∗ f ds. For 1 ≤ p ≤ +∞, and τ > 0, we have
Z
t
∥Wντ ∆ ∗ u∥∞ ≤ Cp
3
−∞
3
1
1
(ν(τ + t − s)) 2 − p
∥f (s, .)∥
Ḃ
−3+ 2
p
ds.
1
Let kp,τ (t) = 1t>0 (t + τ )− 2 + p . Then ∥k1,τ ∥∞ = τ −1/2 and ∥k∞,τ ∥1 = 2τ −1/2 . If 1 < p <
p
p
−1/2
p
+∞, we remark that ( 23 − p1 ) p−1
> 1, so that kp,τ ∈ L p−1 ,1 and ∥kp,τ ∥ p−1
.
,1 = Cp τ
p
L
Since convolution maps L∞ × L1 , L1 × L∞ and L p−1 ,1 × Lp,∞ to L∞ , we find that u belongs
−1
to L∞ Ḃ∞,∞
.
Special Examples of Solutions
299
With this lemma, we may provide a simple exposition of the results of Kyed [290] on
1/2
time-periodic solutions which belong to L∞
∩ L2per Ḣ 3/2 :
t Ḣ
Time-periodic Navier–Stokes equations in Sobolev spaces
Theorem 10.14.
There exists a positive constant η such that: if f⃗per is a time-periodic vector field on
R × R3 (with period T ) such that
RT
• the mean value f⃗0 = T1 0 f⃗per (s, .) ds belongs to Ḣ −3/2 and satisfies
∥Pf⃗0 ∥Ḣ −3/2 < ην
• f⃗per belongs to L2per Ḣ −1/2 with
∥f⃗per ∥L2
per Ḣ
−1/2
√
<η ν
then there exists a time-periodic solution ⃗uper of the Navier–Stokes problem (10.49)
1/2
∩ L2per Ḣ 3/2 .
such that ⃗uper ∈ L∞
t Ḣ
Rt
⃗0 =
Proof. We first study U
−∞
eν(t−s)∆ Pf⃗per ds. We expand Pf⃗per as a time-Fourier series
Pf⃗per =
X
2π
⃗gk (x)e T
ikt
.
k∈Z
We have
Z
T
∥Pf⃗per ∥2Ḣ −1/2 dt = T
0
X
∥⃗gk ∥2Ḣ −1/2 .
k∈Z
⃗ 0 is
The Fourier expansion of U
⃗0 =
U
X
⃗ k (x)e 2π
⃗k =
T ikt ,
W
with W
k∈Z
1
P⃗gk .
ik 2π
T − ν∆
⃗ k ∥ 3/2 ≤ 1 ∥⃗gk ∥ −1/2 , and thus U
⃗ 0 ∈ L2per Ḣ 3/2 . Moreover, ⃗g0 = Pf⃗0 ∈ Ḣ −3/2
We have ∥W
Ḣ
Ḣ
ν
⃗ 0 ∈ Ḣ 1/2 . Let Ω
⃗ k be the Fourier transform of ⃗gk . We have:
so that W
2
⃗ 0 (t, .) −
(2π) ∥U
3
⃗ 0 ∥2 1/2
W
Ḣ
Z
X
|ξ|
=
k̸=0
Z
≤
|ξ| (
ik 2π
T
X
k̸=0
k2
4π 2
T2
1
⃗ k (ξ)e 2π
T ikt
Ω
+ ν|ξ|2
X
1
⃗ k (ξ)|2 ) dξ
)(
|Ω
2
4
+ ν |ξ| k̸=0
If νT |ξ|2 ≤ 1, we write
X
k̸=0
k2
4π 2
T2
dξ
T2 X 1
T2
T
1
≤
=
≤
.
2
2
2
4
4π
k
12
12ν|ξ|2
+ ν |ξ|
k̸=0
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The Navier–Stokes Problem in the 21st Century (2nd edition)
If νT |ξ|2 > 1, we write
X
2
k 2 4π
T2
k̸=0
X
X
1
1
1
1
T
≤ 2(
+
)
2 ) ≤ (4 +
2
4
2
4π
2
4
2
ν |ξ|
2π ν|ξ|2
+ ν |ξ|
k T2
1≤k≤2νT |ξ|2
k>2νT |ξ|2
1/2
⃗ 0 ∈ L∞
Thus, we find that U
. We found more precisely:
t Ḣ
⃗ 0 ∥ ∞ 1/2 ≤ C 1 ∥f⃗0 ∥ −3/2 + C √1 ∥f⃗∥ 2 −1/2
∥U
L Ḣ
Ḣ
Lper Ḣ
ν
ν
and
⃗ 0 ∥ 2 3/2 ≤ C 1 ∥f⃗∥ 2 −1/2 .
∥U
L Ḣ
Lper Ḣ
ν
It is now easy to check that the bilinear operator B
Z t
⃗
⃗
⃗ ⊗V
⃗ ) ds
B(U , V ) =
Wν(t−s) ∗ P div(U
−∞
1/2
⃗ and V
⃗ in E,
is bounded on E = L∞
∩ L2per Ḣ 3/2 . Indeed, we have, for U
t Ḣ
⃗ (t, .) ⊗ V
⃗ (t, .)∥ 1/2 ≤ C(∥U
⃗ (t, .)∥ 1/2 ∥V
⃗ (t, .)∥ 3/2 + ∥V
⃗ (t, .)∥ 1/2 ∥U
⃗ (t, .)∥ 3/2 )
∥U
Ḣ
Ḣ
Ḣ
Ḣ
Ḣ
and
⃗ (t, .) ⊗ V
⃗ (t, .)∥ −1/2 ≤ C∥U
⃗ (t, .)∥ 1/2 ∥V
⃗ (t, .)∥ 1/2
∥U
Ḣ
Ḣ
Ḣ
R
T
⃗ ⊗V
⃗ ) satisfies F⃗ ∈ L2 Ḣ −1/2 and
so that F⃗ = div (U
F⃗ (s, .) ds ∈ Ḣ −3/2 . The proof we
per
0
⃗
⃗
⃗
gave on U0 gives us as well that B(U , V ) ∈ E: we have
Z T
1
⃗,V
⃗ )∥ ∞ 1/2 ≤C 1 ∥ 1
∥B(U
F⃗ (s, .) ds∥Ḣ −3/2 + C √ ∥F⃗ ∥L2 Ḣ −1/2
L Ḣ
per
ν T 0
ν
1
⃗ ∥ ∞ 1/2 ∥V
⃗ ∥ ∞ 1/2
≤ C ′ ∥U
L Ḣ
L Ḣ
ν
1
⃗ ∥ ∞ 1/2 ∥V
⃗ ∥ 2 3/2 + ∥V
⃗ ∥ ∞ 1/2 ∥U
⃗ ∥ 2 3/2 )
+C ′ √ (∥U
L Ḣ
Lper Ḣ
L Ḣ
Lper Ḣ
ν
and
⃗,V
⃗ )∥ 2 3/2 ≤C 1 ∥F⃗ ∥ 2 −1/2
∥B(U
L Ḣ
Lper Ḣ
ν
1 ⃗
⃗ ∥ 2 3/2 + ∥V
⃗ ∥ ∞ 1/2 ∥U
⃗ ∥ 2 3/2 )
≤ C ′ (∥U
∥L∞ Ḣ 1/2 ∥V
Lper Ḣ
L Ḣ
Lper Ḣ
ν
The proof of the theorem is now reduced
to the fixed–point theorem of Picard, by takin
⃗ ∥ 2 3/2 .
⃗ ∥E = ∥U
⃗ ∥ ∞ 1/2 + √ν∥U
ion E the norm ∥U
L Ḣ
L Ḣ
per
We now give another theorem on the existence of time-periodic solutions, based on
Kozono and Nakao’s approach:
Time-periodic Navier–Stokes equations in Morrey spaces
Theorem 10.15.
We consider a Banach space of distributions on R3 , X ⊂ L2loc , such that:
Special Examples of Solutions
301
• (A1) The pointwise product is bounded from L∞ × X to X .
• (A2) The Riesz transforms are bounded on X .
• (A3) The Hardy–Littlewood maximal function is bounded on X .
• (A4) The operator (u, v) 7→
√ 1 (uv)
−∆
• (A5) The operator (u, v) 7→
1
1
(u (−∆)
1/4 v)
(−∆)1/4
is bounded from X × X to X .
is bounded from X × X to X .
Then there exist positive constants ϵX and CX such that: if f⃗per is a time-periodic vector
field (with period T ) on R × R3 such that
• f⃗per belongs to L1per X with
Z
T
∥f⃗per ∥X dt < ϵX ν
0
• the mean value f⃗0 =
with
1
T
RT
0
−3
f⃗per (s, .) ds belongs to Ḃ∞,∞
and satisfies
∥
1 ⃗
∆ f0
∈X
1 ⃗
f0 ∥X < ϵX ν 2
∆
then
• there exists a time-periodic solution ⃗uper of the Navier–Stokes problem (10.49)
such that supt∈R |⃗u(t, x)| ∈ X and ∥ supt∈R |⃗u(t, x)| ∥X < CX ν
⃗ ⊗ ⃗uper ∈ L2,∞ X
• ∇
per
• if ∥⃗u0 ∥X < ϵX ν, there exists a unique solution ⃗u on (0, +∞)×R3 of the problem

⃗
∂t ⃗u + div(⃗u ⊗ ⃗u) = ν∆⃗u + f⃗per − ∇p
(10.50)
div ⃗u = 0

⃗u(0, .) = ⃗u0
such that supt>0 |⃗u(t, .)| ∈ X and ∥ supt>0 |⃗u(t, .)|∥X < CX ν.
• moreover, we have
sup t1/4 ∥(−∆)1/4 (⃗u(t, x) − ⃗uper (x))∥X ≤ CX ν 3/4
(10.51)
t>0
so that ⃗u(t, .) converges to ⃗uper in S ′ (and in L2loc ) as t goes to +∞.
Examples of such Banach spaces X are the Lebesgue space L3 , the Lorentz spaces L3,p
(1 ≤ p ≤ +∞), the Morrey spaces Ṁ p,3 (2 < p < 3) and the multiplier space V 1 =
M(Ḣ 1 7→ L2 ). Recall that L3,1 ⊂ L3 ⊂ L3,∞ ⊂ Ṁ p,3 ⊂ V 1 .
Let us further remark that if f⃗ is divergence-free and belongs to L1per V 1 , and if moreover
−3
the support of f⃗ is bounded: f⃗(t, x) = 0 if |x| ≥ R, then its mean value belongs to Ḃ∞,∞
1 ⃗
and satisfies ∆
f0 ∈ L3,1 . Indeed, f⃗0 ∈ V 1 ⊂ L2loc and the support of f⃗0 is contained in
302
The Navier–Stokes Problem in the 21st Century (2nd edition)
R
the ball B̄(0, R). Thus, f⃗0 ∈ L1 . Since f⃗0 is divergence-free, we find that f⃗0 dx = 0.
Moreover, f⃗0 ∈ L2 and has compact support. Hence, f⃗0 belongs to the Hardy space H1 ,
1 ⃗
hence ∆
f0 ∈ L3,1 .
We may now prove the theorem:
Proof. The solutions ⃗uper and ⃗u = ⃗uper + w
⃗ are solutions of two-fixed point problems: ⃗uper
is solution of
Z t
⃗ =
⃗ ⊗U
⃗ )) ds
U
Wν(t−s) ∗ P(f⃗per − div(U
−∞
and w
⃗ is solution of
⃗ = Wνt ∗ w
W
⃗0 −
t
Z
⃗ ⊗W
⃗ +W
⃗ ⊗ ⃗uper + ⃗uper ⊗ W
⃗ ) ds
Wν(t−s) ∗ P div(W
0
with w
⃗ 0 = ⃗u0 − ⃗uper (0, .).
Let E be the space of divergence free vector fields ⃗u such that supt∈R |⃗u(t, x)| ∈ X . We
are going to prove the existence of ⃗uper by Picard’s iterations in the space
⃗ ∈E /U
⃗ (t + T, x) = U
⃗ (t, x), ∇
⃗ ⊗U
⃗ ∈ L2,∞
⃗
F = {U
per X and div U = 0}
and the existence of w
⃗ by Picard’s iterations in the space
⃗ / sup |W
⃗ (t, x)| ∈ X and sup t1/4 |(−∆)1/4 W
⃗ (t, x)| ∈ X }
G = {W
t>0
t>0
⃗ n and W
⃗ n as
We thus define inductively U
Z t
⃗0 =
⃗ n+1 = U
⃗ 0 − B(U
⃗ n , Un )
U
Wν(t−s) ∗ Pf⃗per ds and U
−∞
where
⃗,V
⃗)=
B(U
t
Z
⃗ ⊗V
⃗ ) ds
Wν(t−s) ∗ P div(U
−∞
and
⃗ 0 = Wνt ∗ w
⃗ n+1 = W
⃗ 0 − B0 ( W
⃗ n, W
⃗ n ) − B0 (W
⃗ n , ⃗uper ) − B0 (⃗uper , W
⃗ n)
W
⃗ 0 and W
where
⃗,V
⃗)=
B0 (U
Z
t
⃗ ⊗V
⃗ ) ds.
Wν(t−s) ∗ P div(U
0
⃗ 0 belongs to F. Let us
We first study the existence of ⃗uper . First, we must check that U
⃗
remark that, when g ∈ X and ∇g ∈ X , we have for every A > 0
|(−∆)1/4 g| ≤
Z
A
|(−∆)3/4 Wt ∗ (−∆)1/2 g| dt +
0
Z
+∞
|(−∆)5/4 Wt ∗ g| dt
A
≤C(A1/4 M(−∆)1/2 g + A−1/4 Mg )
so that
|(−∆)1/4 g(x)| ≤ C
q
M(−∆)1/2 g (x)Mg )(x)
and
1/2
1/2
⃗
∥(−∆)1/4 g∥X ≤ C∥g∥X ∥∇g∥
X .
(10.52)
Special Examples of Solutions
303
Similarly, when g ∈ X and ∆g ∈ X , we have for every A > 0
|(−∆)1/2 g| ≤
A
Z
|(−∆)1/2 Wt ∗ (−∆)g| dt +
Z
0
+∞
|(−∆)3/2 Wt ∗ g| dt
A
≤C(A1/2 M∆g + A−1/2 Mg )
so that
|(−∆)1/2 g(x)| ≤ C
q
M∆g (x)Mg )(x)
and
1/2
1/2
∥(−∆)1/2 g∥X ≤ C∥g∥X ∥∆g∥X .
(10.53)
Rt
⃗ 0 = P(V
⃗0 + V
⃗1 ), with V
⃗j =
We write U
Wν(t−s) ∗ f⃗j (s, .) ds and f⃗1 = f⃗per − f⃗0 . We are
−∞
⃗0 and PV
⃗1 belong to E and that ∇
⃗ ⊗V
⃗0 and ∇
⃗ ⊗V
⃗1 belong to L2,∞
going to show that PV
per X .
1 ⃗
1 1 ⃗
⃗
⃗
First, we have V0 = − ν∆ f0 so that ∥PV0 ∥E = ν ∥ ∆ Pf0 ∥X . Moreover, by inequality (10.53),
we have
r
√
∥ 1 f⃗0 ∥X
1
1 ⃗
ν ∥f⃗0 ∥X
⃗
⃗
⃗
√
∥∇ ⊗ V0 ∥X ≤ C
∥f0 ∥X ∥ f0 ∥X ≤ C
(
+ ∆ 2 )
ν
∆
ν
ν
T
⃗0 ∈ F.
Thus, PV
⃗1 on the period interval (0, T ). We write V
⃗1 = V
⃗2 + V
⃗3 with
We now study V
⃗2 (t, x) =
V
Z
t
Wν(t−s) ∗ 1[−T,T ] (s)f⃗1 (s, .) ds
−∞
and
−T
Z
⃗3 (t, x) =
V
Wν(t−s) ∗ f⃗1 (s, .) ds.
−∞
We have
Z
⃗2 | ≤ C
|PV
T
−T
MPf⃗1 (s, x) ds
so that
⃗2 ∥E ≤ C∥f⃗1 ∥L1 X .
∥PV
par
We have also
⃗ ⊗ V
⃗2 | ≤ C
|∇
Z
T
−T
so that
1
p
M ⃗ (s, x) ds
ν(t − s) f1
⃗ ⊗V
⃗2 ∥L2,∞ ((0,T ),X ) ≤ C √1 ∥f⃗1 ∥L1 X .
∥∇
per
ν
⃗3 , we integrate by parts, writing that f⃗1 = ∂t f⃗2 , where f⃗2 =
For V
periodic with f⃗2 (0, .) = 0 and ∥f⃗2 ∥L∞ X ≤ ∥f⃗1 ∥L1per X . Thus, we find that
⃗3 (t, x) =
V
−T
Z
ν∆Wν(t−s) ∗ f⃗2 (s, .) ds.
−∞
This gives
⃗ ⊗V
⃗3 | ≤ C
|∇
Z
−T
−∞
ν
M ⃗ (s, x) ds
(ν(t − s))3/2 f2
Rt
0
f⃗1 (s, .) ds is T -
304
The Navier–Stokes Problem in the 21st Century (2nd edition)
so that
⃗ ⊗V
⃗3 ∥L2,∞ ((0,T ),X ) ≤ C √1 ∥f⃗1 ∥L1 X .
∥∇
per
ν
⃗ ∥ , we need to perform one more integration by parts. We define the
For the control of ∥PV
R 3 E
Rt
1 T ⃗
⃗
mean value f3 = T 0 f2 (s, .) ds, the fluctuation f⃗4 = f⃗2 − f⃗3 and finally f⃗5 = 0 f⃗4 (s, .) ds.
⃗3 = V
⃗4 + V
⃗5 , with
We write V
−T
Z
⃗4 =
V
ν∆Wν(t−s) ∗ f⃗3 ds = −Wν(T +t) ∗ f⃗3 .
−∞
⃗4 (t, x)| ≤ M ⃗ (x) and PV
⃗4 ∈ E. On the other hand, we have
Thus, |PV
Pf3
⃗5 =
PV
−T
Z
(ν∆)2 Wν(t−s) ∗ Pf⃗5 (s, .) ds
−∞
so that
⃗5 (t, x)| ≤ C
|PV
Z
−T
−∞
with |Pf⃗5 (t, x)| ≤ T T1
RT
0
|Pf⃗4 (s, x)| ds =
1
T
1
M ⃗ (s, .) ds
(t − s)2 Pf5
⃗5 (t, x)| ≤ CMf (x) and
f6 (x); thus, |PV
6
⃗5 ∥E ≤ C∥f6 ∥X ≤ C ′ ∥f⃗∥L1 X .
∥PV
per
Thus, we found that
⃗ 0 ∥E +
∥U
1 ⃗
√
⃗ ⊗U
⃗ 0 ∥ 2,∞ ≤ C(∥f⃗∥L1 X + ∥ ∆ f0 ∥X )
ν∥∇
Lper X
per
ν
(10.54)
⃗ and V
⃗ in F. The control of B(U
⃗,V
⃗ ) in E is easy: the proof follows the proof of
Let U
⃗
⃗ (t, x)|, and
Calderón [78]. We write Umax (x) = supt∈R |U (t, x)| and Vmax (x) = supt∈R |V
⃗,V
⃗ )(t, x)| ≤ C
|B(U
Z
t
Z
−∞
R3
t
Z
Z
≤C
R3
π
=
C
2ν
and thus
Z
1
⃗ (t, y)| |V
⃗ (t, y)| dy ds
|U
−
+ |x − y|4
1
ds
Umax (y) Vmax (y) dy
ν 2 (t − s)2 + |x − y|4
ν 2 (t
−∞
s)2
1
Umax (y) Vmax (y) dy
|x − y|2
⃗,V
⃗ )(t, x)| ≤ C √1 (Umax Vmax )(x)
sup |B(U
ν −∆
t∈R
and
⃗,V
⃗ )(t, x)|∥X ≤ C∥ √1 (Umax Vmax )∥X ≤ C ′ 1 ∥Umax ∥X ∥Vmax ∥X
∥ sup |B(U
ν
ν −∆
t∈R
⃗ ⊗ B(U
⃗,V
⃗ ) is a little more delicate. We write
The control of ∇
⃗ = div(U
⃗ ⊗V
⃗ ).
Z
Special Examples of Solutions
Then we have
⃗ ⊗ B(U
⃗,V
⃗ )∥X ≤ C
∥∇
Z
3
X
∥
j=1
305
t
⃗ ds∥X .
∂j Wν(t−s) ∗ Z
−∞
3/2
3 2
2,∞
2
First, we notice that L∞
per X ⊂ (Lt Lx )loc and Lper X ⊂ (Lt Lx )loc , so that we may write
R
1
⃗ =U
⃗ .∇
⃗V
⃗ . We thus have ∥ √ Z∥
⃗ X ≤ C∥U
⃗ ∥X ∥∇
⃗ ⊗V
⃗ ∥X . Let Z
⃗ 0 = 1 T Z(s,
⃗ .) ds be the
Z
T
−∆
0
⃗ We have
mean value of Z.
1 ⃗
1
∥√
Z0 ∥X ≤
T
−∆
Z
T
∥√
0
1 ⃗
1 ⃗
⃗ ⊗V
⃗ ∥L2,∞ X
Z∥X ds ≤ C √ ∥U
∥L∞ X ∥∇
−∆
T
and thus
Z
t
∂j ⃗
1
⃗ ∥L∞ X ∥∇
⃗ ⊗V
⃗ ∥L2,∞ X
Z0 ∥X ≤ C √ ∥U
ν∆
ν
T
−∞
R
⃗ 1 = Z−
⃗ Z
⃗ 0 . We write Z
⃗2 = t Z
⃗ (s, .) ds,
We now look at the contribution of the fluctuation Z
0 1
⃗ 2 is periodical and satisfies Z
⃗ 2 (kT ) = 0 for every k ∈ Z. Thus, for 0 ≤ t < T , we may
then Z
write
Z t
Z t
⃗ 1 ds =
⃗ 1 ) ds
∂j Wν(t−s) ∗ Z
∂j Wν(t−s) ∗ (1[−T,T ] (s)Z
⃗ 0 ds∥X = ∥
∂j Wν(t−s) ∗ Z
∥
−∞
−∞
+∞ Z −kT
X
−
k=1
⃗ 2 ds.
ν∆∂j Wν(t−s) ∗ Z
−(k+1)T
We then write
1
1
1
⃗=
⃗ ).∇
⃗V
⃗)
Z
((
(−∆)1/4 U
(−∆)1/4
(−∆)1/4 (−∆)1/4
and, using inequality (10.52),
∥
1
⃗ 4/3,∞ ≤ C∥U
⃗ ∥1/2
⃗ ⊗U
⃗ ∥1/2
⃗ ⊗V
⃗ ∥ 2,∞ .
Z∥
∥∇
∥∇
L∞
Lper X
L2,∞
per X
per X
(−∆)1/4 Lper X
⃗ 1 . We then write
The same estimate holds for Z
Z t
⃗ 1 ) ds∥X ≤
∥
∂j Wν(t−s) ∗ (1[−T,T ] (s)Z
−∞
Z
t
C
−∞
1
1
⃗ 1 ∥X ds
1[−T,T ] (s)∥
Z
3/4
(ν(t − s))
(−∆)1/4
and, since L4/3,∞ ∗ L4/3,∞ ⊂ L2,∞ , we find that
Z t
⃗ 1 ) ds∥L2,∞ X ≤
∥
∂j Wν(t−s) ∗ (1[−T,T ] (s)Z
−∞
C ⃗ 1/2
⃗ ⊗U
⃗ ∥1/2
⃗ ⊗V
⃗ ∥ 2,∞ .
∥U ∥L∞
∥∇
∥∇
Lper X
L2,∞
per X
per X
ν 3/4
1
1
1/4
⃗
⃗
Finally, we have ∥ (−∆)
∥ (−∆)
1/4 Z2 ∥L∞ X ≤ T
1/4 Z1 ∥L4/3,∞ X and thus, for 0 ≤ t < T ,
per
Z
−T
∥
−∞
⃗ 2 ds∥X ≤ C
ν∆∂j Wν(t−s) ∗ Z
1
ν 3/4 T 3/4
T 1/4 ∥
1
⃗ 1 ∥ 4/3,∞
Z
Lper X
(−∆)1/4
306
The Navier–Stokes Problem in the 21st Century (2nd edition)
We thus have found (for ∥g∥E = ∥ supt>0 |g ′ t, .)|∥X ) that
⃗,V
⃗ )∥E ≤ C 1 ∥U
⃗ ∥E ∥V
⃗ ∥E
∥B(U
ν
and
⃗ ⊗U
⃗ ∥ 2,∞ ∥∇
⃗ ⊗V
⃗ ∥ 2,∞
⃗ ⊗ B(U
⃗,V
⃗ )∥ 2,∞ ≤ C 1 ∥U
⃗ ∥E ∥∇
⃗ ⊗V
⃗ ∥ 2,∞ + C 1 ∥∇
∥∇
Lper X
Lper X
Lper X
Lper X
ν
ν 1/2
⃗ 0 ∥E +ν 1/2 ∥∇⊗
⃗
Picard’s iterative algorithm will then provide a solution ⃗uper as soon as ∥U
⃗ 0 ∥ 2,∞ will be less than C0 ν for a constant C0 which does not depend on ν nor on T .
U
Lper X
Existence of w
⃗ is now easy: just follow the proof of the end of Theorem 10.11.
10.10
Beltrami Flows
In this final section, we pay a few words on Beltrami flows. Beltrami flows have thoroughly been used as examples of incompressible fluid flows for Euler or Navier–Stokes equations [40, 129].
Recall that we may write the Navier–Stokes equations as
⃗ + ν∆⃗u + f⃗,
∂t ⃗u + ω
⃗ ∧ ⃗u = −∇Q
div ⃗u = 0.
Beltrami flows are defined as flows for which vorticity and velocity are parallel:
ω
⃗ ∧ ⃗u = 0.
The Navier–Stokes equations then reduce to linear equations:
(
⃗ + ν∆⃗u + f⃗
∂t ⃗u = −∇Q
⃗ ∧ ⃗u = λ(t, x)⃗u
∇
or
(
∂t ⃗u = ν∆⃗u + Pf⃗
⃗ ∧ ⃗u = λ(t, x)⃗u
∇
(10.55)
(10.56)
The case f⃗ = 0 and λ constant was first discussed by Trkal [300, 477]; the solutions are
labeled as Strong Beltrami flows in [40].
Trkalian flows
Theorem 10.16.
Let ⃗u be a solution to
∂t ⃗u = ν∆⃗u
⃗ ∧ ⃗u = λ⃗u
∇
where λ ̸= 0. Then
• ∆⃗u = −λ2 ⃗u
2
⃗ ∧ ⃗u0 = λ⃗u0
• ⃗u(t, x) = e−νλ t ⃗u0 with ⃗u0 ∈ D′ (R3 ) and ∇
(10.57)
Special Examples of Solutions
307
If ⃗u0 ∈ S ′ , then the equation ∇ ∧ ⃗u0 = λ⃗u0 is equivalent to the existence of a
⃗ ∈ D′ (S2 ) with
distribution A
⃗
σ.A(σ)
=0
and
Z
⃗
⃗
cos(λx.σ)A(σ)
− sin(λx.σ)σ ∧ A(σ)
dσ
⃗u0 =
S2
The latter equality means that
⃗
⟨⃗u0 |⃗
φ⟩S ′ ,S =⟨A|
Z
φ
⃗ (x) cos(λx.σ) dx⟩D′ (S2 ),D(S2 )
Z
⃗
− ⟨σ ∧ A| φ
⃗ (x) sin(λx.σ) dx⟩D′ (S2 ),D(S2 )
⃗ ∧ ⃗u = λ⃗u, we get that div ⃗u = 0. Then, we have
Proof. From ∇
⃗ ∧ (∇
⃗ ∧ ⃗u) = −λ2 ⃗u.
∆⃗u = −∇
2
Thus, ⃗u(t, x) = e−νλ t ⃗u0 .
⃗ 0 of
Now, we have −∆⃗u0 = λ2 ⃗u0 . If ⃗u0 ∈ S ′ (R3 ), we find that the Fourier transform U
⃗u0 is supported on the sphere |ξ| = |λ| and satisfies
⃗ 0 (ξ) = 0.
(|λ| − |ξ|)U
⃗ 0 (ξ) = B(σ)
⃗
Thus, in spherical coordinates ξ = ρσ, we find that U
⊗ δ(ρ − |λ|):
2⃗
⃗
⃗ 0 (ξ)|θ(ξ)⟩
⃗
⟨U
= ⟨B(σ)|λ
θ(|λ|σ)⟩
or equivalently
⟨⃗u0 (x)|⃗
φ(x)⟩ =
1
⃗ 0 (ξ)|
⟨U
(2π)3
Z
φ
⃗ (x)e−ix.ξ dx⟩ =
λ2 ⃗
⟨B(σ)|
(2π)3
Z
φ
⃗ (x)e−i|λ| x.σ dx⟩
⃗ ∧ ⃗u0 = λ⃗u0 , so that iξ ∧ U
⃗ 0 = λU
⃗ 0 and
Moreover, we want ∇
⃗ 0 = (i λ σ ∧ B(σ))
⃗
⃗
U
⊗ δ(ρ − |λ|) = C(σ)
⊗ δ(ρ − |λ|)
|λ|
⃗
⃗ ∧ ⃗u0 = 1 (⃗u0 + 1 ∇
⃗ ∧ ⃗u0 ), so that
where σ.C(σ)
= 0. We want as well ⃗u0 = λ1 ∇
2
λ
λ
⃗ 0 = 1 (C(σ)
⃗
⃗
U
+ i σ ∧ C(σ))
⊗ δ(ρ − |λ|).
2
|λ|
⃗
⃗
⃗ even and F⃗ odd, we have σ.E
⃗ = σ.F⃗ = 0 and we get,
Writing C(σ)
= E(σ)
+ iF⃗ (σ) with E
1 ⃗
λ
⃗
⃗
with D = 2 (E − |λ| σ ∧ F ),
⃗ 0 = (D
⃗ + i λ σ ∧ D)
⃗ ⊗ δ(ρ − |λ|)
U
|λ|
R
⃗
⃗
We then find the decomposition ⃗u0 = S2 cos(λx.σ)A(σ)
− sin(λx.σ)σ ∧ A(σ)
dσ with
2
⃗ = 2λ D(σ).
⃗
A
(2π)3
308
The Navier–Stokes Problem in the 21st Century (2nd edition)
A classical example of Trkalian flow is the flow associated to


C cos(λx3 ) − B sin(λx2 )
⃗u0 = A cos(λx1 ) − C sin(λx3 )
B cos(λx2 ) − A sin(λx1 )
for three constants A, B, C. ⃗u0 is known as the Arnold-Beltrami-Childress flow [155].
We may easily construct other Trkalian flows. For instance, starting from the ax⃗e3 , we obtain the axisymmetric Trkalian flow associated to
isymmetric flow ⃗v = sin(λ|x|)
|x|
1 ⃗
1⃗
⃗u0 = ⃗v + 2 ∇ div ⃗v + ∇ ∧ ⃗v . This field is smooth and belongs to L3,∞ ∩ L∞ .
λ
λ
Chapter 11
Blow-up?
11.1
First Criteria
Throughout this chapter, we shall consider the Navier–Stokes problem
⃗ u + f⃗ − ∇p
⃗
∂t ⃗u = ν∆⃗u − (⃗u · ∇)⃗
div ⃗u = 0
⃗u|t=0 = ⃗u0
(11.1)
where ⃗u0 ∈ (H 1 (R3 ))3 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), (L2 (R3 )3 ).
Recall the results of Theorem 7.2: there exists a (positive) maximal time TMAX ∈ (0, +∞]
for which one can find a mild solution ⃗u of Equation (11.1) on (0, TMAX )×R3 which satisfies,
for all T < TMAX , ⃗u belongs to C([0, T ], (H 1 )3 ) ∩ L2 ((0, T ), (H 2 )3 ).
Definition 11.1 (Blow-up).
If TMAX is finite, we shall say that the solution ⃗u blows up in finite time and that TMAX is
the blow-up time of ⃗u.
Theorem 7.2 gave us some criteria on the possibility of blow-up:
ˆ If TMAX < +∞, then sup0<t<TMAX ∥⃗u(t, .)∥H 1 = +∞.
RT
ˆ If TMAX < +∞, then 0 MAX ∥⃗u(s, .)∥2Ḣ 3/2 ds = +∞
ˆ There exists a positive constant ϵ0 (independent of ν, ⃗u0 and f⃗), such that, if
R +∞
∥⃗u0 ∥Ḣ 1/2 < ϵ0 ν and 0 ∥f⃗(s, .)∥2 − 1 ds < ϵ20 ν 3 , then TMAX = +∞.
Ḣ
2
The Clay Millennium problem is essentially to answer the following question:
Clay Millennium problem for the Navier–Stokes equations
Do we have global existence (i.e., TMAX = +∞) when f⃗ = 0?
11.2
Blow-up for the Cheap Navier–Stokes Equation
Let us recall that the proof of Theorem 7.2 was based on energy estimates:
DOI: 10.1201/9781003042594-11
309
310
The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ the L2 norm of ⃗u(t, .) is estimated by
Z
Z
Z
Z
d
⃗ ⊗ ⃗u|2 dx + 2 ⃗u · f⃗ dx
|⃗u(t, x)|2 dx = 2 ⃗u · ∂t ⃗udx = −2ν |∇
dt
so that
Z
(11.2)
t
∥⃗u(t, .)∥2 ≤ ∥⃗u0 ∥2 +
∥f⃗(s, .)∥2 ds.
(11.3)
0
ˆ similarly, the Ḣ 1 norm of ⃗u(t, .) is estimated by
d
dt
Z
⃗ ⊗ ⃗u(t, x)|2 dx = −2ν
|∇
Z
|∆⃗u|2 dx−2
3 Z
X
⃗ u dx
∂i ⃗u · ((∂i ⃗u) · ∇)⃗
Zi=1
−2 ∆⃗u · f⃗ dx
(11.4)
⃗ ⊗ ⃗u∥2 ∥∥∆⃗u∥2 ∥⃗u∥ 3/2
≤ −ν∥∆⃗u∥22 + C∥∇
Ḣ
≤
1
+ ∥f⃗∥22
ν
C ⃗
1
∥∇ ⊗ ⃗u∥22 ∥⃗u∥2Ḣ 3/2 + ∥f⃗∥22
4ν
ν
so that
∥⃗u∥2Ḣ 1 ≤ (∥⃗u0 ∥2Ḣ 1 +
1
ν
Z
t
C
∥f⃗∥22 ) e 4ν
Rt
0
∥⃗
u(s,.)∥2
Ḣ 3/2
ds
(11.5)
0
In order to underline the role of those energy estimates, Montgomery–Smith studied a
general form of (pseudo)-differential equation
∂t ⃗u = ν∆⃗u + σ(D)(⃗u ⊗ ⃗u)
(11.6)
(generalizing the Navier–Stokes problem ∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u)), where
ˆ ⃗u(t, x) is defined on (0, T ) × R3 with values in Rd
ˆ σ(D) is a matrix of Fourier multipliers σ(ξ) = (σj,(k,l) (ξ)) with d rows and d2 columns,
such that the coefficients σi,j are smooth functions on R3 which are positively homogeneous of order 1: for λ > 0, σj,(k,l) (λξ) = λσj,(k,l) (ξ).
It is easy to check that, in the case of equation (11.6), the proofs of Fujita and Kato’s
theorem (Theorem 7.1) or of Koch and Tataru’s theorem (Theorem 9.2) still work.
If we consider only the Hilbertian setting, one may even deal with a more general class
of equations:
Proposition 11.1.
Let σp (D) (p = 0, 1, 2) be matrices of Fourier multipliers σp (ξ) = (σp,α,β (ξ)) with respectively d rows and d2 columns (p = 0) or d rows and d columns (p = 1 or p = 2). Assume
that the coefficients σp,α,β are locally bounded functions on R3 \ {0} which are positively
homogeneous of order λp with 0 ≤ λp and λ0 + λ1 + λ2 = 1. Let ν > 0. If ⃗u0 is a function
on R3 with values in Rd and if ⃗u0 belongs to H 1 then:
• there exists T > 0 and a function ⃗u defined on [0, T ] × R3 , with values in Rd such that
⃗u ∈ C([0, T ), H 1 ) ∩ L2 ((0, T ), H 2 ) and such that
∂t ⃗u = ν∆ + σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗u)
(11.7)
Blow-up?
311
• The existence time T satisfies
T ≥ Cν
1
.
∥⃗u0 ∥4Ḣ 1
In particular, let TMAX be the maximal existence time. We have blow-up if and only
if sup0<t<TMAX ∥⃗u(t, .)∥H 1 = +∞.
• if the maximal time of existence TMAX is finite, then
Z
TMAX
∥⃗u(t, .)∥2Ḣ 3/2 ds = +∞.
0
• there exists ϵ0 > 0 such that if ∥⃗u0 ∥Ḣ 1/2 < ϵ0 ν then TMAX = +∞.
Proof. We first solve the problem in L4 ((0, T ), Ḣ 1 ). By the product laws in Sobolev spaces,
we have
∥σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗v )∥Ḣ −1/2 ≤C∥σ1 (D)⃗u ⊗ σ2 (D)⃗v ∥
1
Ḣ λ0 − 2
′
≤C ∥σ(D)⃗u∥Ḣ 1−λ1 ∥σ2 (D)⃗v ∥Ḣ λ0 +λ1
≤C ′′ ∥⃗u∥Ḣ 1 ∥⃗v ∥Ḣ 1 .
Thus, if ⃗u and ⃗v belong to L4 ((0, T ), Ḣ 1 ), we find, for
Z
t
Wν(t−s) ∗ σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗v ) ds,
B(⃗u, ⃗v ) =
0
q
q
∥B(⃗u, ⃗v )∥L4 ((0,T ),Ḣ 1 ) ≤ ∥B(⃗u, ⃗v )∥L∞ ((0,T ),Ḣ 1/2 ) ∥B(⃗u, ⃗v )∥L2 ((0,T ),Ḣ 3/2 )
≤Cν −3/4 ∥σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗v )∥L2 ((0,T ),Ḣ −1/2 )
≤C ′ ν −3/4 ∥⃗u∥L4 ((0,T ),Ḣ 1 ) ∥⃗v ∥L4 ((0,T ),Ḣ 1 ) .
⃗ 0 = Wνt ∗ ⃗u0 and U
⃗ n+1 = U
⃗ 0 + B(U
⃗ n, U
⃗ n ) will converge to a
Thus, the Picard iterates U
3/4
ν
4
1
⃗ 0∥ 4
solution in L ((0, T ), Ḣ ) as long as ∥U
L ((0,T ),Ḣ 1 ) ≤ 4C ′ , hence if
T ≤
ν3
.
∥⃗u0 ∥4Ḣ 1 (4C ′ )4
Similarly, we have
√
√
∥B(⃗u, ⃗u)∥L∞ H 1 + ν∥B(⃗u, ⃗u)∥L2 Ḣ 1 + ν∥B(⃗u, ⃗u)∥L2 Ḣ 2
≤Cν −1/2 ∥σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗u)∥L2 ((0,T ),Ḣ −1 )
+ Cν −1/2 ∥σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗u)∥L2 ((0,T ),L2 )
≤Cν −1/2 ∥⃗u∥L4 Ḣ 1 (∥⃗u∥L4 ((0,T ),Ḣ 1/2 ) + ∥⃗u∥L4 ((0,T ),Ḣ 3/2 ) )
√
√
≤C ′ ν −3/4 ∥⃗u∥L4 Ḣ 1 (∥⃗u∥L∞ H 1 + ν∥⃗u∥L2 Ḣ 1 + ν∥⃗u∥L2 Ḣ 2 ).
Thus, we find that the solution will belong to L∞ ((0, T ), H 1 ) ∩ L2 ((0, T ), Ḣ 2 ).
∗
Moreover, we have, for 0 < T0 < t < TMAX
,
∥⃗u(t, .)∥Ḣ 1 ≤ ∥⃗u(T0 , .)∥Ḣ 1 + Cν −3/4 sup ∥⃗u(s, .)∥Ḣ 1 ∥⃗u∥L2 ((T0 ,t),Ḣ 3/2 ) .
T0 <s<t
312
The Navier–Stokes Problem in the 21st Century (2nd edition)
RT
This gives that, if 0 MAX ∥⃗u(s, .)∥2Ḣ 3/2 ds < +∞, then ∥⃗u∥Ḣ 1 remains bounded on (0, TMAX ),
so that TMAX = +∞.
Finally, we write
√
∥B(⃗u, ⃗u)∥L∞ Ḣ 1/2 + ν∥B(⃗u, ⃗u)∥L2 Ḣ 3/2
≤Cν −1/2 ∥σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗u)∥L2 ((0,T ),Ḣ −1/2 )
√
≤Cν −1 sup ∥⃗u∥Ḣ 1/2 ν∥⃗u∥L2 ((0,T ),Ḣ 3/2 ) .
0<s<T
⃗ 0∥ ∞
Thus, if ∥U
L ((0,T ),Ḣ 1/2
√ ⃗
+ ν∥U
0 ∥L2 ((0,T ),Ḣ 3/2 <
∥⃗u∥L∞ ((0,T ),Ḣ 1/2 +
ν
4C ′ ,
we have
√
ν
ν∥⃗u∥L2 ((0,T ),Ḣ 3/2 <
.
2C ′
In particular, if ⃗u0 is small enough in Ḣ 1/2 , we find that TMAX = +∞.
The Navier–Stokes equations may be writen in the form of equations (11.7) in two ways.
⃗ The Leray projection operator P may be written as Pf⃗ = R
⃗ ∧ (R
⃗ ∧ f⃗).
⃗ = √ 1 ∇.
Let R
−∆
From the equations
⃗ u − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
we get
⃗ ∧ (R
⃗ ∧ div(⃗u ⊗ ⃗u)) = ν∆⃗u + σ0 (D)(⃗u ⊗ ⃗u)
∂t ⃗u = ν∆⃗u − R
⃗ ∧ (R
⃗ ∧ div):
where σ0 (D) is the matrix of Fourier multipliers associated with −R
σ0,j,(k,l) (ξ) = −iδj,l ξk − iξj
ξk ξl
.
|ξ|2
On the other hand, from the equations
⃗
∂t ⃗u = ν∆⃗u − ω
⃗ ∧ ⃗u − ∇q,
we get
⃗ ∧ (R
⃗ ∧ (⃗
∂t ⃗u = ν∆⃗u − R
ω ∧ ⃗u)) = ν∆⃗u + σ0 (D)((σ1 (D)⃗u ⊗ ⃗u)
where σ0 (D) is the matrix of Fourier multipliers described by the cycle γ : 1 → 2 → 3 → 1
as:
σ0,j,(k,l) (ξ)
= −δk,γ(j) δ
l,γ 2 (j)
+δ
k,γ 2 (j)
δl,γ(j) +
3
X
ξj ξq
q=1
|ξ|2
(−δk,γ(q) δl,γ 2 (q) + δk,γ 2 (q) δl,γ(q) )
⃗
and σ1 (D) = ∇∧:

0
σ1 (ξ) = i  ξ3
−ξ2
−ξ3
0
ξ1

ξ2
−ξ1  .
0
Many problems of the form (11.7) have been studied as models for blow ups (or no blow
up):
• Montgomery–Smith proved
√ blow-up in the case of the cheap Navier–Stokes equation
where d = 1 and σ(D) = −∆ [369]:
√
∂t u = ν∆ + −∆(u2 ).
We will describe below the result of Montgomery–Smith.
Blow-up?
313
• The cheap equation has been recently adapted by Gallagher and Paicu [200] into a
vector equation (d = 3) which preserves the divergence-free condition:
∂t ⃗u = ν∆⃗u + PQ(u, u) = ν∆⃗u + σ(D)(⃗u ⊗ ⃗u)
with
σj,(k,l) (ξ) = 1E (ξ)
1
(|ξ|2 − ξk ξl δj,l )
|ξ|
and
E = {ξ / ξ1 ξ2 < 0, ξ1 ξ3 < 0, |ξ2 | < min(|ξ1 |, |ξ2 |)}.
The key point is the fact that, similarly to the case of the cheap equation, when the
components of the Fourier transform of ⃗u0 are non-negative then the components of
the Fourier transform of the solution ⃗u remain non-negative.
• If we look for a complex–valued solution of the Navier–Stokes problem
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u),
it is equivalent to find real-valued solutions (⃗v , w)
⃗ of the system
∂t⃗v = ν∆⃗v − P div(⃗v ⊗ ⃗v − w
⃗ ⊗ w)
⃗
∂t w
⃗ = ν∆w
⃗ − P div(⃗v ⊗ w
⃗ +w
⃗ ⊗ ⃗v )
which is of the form (11.7). Blow-up for this equation has been proved by Li and Sinai
[334] in a difficult paper based on tools in renormalization group theory and on the
theory of linear hydrodynamic instability.
• On the other hand, Wang [496] gave an example where no blow up occurs, namely
the equations
⃗ ∧ (⃗
⃗ ∧ ⃗u) = ν∆u + σ0 (D)(σ1 (D)⃗u ⊗ σ2 (D)⃗u)
∂t ⃗u = ν∆⃗u − R
ω ∧ (R
with
σ0,j,(k,l) (ξ) = i(
and

0
σ1 (ξ) = i  ξ3
−ξ2
−ξ3
0
ξ1
ξk
ξl
δj,k −
δj,l )
|ξ|
|ξ|

ξ2
1
−ξ1  , σ2 (ξ) =
σ1 (ξ).
|ξ|
0
We now present the result of Montgomery–Smith (blow up), then the result of Wang
(no blow up):
Cheap Navier–Stokes equation
Theorem 11.1.
There exists a positive constant Aν such that if u is a solution of
√
∂t u = ν∆u + −∆(u2 )
with u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and u(0, .) = u0 satisfies
314
The Navier–Stokes Problem in the 21st Century (2nd edition)
• u0 ∈ H 1
• the Fourier transform û0 is non-negative
R
• for some ξ0 ∈ R3 with |ξ0 | = 1, |ξ−ξ0 |<1/3 |û0 (ξ)| dξ > Aν
then we have
TMAX ≤ 1.
Proof. First, we check that û ≥ 0. Indeed, this is true at time t = 0. Let T0 = sup{T ≥
0 / û ≥ 0 on [0, T ] × R3 }. If T0 < TMAX , then, by continuity, we find that û(T0 , .) ≥ 0.
Moreover, there exists a small time T1 such that on [T0 , T0 + T1 ] u may be constructed
by Picard’s iterative scheme. It is easy to check that every Picard iterate has its Fourier
transform non-negative, and so does their limit u; thus, we find û ≥ 0 on [0, T0 + T1 ], in
contradiction with the definition of T0 . Thus, T0 = TMAX .
We now start from Duhamel’s formula
Z t
2
−νt|ξ|2
û(t, ξ) = e
û0 (ξ) +
e−ν(t−s)|ξ| |ξ| (û(s, .) ∗ û(s..))(ξ) ds
0
Let w0 (ξ) = 1B(ξ0 ,1/3) û0 (ξ). Define wn (ξ) by induction as wn+1 = wn ∗ wn . wn is then
supported in B(2n ξ0 , 13 2n ) ⊂ {ξ / 2n+1 < 3|ξ| < 2n+2 }. Thus, if û(t, ξ) ≥ αn (t)wn (ξ), we
have
Z
+∞
X
|ξ|2 (û(t, ξ))2 dξ ≥
4n+1 ∥wn (ξ)∥22 αn (t)2
n=0
On the other hand, we have
n
A2ν
Z
≤
wn (ξ) dξ ≤ C0 23n/2 ∥wn ∥2
and thus
Z
|ξ|2 (û(t, ξ))2 dξ ≥
+∞
4 X −n 2n+1
2
Aν αn (t)2 .
C02 n=0
2
We now turn to the estimation of αn (t). As û(t, ξ) ≥ e−νt|ξ| û0 (ξ), we find that
16
α0 (t) ≥ e− 9 νt .
Further, we have
Z
αn+1 (t) ≥
min
2n+1 <3|ξ|<2n+2
t
e
−ν(t−s)|ξ|2
|ξ|αn2 (s) ds
0
2
≥ 2n
3
Z
t
0
Now, we define
βn =
min
1−4−n ≤t≤1
αn (t).
For 1 − 4−n−1 ≤ t ≤ 1, we have
2
αn+1 (t) ≥ 2n
3
Z
t
t−4−n−1
16
n
16
n
e− 9 ν(t−s)4 αn2 (s) ds
e− 9 ν(t−s)4 αn2 (s) ds
Blow-up?
315
which gives
1 −4ν 2
e 9 βn .
6
βn+1 ≥
Recall that
Z
|ξ|2 (û(1, ξ))2 dξ ≥
+∞
4 X −n 2n+1 2
2
Aν βn .
C02 n=0
n
n+1
Assume that Aν > 2; then 2−n A2ν ≥ 1, so that 2−n A2ν
n+1
A2ν
2
βn+1
≥
n
βn2 ≥ A2ν βn2 ; moreover
1 − 8 ν 2n 2 2
e 9 Aν βn
36
n
40
8
Hence, if Aν > 36 e 9 ν , we find by induction on n that A2ν βn2 > 36 e 9 ν and finally that
∥u(1, .)∥H 1 = +∞. Thus TMAX ≤ 1.
Proposition 11.2.
Let ⃗u be the solution in C([0, TMAX , H 1 ) ∩ ∩T <TMAX L2 ([0, T ], Ḣ 2 ) of the Cauchy problem for
the equations
⃗ ∧ (⃗
⃗ ∧ ⃗u)))
∂t ⃗u = ν∆⃗u − R
ω ∧ (R
(where ω
⃗ = curl ⃗u) with initial value ⃗u0 ∈ H 1 . Then TMAX = +∞.
Proof. We have
Z
√
d
∥⃗u(t, .)∥2Ḣ 1/2 =2 ∂t ⃗u · −∆⃗u dx
dt
= − 2ν∥⃗u∥2Ḣ 3/2 ,
since
Z
√
⃗ ∧ (⃗
⃗ ∧ ⃗u)) dx =
−∆⃗u · R
ω ∧ (R
Z
⃗ ∧ ⃗u)) dx = 0.
ω
⃗ · (⃗
ω ∧ (R
Thus, we have
Z
0
TMAX
∥⃗u(s, .)∥2Ḣ 3/2 ds ≤
1
∥⃗u0 ∥Ḣ 1/2 .
2ν
In Wang’s example, we have as well energy conservation:
d
∥⃗u∥22 = −2ν∥⃗u∥2Ḣ 1 .
dt
This is not necessary: we could have dealt with the equation
⃗ ∧ (⃗
∂t ⃗u = ν∆⃗u − R
ω ∧ ⃗u)).
On the other hand, Tao [462] considered the problem of blow-up in presence of energy
conservation. More precisely, he considered the abstract problem ∂t ⃗u = ν∆⃗u − B(⃗u, ⃗u),
where B would mimick the operator BN S (⃗u, ⃗v ) = P div(⃗u ⊗ ⃗v ) on three points:
ˆ div B(⃗u, ⃗v ) = 0
⃗ ⊗ ⃗v ∥4 + ∥⃗v ∥4 ∥∇
⃗ ⊗ ⃗u∥4 )
ˆ ∥B(⃗u, ⃗v )∥L2 (R3 ≤ C(∥⃗u∥4 ∥∇
R
ˆ B(⃗u, ⃗u).⃗u dx = 0 for ⃗u ∈ H 2 with div ⃗u = 0
He constructed an example of such an operator B for which blow-up occurs, thus invalidating
the abstract Hilbertian approach of Otelbaev [389].
316
11.3
The Navier–Stokes Problem in the 21st Century (2nd edition)
Serrin’s Criterion
Recall that the proof of Theorem 7.2 relied on the differential equalities
Z
Z
Z
d
⃗ ⊗ ⃗u|2 dx + 2 ⃗u · f⃗ dx
|⃗u(t, x)|2 dx = −2ν |∇
dt
(11.8)
and
d
dt
Z
⃗ ⊗ ⃗u(t, x)|2 dx = −2ν
|∇
Z
|∆⃗u|2 dx + 2
Z
Z
−2
⃗ u) dx
∆⃗u · (⃗u · ∇⃗
(11.9)
∆⃗u · f⃗ dx
The first one allows to control the L2 norm:
Z
∥⃗u(t, .)∥2 ≤ ∥⃗u0 ∥2 +
t
∥f⃗(s, .)∥2 ds.
(11.10)
0
The second one aims to control the Ḣ 1 norm.
Serrin [435] gave a very simple criterion to ensure the control of the Ḣ 1 norm of ⃗u
through Equation (11.9):
Serrin’s criterion
Theorem 11.2.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), L2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . Then, if
2/p + 3/q = 1 with 2 ≤ p < +∞, we have
∥⃗u(T, .)∥2Ḣ 1 ≤ (∥⃗u0 ∥2Ḣ 1 +
R
1−p T
1 ⃗ 2
∥⃗
u∥p
q dt
0
∥f ∥L2 L2 )eC0 ν
ν
(11.11)
where the constant C0 does not depend on T .
In particular, if the maximal existence time TMAX satisfies TMAX < +∞, then
R TMAX
∥⃗u∥pq dt = +∞.
0
Proof. From (11.9), we get
d
⃗ u∥2 + 2∥∆⃗u∥2 ∥f⃗∥2
∥⃗u∥2Ḣ 1 ≤ −2ν∥∆⃗u∥22 + 2 ∥∆⃗u∥2 ∥⃗u · ∇⃗
dt
We then write for
1
r
=
1
2
−
1
q
and σ = 3( 21 − 1r ) = 1 − p2 :
⃗ u∥2 ≤ ∥⃗u∥q ∥∇
⃗ ⊗ ⃗u∥r ≤ C∥⃗u∥q ∥∇
⃗ ⊗ ⃗u∥ σ ≤ C∥⃗u∥q ∥⃗u∥1−σ ∥∆⃗u∥σ
∥⃗u · ∇⃗
2
Ḣ
Ḣ 1
and thus
2
1+σ
1+σ
1
d
1
∥⃗u∥2Ḣ 1 ≤ ∥f⃗∥22 + Cσ ν − 1−σ ∥⃗u∥q1−σ ∥⃗u∥2Ḣ 1 = ∥f⃗∥22 + Cσ ν − 1−σ ∥⃗u∥pq ∥⃗u∥2Ḣ 1
dt
ν
ν
and we conclude by Grönwall’s lemma.
Blow-up?
317
As we shall see in Chapter 15, a theorem by Escauriaza, Seregin and Šverák [163] proves
that the endpoint case p = +∞, q = 3 of the Serrin criterion holds: if TMAX < +∞, then
sup0<t<TMAX ∥⃗u(t, .)∥3 = +∞.
A former result of Kozono and Sohr [275] stated that, if TMAX < +∞ and ⃗u remained
bounded in L3 as t → TMAX , then there was a discontinuity of ∥⃗u∥3 at time TMAX : there
exists a positive constant γ such that
lim sup ∥⃗u(t, .) − ⃗u(TMAX , .)∥3 ≥ γν
−
t→TMAX
Indeed, we split ⃗u(TMAX , .) into ⃗v + w,
⃗ where ⃗v ∈ L∞ and ∥w∥
⃗ 3 is small. We get
⃗ u∥2 ≤ (∥⃗u − ⃗u(TMAX , .)∥3 + w∥
⃗ ⊗ ⃗u∥6 + ∥⃗v ∥∞ ∥⃗u∥ 1
∥⃗u · ∇⃗
⃗ 3 )∥∇
Ḣ
so that
⃗ u∥2 ≤ C0 (∥⃗u − ⃗u(TMAX , .)∥3 + w∥
2 ∥∆⃗u∥2 ∥⃗u · ∇⃗
⃗ 3 )∥∆⃗u∥22 +
If we choose w
⃗ such that ∥w∥
⃗ 3<
that on (T1 , TMAX ) we have
ν
4C0
2
ν
∥∆⃗u∥22 + ∥⃗v ∥2∞ ∥⃗u∥2Ḣ 1 .
2
ν
and if supT1 <t<TMAX ∥⃗u − ⃗u(TMAX , .)∥3 <
ν
4C0 ,
we find
1
d
2
∥⃗u∥2Ḣ 1 ≤ ∥f⃗∥22 + ∥⃗v ∥2∞ ∥⃗u∥2Ḣ 1 .
dt
ν
ν
Grönwall’s lemma then gives the control on the Ḣ 1 norm of ⃗u, which is in contradiction
with TMAX < +∞.
Theorem 11.2 has been generalized to the setting of Besov spaces, with the condition
σ
with 1 ≤ p ≤ +∞, −1 ≤ σ ≤ +1 and p2 = 1 + σ. The case 2 < p < +∞
⃗u ∈ Lp Ḃ∞,∞
was treated by Kozono and Shimada [274]; the case 1 < p ≤ 2 may be found in the
paper by Chen and Zhang [116]; the case p = +∞ has been first discussed by May [354]
as a generalization of the result of Kozono and Sohr (see also the more recent paper of
Cheskidov and Shvydkoy [120]). The case p = 1 goes back to the criterion of Beale, Kato
RT
and Majda [27] which stated TMAX < +∞ ⇒ 0 /rmM AX ∥ curl ⃗u∥∞ dt = +∞; the L∞ norm
was replaced by the weaker norm ∥ curl ⃗u∥BM O by Kozono and Taniuchi [278], then by the
still weaker norm ∥ curl ⃗u∥Ḃ 0
by Kozono, Ogawa and Taniuchi [273].
∞,∞
Serrin’s criterion and Besov spaces
Theorem 11.3.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), L2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . Then
• if 1 < p < +∞, −1 < σ < 1 and
2
p
= 1 + σ, we have
C0 ν 1−p 0T ∥⃗
u∥p σ
dt
1 ⃗ 2
Ḃ∞,∞
(11.12)
∥f ∥L2 L2 )e
ν
where the constant C0 does not depend on T .
In particular, if the maximal existence time TMAX satisfies TMAX < +∞, then
R TMAX
∥⃗u∥pḂ σ dt = +∞.
0
R
∥⃗u(T, .)∥2Ḣ 1 ≤ (∥⃗u0 ∥2Ḣ 1 +
∞,∞
318
The Navier–Stokes Problem in the 21st Century (2nd edition)
• case p = +∞: there exists a positive constant γ such that, if TMAX <
−1
+∞ and if sup0<t<TMAX ∥⃗u(t, .)∥Ḃ∞,∞
< +∞, then lim supt→T − ∥⃗u(t, .) −
MAX
−1
⃗u(TMAX , .)∥Ḃ∞,∞
≥ γν.
• case p = 1: we have
u∥Ḟ 1
dt
C0 T ∥⃗
1
∞,2
(11.13)
≤
+ ∥f⃗∥2L2 L2 )e 0
ν
where the constant C0 does not depend on T .
In particular, if the maximal existence time TMAX satisfies TMAX < +∞, then
R TMAX
∥ curl ⃗u∥BM O dt = +∞.
0
∥⃗u(T, .)∥2Ḣ 1
R
(∥⃗u0 ∥2Ḣ 1
• case p = 1 (continued): if f⃗ ∈ L2 H 2 , then ⃗u ∈ C((0, TMAX , H 3 ). Moreover we
have, for any δ ∈ (0, TMAX ) and δ < T < Tmax
∥⃗u(T, .)∥2Ḣ 3
≤ C0 e
R
C0 δT ∥⃗
u∥ 1
dt
Ḃ∞,∞
(∥⃗
u(δ,)∥2H 3 + ν1 ∥f⃗∥2L2 H 2 )e
(11.14)
where the constant C0 does not depend on T .
In particular, if the maximal existence time TMAX satisfies TMAX < +∞, then
R TMAX
∥ curl ⃗u∥Ḃ 0 dt = +∞.
0
∞,∞
Proof. The proof is based on the Littlewood–Paley decomposition1 and on the use of paraproducts. If u and v belong to L2 , we write
X
X
X X
uv =
Sj−2 u∆j v +
Sj−2 v∆j u +
∆j u∆k v
j∈Z
j∈Z
j∈Z |k−j|≤2
where ∆j u is the j-th dyadic block of the Littlewood–Paley decomposition of ⃗u: the Fourier
transform F(∆j u) is given by
F(∆j u)(ξ) = ψ(
ξ
)û(ξ)
2j
where ψ is a smooth function supported in {ξ / 12 ≤ |ξ| ≤ 2} and such that, for ξ =
̸ 0,
P
P
P
(ξ
ψ(
))
=
1,
while
S
u
=
∆
u.
The
term
π(u,
v)
=
S
u∆
v
is
called
j
k
j
j∈Z
k<j
j∈Z j−2
2j
P
P
the paraproduct of u and v; we shall write R(u, v) = + j∈Z |k−j|≤2 ∆j u∆k v, so that
uv = π(u, v) + π(v, u) + R(u, v).
(11.15)
The important point is that the constituents of π(u, v) and of R(u, v) are localized in
frequency variable: the support of F(Sj−2 u∆j v) is contained in {ξ / 14 2j ≤ |ξ| ≤ 94 2j }
while, for |k − j| ≤ 2, the support of F(∆j u∆k v) is contained in {ξ / |ξ| ≤ 10 2j }.
1 See
Lemarié-Rieusset [313] or Bahouri, Chemin and Danchin [15] for definitions and notations.
Blow-up?
319
Case 1 < p < +∞:
We start from
Z
Z
Z
d
⃗ ⊗ ⃗u(t, x)|2 dx = −2ν |∆⃗u|2 dx−2 ∆⃗u · f⃗ dx
|∇
dt
3 Z
X
⃗ u dx
−2
∂i ⃗u · ((∂i ⃗u) · ∇)⃗
(11.16)
i=1
σ
and we estimate ∂i u∂j v∂k w dx for u, v, w ∈ Ḣ 1 ∩ Ḣ 2 ∩ Ḃ∞,∞
, with −1 < σ < 1. Let
2r = 3 − σ, so that 1 < r < 2. We then write
Z
| ∂i u∂j v∂k w dx| ≤C∥∂i u∥Ḣ r−1 (∥π(∂j v, ∂k w∥Ḣ 1−r + ∥π(∂k w, ∂j v∥Ḣ 1−r )
R
σ−1 ∥R(∂j v, ∂k w)∥ 1−σ .
+ C∥∂i u∥Ḃ∞,∞
Ḃ
1,1
We have 1 − r = r + σ − 2 and σ − 1 < 0 so that
X
∥π(∂j v, ∂k w)∥2Ḣ 1−r ≤C
22j(r+σ−2) ∥Sj−2 (∂j v)∥2∞ ∥∆j (∂k w)∥22
j∈Z
X
≤C sup 22j(σ−1) ∥Sj−2 (∂j v)∥2∞
j∈Z
≤C
′
22j(r−1) ∥∆j (∂k w)∥22
j∈Z
∥∂j v∥2Ḃ σ−1 ∥∂k w∥2Ḣ r−1
∞,∞
≤C ′ ∥v∥2Ḃ σ
∞,∞
∥w∥2Ḣ r .
We have, of course, the similar estimate
∥π(∂k w, ∂j v)∥2Ḣ 1−r ≤ C∥w∥2Ḃ σ
∞,∞
∥v∥2Ḣ r .
On the other hand, we have 1 − σ = 2r − 2 and 1 − σ > 0, so that
X
X
∥R(∂j v, ∂k w)∥Ḃ 1−σ ≤C
2j(2r−2) ∥
∆j (∂j v)∆k (∂k w)∥1
1,1
j∈Z
|k−j|≤2
X
X
≤C (
22j(r−1) ∥∆j (∂j v)∥22 )1/2 (
22j(r−1) ∥∆j (∂k w)∥22 )1/2
′
j∈Z
j∈Z
′′
≤C ∥∂j v∥Ḣ r−1 ∥∂k w∥Ḣ r−1
≤C ′′ ∥v∥Ḣ r ∥w∥Ḣ r .
Thus, we find
|2
3 Z
X
⃗ u dx| ≤C∥⃗u∥ σ
∂i ⃗u.((∂i ⃗u) · ∇)⃗
Ḃ
∞,∞
∥⃗u∥2Ḣ r
i=1
≤C∥⃗u∥Ḃ σ
∥⃗u∥4−2r
∥⃗u∥2r−2
Ḣ 1
Ḣ 2
=C∥⃗u∥Ḃ σ
∥⃗u∥1+σ
∥⃗u∥1−σ
Ḣ 1
Ḣ 2
∞,∞
∞,∞
(11.17)
This gives
2
1+σ
d
1
1
∥⃗u∥2Ḣ 1 ≤ ∥f⃗∥22 + Cσ ν − 1−σ ∥⃗u∥Ḃ1−σ
∥⃗u∥2Ḣ 1 = ∥f⃗∥22 + Cσ ν 1−p ∥⃗u∥pḂ σ ∥⃗u∥2Ḣ 1
σ
∞,∞
∞,∞
dt
ν
ν
and we conclude by Grönwall’s lemma.
320
The Navier–Stokes Problem in the 21st Century (2nd edition)
Case p = +∞:
For σ = −1, we get a similar estimate
|2
3 Z
X
⃗ u dx| ≤ C∥⃗u∥ −1 ∥⃗u∥2 2
∂i ⃗u · ((∂i ⃗u) · ∇)⃗
Ḃ∞,∞
Ḣ
(11.18)
i=1
but the end of the proof would work only if ⃗u was small enough to grant that
−1
(−ν + C∥⃗u∥Ḃ∞,∞
)∥∆⃗u∥22 < 0.
In Theorem 11.3, ⃗u is not assumed to be small, but only to have a small jump at time
t = TMAX . Let us make this statement more precise. First, we assume that
sup
0<t<TMAX
−1
∥⃗u(t, .)∥Ḃ∞,∞
< +∞.
We shall see in Theorem 12.2 that the Navier–Stokes problem with initial value ⃗u0 and
forcing term f⃗ admits global weak Leray solutions; moreover, from Theorem 12.3, those
weak solutions will coincide with ⃗u on (0, TMAX ).
In particular, the map t 7→ ⃗u(t, .) can be extended as a map from [0, +∞) to L2 which is
−
weakly continuous. Thus, ⃗u(t, .) has a limit ⃗u(TMAX , .) when t → TMAX
(if TMAX < +∞).
−1
−1
Moreover, as Ḃ∞,∞
is a dual space, we find that ⃗u(TMAX , .) ∈ Ḃ∞,∞
and
−1
−1
∥⃗u(TMAX , .)∥Ḃ∞,∞
≤ lim −inf ∥⃗u(t, .)∥Ḃ∞,∞
.
t→TMAX
Now, we want to prove that, if TMAX < +∞, then
−1
lim sup ∥⃗u(t, .) − ⃗u(TMAX , .)∥Ḃ∞,∞
≥ γν.
−
t→TMAX
−1
Indeed, let ϵ = lim supt→T − ∥⃗u(t, .) − ⃗u(TMAX , .)∥Ḃ∞,∞
, and let η > ϵ. There is an interval
MAX
−1
[T0 , TMAX) on which ∥⃗u(t, .) − ⃗u(TMAX , .)∥Ḃ∞,∞
< η. Moreover, chossing T0 > 0, ⃗u(T0 , .) ∈
−1
H 2 ⊂ L∞ , while ∥⃗u(t, .) − ⃗u(T0 , .)∥Ḃ∞,∞
< 2η on (T0 , TMAX ).
R
The next step is to estimate ∂i u∂j v∂k w dx for u, v, w ∈ Ḣ 1 ∩ Ḣ 2 , with u = u1 + u2 ,
−1
u1 ∈ L∞ and u2 ∈ Ḃ∞,∞
, and the same for v = v1 + v2 and w = w1 + w2 . We write
Z
|
∂i u∂j v∂k w dx| ≤C∥∂i u∥2 (∥π(∂j v1 , ∂k w∥2 + ∥π(∂k w1 , ∂j v∥2 )
+C∥∂i u∥Ḣ 1 (∥π(∂j v2 , ∂k w∥Ḣ −1 + ∥π(∂k w2 , ∂j v∥Ḣ −1 )
+ C∥u1 ∥∞ ∥∂i R(∂j v, ∂k w)∥1
−2
+ C∥∂i u2 ∥Ḃ∞,∞
∥R(∂j v, ∂k w)∥Ḃ 2
1,1
≤C ′ ∥u∥Ḣ 1 (∥v1 ∥∞ ∥w∥Ḣ 2 + ∥w1 ∥∞ ∥v∥Ḣ 2 )
−1
−1
+C ′ ∥u∥Ḣ 2 (∥v2 ∥Ḃ∞,∞
∥w∥Ḣ 2 + ∥w2 ∥Ḃ∞,∞
∥v∥Ḣ 2 )
+ C ′ ∥u1 ∥∞ (∥v∥Ḣ 1 ∥w∥Ḣ 2 + ∥v∥Ḣ 2 ∥w∥Ḣ 1 )
−1
+ C ′ ∥u2 ∥Ḃ∞,∞
∥v∥Ḣ 2 ∥w∥Ḣ 2 .
Thus, we find on (T0 , TMAX ):
|2
3 Z
X
i=1
⃗ u dx| ≤C0 η∥⃗u∥2 2 + C∥⃗u(T0 , .)∥∞ ∥⃗u∥ 1 ∥⃗u∥ 2
∂i ⃗u · ((∂i ⃗u) · ∇)⃗
Ḣ
Ḣ
Ḣ
Blow-up?
321
and thus
1
ν
1
d
∥⃗u∥2Ḣ 1 ≤ ∥f⃗∥22 + C ∥⃗u(T0 , .)∥2∞ ∥⃗u∥2Ḣ 1 + (− + C0 η)∥⃗u∥2Ḣ 2 .
dt
ν
ν
2
ν
If η < 2C
, we may apply Grönwall’s lemma and get that ⃗u remains bounded in Ḣ 1 ,
0
which contradicts TMAX < +∞.
Case p = 1:
For σ = 1, both estimates
∥π(∂j v, ∂k w)∥22 ≤ C∥∂j v∥2Ḃ 0
∞,∞
∥∂k w∥22
and
∥R(∂j v, ∂k w)∥Ḃ 0 ≤ C∥∂j v∥2 ∥∂k w∥2 .
1,1
fail. However, we may use the div-curl lemma of Coifman, Lions Meyer and Semmes [124]
since div ⃗u = 0 and write
⃗ u∥H1 ≤ C∥⃗u∥2 1
∥∂j ⃗u · ∇⃗
Ḣ
0
where H1 is the Hardy space (whose dual is BM O = Ḟ∞,2
). Thus, we get
|2
3 Z
X
⃗ u dx| ≤ C∥⃗u∥ 1 ∥⃗u∥2 2
∂i ⃗u · ((∂i ⃗u) · ∇)⃗
Ḟ
Ḣ
∞,2
(11.19)
i=1
and we conclude by Grönwall’s lemma.
Case p = 1 (continued):
Let f⃗ ∈ L2 H 2 . For every 0 < δ < T < TMAX , we know that ⃗u will belong to C([T0 , T ], H 3 ) ∩
L2 ((T0 , T ), H 4 ).
We want to estimate
Z
2
=
|(−∆)3/2 ⃗u|2 dx.
∥⃗u∥Ḣ
3
We write
d
(∥⃗u∥2Ḣ 3 ) =2⟨(−∆)3 ⃗u|∂t ⃗u⟩H −3 ,H 3
dt
⃗ u⟩H −2 ,H 2
= −2ν∥⃗u∥2Ḣ 4 − 2⟨(−∆)3 ⃗u|⃗u · ∇⃗
+2⟨(−∆)3 ⃗u|f⃗⟩H −2 ,H 2
P
We have (−∆)3 = |α|=3 cα ∂ α ∂ α . Integration by parts and Leibnitz rule give then
Z
Z
X
⃗ u) dx =
⃗ γ ⃗u) dx.
∂ α ∂ α ⃗u · (⃗u · ∇⃗
cβ,γ ∂ α ⃗u · (∂ β ⃗u · ∇∂
β+γ=α
As div ⃗u = 0, we have
Z
⃗ α ⃗u) dx = 0.
∂ α ⃗u · (⃗u · ∇∂
Thus, we need to estimate integrals
Z
Iα,β,γ (u, v, w) =
∂ α u ∂ β v ∂ γ w dx
1
with |α| = 3, |β| + |γ| = 4 and |β| ≥ 1, |γ ≥ 1, for u, v, w ∈ Ḣ 3 ∩ Ḃ∞,∞
.
(11.20)
322
The Navier–Stokes Problem in the 21st Century (2nd edition)
We have, for |β| = 1,
|Iα,β,γ (u, v, w)| ≤ ∥u∥Ḣ 3 ∥∂ β v∥∞ ∥w∥Ḣ 3 .
For |β| = 2, ∂ β = ∂i ∂j , we use the Gagliardo–Nirenberg inequality ∥∂ β v∥4 ≤
1/2
1/2
C∥∂i v∥∞ ∥∂i v∥Ḣ 2 . This inequality is easily established through Hedberg’s inequality: write
R +∞
∂ β v = − 0 Wt ∗ ∆∂ β v dt; then write
1
1
|Wt ∗ ∆∂ β v)(x)| ≤ C min( √ M∆∂i v (x), 3/2 ∥∂i v∥∞ )
t
t
to get
|∂ β v(x)| ≤ C∥∂i v∥1/2
∞
p
M∆∂i v (x).
Thus, we get
⃗ 1/2 ∥v∥ 3 ∥∇w∥
⃗ 1/2 ∥w∥ 3 .
|Iα,β,γ (u, v, w)| ≤ C∥u∥Ḣ 3 ∥∇v∥
∞
∞
Ḣ
Ḣ
Finally, we get
⃗ u⟩H −2 ,H 2 | ≤ C∥∇
⃗ ⊗ ⃗u∥∞ ∥⃗u∥2 3
|⟨(−∆)3 ⃗u|⃗u · ∇⃗
Ḣ
We then use the logarithmic Sobolev inequality
⃗ ⊗ ⃗u∥∞ ≤ C ∥⃗u∥2 + 1 + ∥⃗u∥ 1
∥∇
Ḃ
∞,∞
ln(e + ∥⃗u∥2Ḣ 3 )
To establish this well-known inequality, just write ∂j v = −
R +∞
0
(11.21)
Wt ∗ ∆∂j v dt, with
1
1
1
∥v∥Ḣ 3 , ∥v∥Ḃ 1 , 9/4 ∥v∥2 );
∞,∞ t
t
t3/4
R1 1
R +∞ 1
≤ ∥v∥Ḃ 1 , conclude by integrating 0 t3/4
∥v∥Ḣ 3 dt + 1
∥v∥2 dt; if ∥v∥Ḣ 3 >
t9/4
∞,∞
4
∥v∥Ḃ 1
∞,∞
, let A =
and integrate
∥v∥ 3
|(Wt ∗ ∆∂j v)(x)| ≤ C min(
if ∥v∥Ḣ 3
∥v∥Ḃ 1
∞,∞
Ḣ
A
Z
0
t
Z +∞
1
1
∥v∥Ḃ 1 dt +
∥v∥2 dt.
9/4
∞,∞
t
t
A
1
∥v∥
(1 + ln+ ∥v∥ 1Ḣ 3
); finally, if ∥v∥Ḃ 1
Z
1
∥v∥Ḣ 3 dt +
3/4
1
We obtain ∥∂j v∥∞ ≤ C(∥v∥2 + ∥v∥Ḃ 1
∞,∞
∥v∥
ln+ ∥v∥ 1Ḣ 3
≤ ln(e + ∥v∥Ḣ 3 ), if ∥v∥Ḃ 1
∞,∞
B∞,∞
∥v∥Ḃ 1
∞,∞
+
ln
∥v∥Ḣ 3
1
∥v∥B∞,∞
∞,∞
B∞,∞
≥ 1, write
≤ min(1, ∥v∥Ḣ 3 ), write
!
=∥v∥Ḃ 1
∞,∞
(ln(∥v∥Ḣ 3 − ln(∥v∥Ḃ 1
∞,∞
))
1 1
≤ + ∥v∥Ḃ 1 ln(e + ∥v∥2Ḣ 3 )
∞,∞
e 2
Thus far, we have obtained
d
1
1
(∥⃗u∥2Ḣ 3 ) ≤ ∥f⃗∥2H 2 + C(1 + ∥⃗u∥2 )∥⃗u∥2Ḣ 3 + C∥⃗u∥B∞,∞
∥⃗u∥2Ḣ 3 ln(e + ∥⃗u∥2Ḣ 3 )
dt
ν
If Φ(t) = ln(e + ∥⃗u∥2Ḣ 3 ), we obtain
d
1
1
Φ ≤ ∥f⃗∥2H 2 + C(1 + ∥⃗u∥2 ) + C∥⃗u∥B∞,∞
Φ(t).
dt
ν
We then conclude by applying Grönwall’s lemma.
Blow-up?
323
σ
In spite of the maximality of the Besov spaces Ḃ∞,∞
(if E is a Banach space of distributions such that its norm if shift-invariant [ ∥f (x − x0 )∥E = ∥f ∥E ] and homogeneous
σ
[ ∥f (λx)∥E = λσ ∥|f ∥E ], then E ⊂ Ḃ∞,∞
), Theorem 11.3 may still be extended to some
criteria based on weaker norms. For instance, Planchon [400] discussed how to replace the
RT
1
1
norm in L1 Ḃ∞,∞
(∥⃗u∥L1 ((0,T ),Ḃ 1 ) = 0 supj∈Z 2j ∥∆j ⃗u∥∞ dt) by the norm in L˜1 Ḃ∞,∞
∞,∞
R
T
s
(∥⃗u∥L˜1 ((0,T ),Ḃ 1 ) = supj∈Z 2j 0 ∥∆j ⃗u∥∞ dt). Spaces L˜p Ḃq,r
often occur in critical esti∞,∞
mates for the Navier–Stokes equations, since the seminal paper of Chemin and Lerner [113]
on quasi-Lipschitz flows.
Another way of extending Serrin’s criterion is the remark by Montgomery-Smith [370]
that the proof of Theorem 11.3 is based on the Grönwall lemma applied to (sub)linear
estimates on the Ḣ 1 norm of ⃗u, while Grönwall’s lemma applies to a slightly larger class of
estimates:
Lemma 11.1 (Grönwall’s lemma).
If u ≥ 0 is defined on [0, T ) and satisfies
Z
u(t) ≤ a0 +
t
Φ(u(s))ω(s) ds
0
with Φ ≥ 0 a non-decreasing function such that
Z +∞
dt
= +∞
Φ(t)
1
and ω ≥ 0 with ω ∈ L1 ((0, T )), then sup0<t<T u(t) ≤ A∗ , where A∗ is defined by
RT
R A∗
ds
= 0 ω(s) ds.
max(1,a0 ) Φ(s)
Rt
d
Proof. Let a1 = max(1, a0 ) and define A(t) = a1 + 0 Φ(u(s))ω(s) ds. We have dt
A =
d
d
Rt
R A(t) ds
R t dt
R t dt
A
A
ωΦ(u) ≤ ωΦ(A), so that 0 Φ(A) ds ≤ 0 ω(s) ds. We write 0 Φ(A) ds = a1 Φ(s) . Thus,
we find sup0<t<T u(t) ≤ sup0<t<T A(t) ≤ A∗ .
We may now state Montgomery–Smith’s result:
Proposition 11.3.
Let 1 < p < +∞ and σ ∈ (−1, 1) with p2 = 1 + σ. Let Θ ≥ 1 be a non-decreasing function
R +∞ ds
1
on (0, +∞) such that 1
sΘ(s) ds = +∞. Let k ∈ {0, 1, 2} with k > σ + 2 .
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H k ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . If the maximal
p
R TMAX ∥⃗u∥Ḃ∞,∞
σ
existence time TMAX satisfies TMAX < +∞, then 0
Θ(∥⃗
u∥ σ
) dt = +∞.
Ḃ∞,∞
Proof. For every 0 < T0 < T < TMAX , we know that ⃗u will belong to C([T0 , T ], H k+1 ) ∩
L2 ((T0 , T ), H k+2 ).
We start from
Z
2
∥⃗u∥Ḣ k+1 = |(−∆)(k+1)/2 ⃗u|2 dx
324
The Navier–Stokes Problem in the 21st Century (2nd edition)
so that
d
(∥⃗u∥2Ḣ k+1 ) =2⟨(−∆)k+1 ⃗u|∂t ⃗u⟩H −k ,H k
dt
⃗ u⟩H −k ,H k
= −2ν∥⃗u∥2Ḣ k+2 − 2⟨(−∆)k+1 ⃗u|⃗u · ∇⃗
(11.22)
+2⟨(−∆)k+1 ⃗u|f⃗⟩H −k ,H k
Integration by parts and Leibnitz rule give then, for |α| = k + 1,
Z
Z
X
⃗ u) dx =
⃗ γ ⃗u) dx
∂ α ∂ α ⃗u · (⃗u · ∇⃗
cβ,γ ∂ α ⃗u · (∂ β ⃗u · ∇∂
β+γ=α
As div ⃗u = 0, we have
Z
⃗ α ⃗u) dx = 0.
∂ α ⃗u · (⃗u · ∇∂
Thus, we need to estimate integrals (in the sense of duality brackets)
Z
Iα,β,γ (u, v, w) = ∂ α u ∂ β v ∂ γ w dx
σ
with |α| = k + 1, |β| + |γ| = k + 2 and |β| ≥ 1, |γ ≥ 1, for u, v, w ∈ Ḣ k+1 ∩ Ḣ k+2 ∩ Ḃ∞,∞
,
with −1 < σ < 1.
Let 2r = 2k + 3 − σ, so that k + 1 < r < k + 2. We then write
|Iα,β,γ (u, v, w)| ≤C∥∂ α u∥Ḣ r−k−1 (∥π(∂ β v, ∂ γ w)∥Ḣ 1+k−r + ∥π(∂ γ w, ∂β v∥Ḣ 1+k−r )
β
γ
+ C∥∂ α u∥Ḃ∞,∞
σ−k−1 ∥R(∂ v, ∂ w)∥ 1+k−σ .
Ḃ
1,1
We have 1 + k − r = r + σ − k − 2 and σ − |β| < 0 so that
X
∥π(∂ β v, ∂ γ w)∥2Ḣ 1+k−r ≤C
22j(r+σ−k−2) ∥Sj−2 (∂ β v)∥2∞ ∥∆j (∂ γ w)∥22
j∈Z
≤C sup 22j(σ−|β|) ∥Sj−2 (∂ β v)∥2∞
j∈Z
′
≤C ∥∂
β
X
22j(r−|γ|) ∥∆j (∂ γ w)∥22
j∈Z
v∥2Ḃ σ−|β| ∥∂ γ w∥2Ḣ r−|γ|
∞,∞
≤C ′ ∥v∥2Ḃ σ
∞,∞
∥w∥2Ḣ r .
We have, of course, the similar estimate
∥π(∂ γ w, ∂ β v)|2Ḣ 1+k−r ≤ C∥w∥2Ḃ σ
∞,∞
∥v∥2Ḣ r .
On the other hand, we have 1 + k − σ = 2r − k − 2 and 1 + k − σ > 0, so that
X
X
∥R(∂ β v, ∂ γ w)∥Ḃ 1+k−σ ≤C
2j(2r−k−2) ∥
∆j (∂ β v)∆k (∂ γ w)∥1
1,1
j∈Z
|k−j|≤2
X
X
≤ C ′ ( 22j(r−|β|) ∥∆j (∂ β v)∥22 )1/2 (
22j(r−|γ|) ∥∆j (∂ γ w)∥22 )1/2
j∈Z
j∈Z
′′
β
γ
≤C ∥∂ v∥Ḣ r−|β| ∥∂ w∥Ḣ r−|γ|
≤C ′′ ∥v∥Ḣ r ∥w∥Ḣ r .
Blow-up?
325
This gives
d
1
∥⃗u∥2Ḣ k+1 ≤ ∥f⃗∥2H k + Cσ ν 1−p ∥⃗u∥pḂ σ ∥⃗u∥2Ḣ k+1
∞,∞
dt
ν
Next, we write
∥⃗u∥Ḃ σ
∞,∞
≤ Dσ ∥⃗u∥H k+1 ≤ Dσ (∥⃗u∥2 + ∥⃗u∥Ḣ k+1 ).
If TMAX < +∞ and A0 = sup0<t<TMAX ∥⃗u(t, .)∥2 , we find that
∥⃗u∥Ḃ σ
∞,∞
≤ 2Dσ
A20 + ∥⃗u∥2Ḣ k+1
so that
Θ(∥⃗u∥Ḃ σ
∞,∞
Let us write B(t) = 2Dσ
A20 +∥⃗
u∥2Ḣ k+1
A0
2Dσ
B(t) ≤ B(T0 ) +
A0 ν
TMAX
Z
0
) ≤ Θ(2Dσ
A0
A20 + ∥⃗u∥2Ḣ k+1
A0
).
, for 0 < T0 < t < TMAX , We have
1−p
ν
∥f⃗∥2H k ds + Cσ
A0
t
∥⃗u∥pḂ σ
T0
Θ(∥⃗u∥Ḃ σ
Z
∞,∞
∞,∞
)
B(s)Θ(B(s)) ds
and we conclude by Grönwall’s lemma.
Montgomery–Smith’s result paved the way to numerous works on “logarithmic improvements” of Serrin’s criterion. Many of them were quite uninspired, but some of them were
very interesting. For instance, we shall describe Chan and Vasseur’s result [103] (in a slighly
more general statement):
Proposition 11.4.
Let 1 < p < +∞, 3 < q < +∞ with p2 + 3q = 1. Let Θ ≥ 1 be a non-decreasing function on
R +∞ ds
(0, +∞) such that 1
sΘ(s) ds = +∞.
1
Let ⃗u0 ∈ H with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 1 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . If the maximal
RT
existence time TMAX satisfies TMAX < +∞, then 0 MAX ∥ Θ(|⃗u⃗u|1/p ) ∥pṀ 2,q dt = +∞.
Proof. For every 0 < T0 < T < TMAX , we know that ⃗u will belong to C([T0 , T ], H 2 ) ∩
L2 ((T0 , T ), H 3 ). Thus, ⃗u is bounded on (T0 , T ) × R3 . We are going to prove that
∥⃗u(t, .)∥∞ ≤Cν (∥⃗u(T0 , .)∥∞ + (1 +
Z
p
T − T0 )(
t
∥f⃗(s, .)∥2H 1 ds)1/2 )
T0
Z
(11.23)
T
+ Cν
T0
∥⃗u∥p+1
Ṁ 2(p+1)/p,(p+1)q/p
ds
This inequality may be proved in a very simple way: we write
Z t
Z t
⃗
⃗u(t, .) = Wν(t−T0 ) ∗ ⃗u(T0 , .) +
Wν(t−s) ∗ Pf ds −
Wν(t−s) ∗ P div(⃗u ⊗ ⃗u) ds.
T0
T0
326
The Navier–Stokes Problem in the 21st Century (2nd edition)
We have, on (T0 , T )
sup ∥Wν(t−T0 ) ∗ ⃗u(T0 , .)∥∞ ≤ ∥⃗u(T0 , .)∥∞
T0 <t<T
and
Z
t
Wν(t−s) ∗ Pf⃗ ds∥∞ ≤ Cν (1 +
sup ∥
T0 <t<T
Z
p
T − T0 )(
T0
T
∥f⃗∥2H 1 ds)1/2
0
The key point is the estimation of the last term
Z t
I(t) =
Wν(t−s) ∗ P div(⃗u ⊗ ⃗u) ds.
T0
We have
1
∥Wν(t−s) ∗ P div(⃗u ⊗ ⃗u)∥∞ ≤ C p
∥⃗u(s, .)∥2∞
ν(t − s)
and
∥Wν(t−s) ∗ P div(⃗u ⊗ ⃗u)∥∞ ≤
C
3p
1
2 + q(p+1)
(ν(t − s))
∥⃗u(s, .)∥2
Ṁ
2(p+1) (p+1)q
,
p
p
so that, for every positive A, we have
Z t
1
|I(t)| ≤C
1s>T0 p
∥⃗u(s, .)∥2∞ ds
ν(t − s)
t−A
Z t−A
1
+C
1s>T0
∥⃗u(s, .)∥2Ṁ 2(p+1)/p,(p+1)q/p ds
3p
1
−∞
(ν(t − s)) 2 + q(p+1)
√
≤Cν A ( sup ∥⃗u(s, .)∥∞ )2
T0 <s<t
Z
+∞
+Cν (
A
1
3p
p+1
( 12 + q(p+1)
) p−1
ds)
2
1− p+1
Z
t
(
(t − s)
T0
2
∥⃗u(s, .)∥p+1
ds) p+1
Ṁ 2(p+1)/p,(p+1)q/p
The first thing now is to check that
1
3p
p+1
( +
)
> 1.
2 q(p + 1) p − 1
But, recalling that
3
q
= 1 − p2 , we find that ( 12 +
3p
p+1
q(p+1) ) p−1
= 32 . Thus, we find that
√
p
sup |I(t)| ≤ Cν A( sup ∥⃗u∥2∞ + A− p+1 ∥⃗u∥2Lp+1 Ṁ 2(p+1)/p,(p+1)q/p ).
T0 <t<T
T0 <t<T
Optimizing the choice of A, we get
sup |I(t)| ≤ Cν ( sup ∥⃗u∥∞ )
T0 <t<T
T0 <t<T
p−1
p
(∥⃗u∥Lp+1 Ṁ 2(p+1)/p,(p+1)q/p )
p+1
p
so that, by Young’s inequality,
sup |I(t)| ≤
T0 <t<T
1
sup ∥⃗u∥∞ + Cν′ (∥⃗u∥Lp+1 Ṁ 2(p+1)/p,(p+1)q/p )p+1
2 T0 <t<T
Thus, we have proved (11.23).
Blow-up?
327
We now easily finish the proof. Recall that
∥u∥Ṁ 2(p+1)/p,(p+1)q/p ≈
sup
x∈R3 ,ρ>0
ρ
3(1− q2 )
and
∥v∥Ṁ 2,q ≈
∥u∥
Ṁ
2(p+1) (p+1)q
,
p
p
≈
ρ
x∈R3 ,ρ>0
|u(y)|2 dy
B(x,ρ)
Z
1
ρ
3(1− q2 )
1
p+1
≤ ∥u∥∞ ∥Θ(|u|)∥∞
|u|2/p Θ(|u|)2/p |v|2 dy
1
sup
ρ
3(1− q2 )
and thus
∥⃗u∥p+12(p+1) , (p+1)q ≤ C∥⃗u∥∞ Θ(∥⃗u∥∞ )∥
Ṁ
p/(2(p+1))
B(x,ρ)
x∈R3 ,ρ>0
p
1/2
we find
sup
1
p+1
p/(2(p+1))
B(x,ρ)
3(1− q2 )
x∈R3 ,ρ>0
|u(y)|2(p+1)/p dy
Z
1
sup
|u|
,
Θ(|u|)1/p
In particular, writing v =
Z
1
p
Z
|v(y)|2 dy
p/(2(p+1))
B(x,ρ)
⃗u
∥p .
Θ(|⃗u|)1/p Ṁ 2,q
(11.24)
From inequalities (11.23) and (11.24), and from Grönwall’s lemma, we conclude that, if
R TMAX
∥ Θ(|⃗u⃗u|)1/p ∥pṀ 2,q < +∞, then ∥⃗u(t, .)∥∞ remains bounded on (T0 , TMAX ), so that ⃗u ∈
T0
L2 ((T0 , TMAX , L∞ ); but this contradicts TMAX < +∞.
11.4
A Remark on Serrin’s Criterion and Leray’s Criterion
Serrin’s criterion [435] is a very simple criterion to ensure the control of the Ḣ 1 norm
of ⃗u through Equation (11.9). However, if the force is slightly more regular, it is a simple
consequence of a remark done by Leray in his 1934 paper [328]:
Leray’s criterion
Theorem 11.4.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L∞ ((0, +∞), L2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with u ∈ C([0, T ], H 1 )∩L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . If q > 3, there
exists a positive constant Cq such that, if the maximal existence time TMAX satisfies
TMAX < +∞, then
1
3
lim inf (T ∗ − T ) 2 (1− q ) ∥⃗u(T, .)∥q > Cq .
T →TMAX
328
The Navier–Stokes Problem in the 21st Century (2nd edition)
Similarly, Kozono and Shimada’s criterion [274] can be treated through Leray’s criterion:
Leray’s criterion and Besov spaces
Theorem 11.5.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L∞ ((0, +∞), L2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 .
If 0 < σ < 1, there exists a positive constant Cσ such that, if the maximal existence
time TMAX satisfies TMAX < +∞, then
1
−σ
> Cσ .
lim inf (T ∗ − T ) 2 (1−σ) ∥⃗u(T, .)∥Ḃ∞,∞
T →TMAX
−σ
, then
Proof. If ⃗u(T, .) ∈ Ḃ∞,∞
−σ .
sup tσ/2 ∥et∆ ⃗u(T, .)∥∞ ≤ C∥⃗u(T, .)∥Ḃ∞,∞
t>0
On the other hand,
sup tσ/2 ∥
0<t<t0
Z
t
σ
e(t−s)∆ Pf⃗(T + s, .) ds∥∞ ≤ Ct02
+ 14
∥f⃗∥L∞ L2
0
(where the constant C does not depend on t0 ). We have
σ/2
sup t
Z
∥
t
e(t−s)∆ P(⃗u(T + s, .) ⊗ ⃗v (T + s, .)) ds∥∞
0<t<t0
0
1−σ
2
σ/2
≤ Ct0
sup t
∥⃗u(T + t, .)∥∞ sup tσ/2 ∥⃗v (T + t, .)∥∞
0<t<t0
0<t<t0
This gives a solution ⃗vT of the Cauchy problem for the Navier–Stokes problem on [T, T + t0 ]
with initial value ⃗u(T, .), provided that
1−σ
σ
−σ
Cσ t0 2 (∥⃗u(T, .)∥Ḃ∞,∞
+ t02
+ 14
∥f⃗∥L∞ L2 ) ≤ 1.
By uniqueness of mild sollutions, this solution ⃗vT will coincide with ⃗u on [T, min(T ∗ , T +t0 )).
This implies, if T ∗ < +∞, that t0 < T ∗ − T , as ⃗u(t, .) cannot remain bounded when t is
approaching T ∗ . Thus,
1 ≤ Cσ (T ∗ − T )
11.5
1−σ
2
σ
1
−σ
(∥⃗u(T, .)∥Ḃ∞,∞
+ (T ∗ − T ) 2 + 4 ∥f⃗∥L∞ L2 ).
Some Further Generalizations of Serrin’s Criterion
Let us review what we have seen so far.
Blow-up?
329
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), L2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . Let us assume
that the maximal existence time TMAX satisfies TMAX < +∞. Then
RT
• Serrin proved in 1963 [435] that, for 2/p + 3/q = 1 with 2 ≤ p < ∞, 0 MAX ∥⃗u∥pq dt =
+∞.
Escauriaza, Seregin and Šverák [163] proved in 2003 that the endpoint case p = +∞,
q = 3 of the Serrin criterion holds.
• Serrin’s criterion was extended in 2004 by Kozono and Shimada [274] who proved that
R TMAX
∥⃗u∥pḂ σ dt = +∞ with 2/p = 1 + σ, 2 < p < +∞ and −1 < σ < 0 (note that
0
∞,∞
−3
q
Lq ⊂ Ḃ∞,∞
).
• Beirão da Vega [29] proved in 1995 that, for 2/p+3/r = 2 with 1 < p < ∞,
⃗u∥pr dt = +∞.
R TMAX
0
⃗
∥∇⊗
⃗ ⊗ ⃗u ∈
• For 2 < p, Beirão da Vega’s criterion is a consequence of Serrin’s criterion, as ∇
p r
p q
L L ⇒ ⃗u ∈ L L . This is no longer the case for p = 2, as we do not have the
⃗ ⊗ ⃗u ∈ L2 L3 ⇒ ⃗u ∈ L2 L∞ , but only the implication ∇
⃗ ⊗ ⃗u ∈ L2 L3 ⇒
implication ∇
2
⃗u ∈ L BM O. However, this was generalized in 2000 by Kozono and Taniuchi [278]
RT
who proved that 0 MAX ∥⃗u∥2BM O dt = +∞.
• Beirão da Vega’s criterion was fully generalized in 2008 by Chen and Zhang [116] who
RT
proved that 0 MAX ∥⃗u∥pḂ σ dt = +∞ with 2/p = 1 + σ, 1 < p < +∞ and −1 < σ < 1
∞,∞
σ
⃗ ⊗ ⃗u ∈ Lp Lr =⇒ ⃗u ∈ Lp Ḃ∞,∞
with σ = 1 − 3r ).
(remark that ∇
RT
• In 1984, Beale, Kato and Majda [27] proved that 0 MAX ∥ curl ⃗u∥∞ dt = +∞; this was
RT
⃗ ⊗ ⃗u∥BM O = +∞ and
generalized in 2000 by Kozono and Taniuchi [278] to 0 MAX ∥∇
finally (if f⃗ ∈ L2 H 1 ) in 2002 by Kozono, Ogawa and Taniuchi [273] who proved that
R TMAX
∥⃗u∥Ḃ 1 dt = +∞.
0
∞,∞
One could think that every possible criteria have been proposed (up to logarithmic
⃗ ⊗ ⃗u. However, several generalizations of
improvements) in terms of the size of ⃗u or of ∇
Beirão da Vega’s criterion were proposed:
RT
⃗ 3 ∥pr dt = +∞ with 2 + 3 = 3 and 2 ≤ p <
ˆ control of just one component: 0 MAX ∥∇u
p
r
2
+∞ (Neustupa, Novotný and Penel [374], He [231], Pokorný [404], Zhou [511])
RT
ˆ control of just one derivative: 0 MAX ∥∂3 ⃗u∥pr dt = +∞ with p2 + 3r = 2 and 2 ≤ p ≤ 3
(Penel and Pokorný [394], Kukavica and Ziane [289])
R
1
2
3
p
⃗ u), TMAX ∥∇ϖ∥
⃗
ˆ control of the pressure: for ϖ = − ∆
div(⃗u·∇⃗
q dt = +∞ with p + q = 3
0
2
and 3 < p < +∞ (Berselli and Galdi [39], Zhou [513, 512], Struwe [456]).
We are going to prove those three results. (Of course, they can be “logarithmically
improved” in the spirit of Proposition 11.3.) Notice that recently Cao and Titi [88] have
studied a global regularity criterion involving just one entry ∂i uj of the velocity gradient
tensor.
330
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proposition 11.5.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 1 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . If the maximal
RT
⃗ 3 ∥pr dt = +∞ with 2 + 3 = 3
existence time TMAX satisfies TMAX < +∞, then 0 MAX ∥∇u
p
r
2
and 2 ≤ p < +∞.
Proof. As f⃗ ∈ L2 L2 , we know that, for every 0 < T0 < T < TMAX , ⃗u will belong to
C([T0 , T ], H 1 )∩L2 ((T0 , T ), H 2 ). Thus, we may estimate at each positive time t the quantities
R
Z ⃗
( |ω3 |2 dx)2
|∇ ⊗ ⃗u|2
I(t) =
dx and J(t) = 1 +
2
4
where ω
⃗ = curl ⃗u. We introduce as well two other quantities
⃗ 3 ∥22 .
A(t) = ∥∆⃗u∥22 and B(t) = ∥ω3 ∥22 ∥∇ω
We shall first study J, in order to explain the choice of the exponent. Recall that we
have ⃗u ∈ L∞ L2 ∩ L2 Ḣ 1 :
Z TMAX
∥⃗u(t, .)∥2 ≤ ∥⃗u0 ∥2 +
∥f⃗∥2 dx
(11.25)
0
and
Z
TMAX
⃗ ⊗ ⃗u(s, .)∥22 ds ≤
∥∇
0
1
(∥⃗u0 ∥2 +
ν
We shall write
Z
TMAX
N0 = ∥⃗u0 ∥2 +
Z
TMAX
∥f⃗∥2 dx)2
(11.26)
0
∥f⃗∥2 dx.
0
In order to estimate J, we write and
⃗ω+ω
⃗ u + curl f⃗
∂t ω
⃗ = ν∆⃗
ω − ⃗u · ∇⃗
⃗ · ∇⃗
which gives
d
J(t) =∥ω3 ∥22
dt
Z
ω3 ∂t ω3 dx
Z
Z
2
2
2
⃗
= − ν∥ω3 ∥2 |∇ω3 | dx + ∥ω3 ∥2 f1 ∂2 ω3 − f2 ∂1 ω3 dx
Z
Z
⃗ 3 dx + ∥ω3 ∥22 ω3 ω
⃗ 3 dx
− ∥ω3 ∥22 ω3 ⃗u · ∇ω
⃗ · ∇u
As div ⃗u = 0, we have
Thus, we find
R
1/2
3r
2r−3
1/2
⃗ 3 dx = 0. Moreover, we have ∥ω∥3 ≤ C∥⃗u∥ ∥∆⃗u∥ .
ω3 ⃗u · ∇ω
2
Ḣ 1
d
⃗ 3 ∥22 + ∥ω3 ∥22 ∥∇ω
⃗ 3 ∥2 ∥f⃗∥2
J(t) ≤ − ν∥ω3 ∥22 ∥∇ω
dt
⃗ 3 ∥r ∥⃗u∥1/2 ∥∆⃗u∥1/2 ∥ω3 ∥ 3r
+ C∥ω3 ∥22 ∥∇u
2
Ḣ 1
2r−3
with 2 ≤
(11.27)
< 6. We have the interpolation inequality, for ρ ∈ [2, 6],
3
∥ω3 ∥ρ ≤ ∥ω3 ∥2ρ
− 12
3
∥ω3 ∥62
3
−ρ
Blow-up?
so that
3
3r
≤ ∥ω3 ∥22
∥ω3 ∥ 2r−3
331
− r3
3
∥ω3 ∥6r
− 12
.
This gives (since J ≥ 1, hence J 1/4 ≤ J 1/2 )
√
1
1
d
J(t) ≤ − νB(t) + 2B(t) 2 J(t) 2 ∥f⃗∥2
dt
⃗ 3 ∥r ∥⃗u∥1/2 A(t) 41 B(t) 2r3 − 41 J 1− 2r3
+ C∥∇u
Ḣ 1
(11.28)
3
3
− 14 ) + (1 − 2r
) = 1, we shall be able to use Grönwall’s lemma.
As ( 41 ) + ( 2r
Now, we rewrite the Navier–Stokes equations as
2
⃗ + |⃗u| )
∂t ⃗u = ν∆⃗u − ω
⃗ ∧ ⃗u + f⃗ − ∇(p
2
(with div ⃗u = 0) and obtain
Z
d
I(t) = − ∆⃗u · ∂t ⃗u dx
dt
Z
Z
Z
= − ν |∆⃗u|2 dx − ∆⃗u · f⃗ dx + ∆⃗u · (⃗
ω ∧ ⃗u) dx
(11.29)
which we expand into
Z
Z
d
2
I(t) = − ν |∆⃗u| dx − ∆⃗u · f⃗ dx
dt
Z
+ ∆u1 ω2 u3 − ∆u1 ω3 u2 dx
Z
+ ∆u2 ω3 u1 − ∆u2 ω1 u3 dx
Z
+ ∆u3 ω1 u2 − ∆u3 ω2 u1 dx.
R
u3 appears everywhere except in the term ∆u2 ω3 u1 − ∆u1 ω3 u2 dx. For this last term, we
may replace the control on u3 by the control on ω3 (given by (11.28)).
⃗ 3 would have been 2 + 3 = 2 as for Beirão da Vega’s criterion
The natural scaling for ∇u
p
r
d
[29]. As a matter of fact, in our decomposition of dt
I(t), the scaling p2 + 3r = 2 would be
enough to control the terms involving u3 ; the scaling p2 + 3r = 32 is necessary only for the
terms involving ω3 .
We thus define p10 = 12 ( p1 + 12 ) and r10 = 21 ( 1r + 12 ), so that p20 + r30 = 2. We define Xr0
as Lr0 if 2 < r < 6 (hence 2 < r0 < 3) and L3,1 if r = 6 (r0 = 3). Notice that:
ˆ 2 < r ≤ 6, hence 2 < r0 ≤ 3
ˆ Xr0 ⊂ Lr0
p0
p0
RT
R
R
⃗ 3 ∥ 2 dt, and thus TMAX
⃗ 3 ∥p0 dt ≤ C TMAX ∥∇u
⃗ 3 ∥ 2 ∥∇u
ˆ we have 0 MAX ∥∇u
2
X
Xr 0
0
0
r
p
R TMAX
N0 p40
⃗ 3 ∥p0 dt ≤ C( √
⃗ 3 ∥pr dt) 2p0
∥∇u
)
(
∥
∇u
Xr
0
ν
0
⃗ 3 ∈ Lp0 Xr gives u3 ∈ Lp0 Lq0 with
ˆ the assumption ∇u
0
1
q0
=
1
r0
− 31 .
332
The Navier–Stokes Problem in the 21st Century (2nd edition)
We then write
Z
Z
∆u3 ω1 u2 − ∆u3 ω2 u1 dx = −
⃗ 3 · ∇(ω
⃗ 1 u2 − ω2 u1 ) dx
∇u
and
d
I(t) ≤ − ν∥∆⃗u∥22 + ∥∆⃗u∥2 ∥f⃗∥2
dt
⃗ ⊗ ⃗u∥ 2q0 ∥u3 ∥q + C∥∆⃗u∥2 ∥ω3 ∥3 ∥⃗u∥6
+ C∥∆⃗u∥2 ∥∇
0
q0 −2
⃗ 3 ∥r ∥∆⃗u∥2 ∥⃗u∥
+ C∥∇u
0
0
Notice that 2 ≤ q2q
< 3, 6 ≤
0 −2
interpolation inequalities:
2r0
r0 −2
2r0
r0 −2
⃗ 3 ∥r ∥∇
⃗ ⊗ ⃗u∥22r0
+ C∥∇u
0
r0 −1
< +∞ and 3 ≤
2r0
r0 −1
< 4. We then use the following
3
3
3
− 14
⃗ ⊗ ⃗u∥ρ ≤ CI(t) 2ρ
ˆ for ρ ∈ [2, 6], ∥∇
A(t) 4 − 2ρ
6
6
1− σ
ˆ for σ ∈ [6, +∞], ∥⃗u∥σ ≤ ∥⃗u∥6σ ∥⃗u∥∞
1
1
1
3
1
3
≤ CI(t) 4 + 2σ A(t) 4 − 2σ
1
ˆ ∥ω3 ∥3 ≤ ∥ω3 ∥22 ∥ω3 ∥62 ≤ CB(t) 4
1
1
1
1
ˆ ∥⃗u∥6 ≤ C∥⃗u∥Ḣ 1 ≤ C∥⃗u∥22 ∥∆⃗u∥22 ≤ C∥⃗u∥22 A(t) 4
This gives
1
d
I(t) ≤ − νA(t) + A(t) 2 ∥f⃗∥2
dt
3
⃗ 3 ∥X A(t) 2r0 I(t)1− 2r3 0
+ C∥∇u
r0
1
2
3
(11.30)
1
+ C∥u(t, .)∥2 A(t) 4 B(t) 4
Thus, we obtain
d
ν
1
I(t) ≤ − A(t) + C1 ∥f⃗∥22
dt
2
ν
2r0
3
⃗ 3 ∥ 2r0 −3 ν − 2r0 −3 I(t)
+ C1 ∥∇u
Xr
0
+ C1 N02
(with
2r0
2r0 −3
1
B(t)
ν3
= p0 ) and (for ϵ > 0)
d
ν
1
J(t) ≤ − B(t) + C J(t)∥f⃗∥22
dt
2
ν
4
2
⃗ 3 ∥r3 ∥⃗u∥ 3 B(t) r2 − 13 J(t) 43 − r2
+ ϵ4 νA(t) + Cϵ−4/3 ν −1/3 ∥∇u
Ḣ 1
≤−
ν
1
B(t) + C2 J(t)∥f⃗∥22
4
ν
2r
r
r
2r
⃗ 3 ∥r2r−3 ∥⃗u∥ 2r−3 J(t)
+ ϵ4 νA(t) + C2 ϵ− 2r−3 ν − 2(2r−3) ∥∇u
Ḣ 1
Let
K(t) = I(t) + λJ(t).
Blow-up?
We take λ =
4C1 N02
ν4
and ϵ4 =
1
2λ .
333
We obtain
d
1
⃗ 3 ∥p0 ν 1−p0 K(t) + C2 1 ∥f⃗∥22 K(t)
K(t) ≤C1 ∥f⃗∥22 + C1 ∥∇u
Xr0
dt
ν
ν
2r
r
r
2r
− 2(2r−3)
− 2r−3
2r−3
2r−3
⃗ 3 ∥r ∥⃗u∥
ν
∥∇u
+ C2 ϵ
K(t)
Ḣ 1
We then conclude by using Grönwall’s lemma, as
RT
ˆ 0 MAX ∥f⃗∥2 dt < +∞ by assumption on f⃗
p
RT
RT
N0 p40
⃗ 3 ∥p0 dt ≤ C( √
⃗ 3 ∥pr dt) 2p0
) ( 0 MAX ∥∇u
ˆ 0 MAX ∥∇u
Xr
ν
0
ˆ
R TMAX
0
4r
3r−2 .
2r
2r−3
⃗ 3 ∥r
∥∇u
r
2r−3
∥⃗u∥Ḣ
dt ≤ (
1
Thus, if ⃗u blows up, we must have
RT
r
N02 2(2r−3)
( 0 MAX
ν )
R TMAX
0
4r
3(r−2)
⃗ 3 ∥r3r−2 dt) 2(2r−3) with p =
∥∇u
⃗ 3 ∥pr dt = +∞.
∥∇u
Proposition 11.6.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 1 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . If the maximal
RT
existence time TMAX satisfies TMAX < +∞, then 0 MAX ∥∂3 ⃗u∥pr dt = +∞ with p2 + 3r = 2
and 2 ≤ p ≤ 3.
Proof. We follow the proof in Kukavica and Ziane [289]. As f⃗ ∈ L2 L2 , we know that, for
every 0 < T0 < T < TMAX , ⃗u will belong to C([T0 , T ], H 1 ) ∩ L2 ((T0 , T ), H 2 ). Thus, we may
estimate at each positive time t the quantity
I(t) = 1 +
(∥u1 ∥2Ḣ 1 + ∥u2 ∥2Ḣ 1 )3
2
+
∥u3 ∥66
.
6
We define
J(t) = ∥u1 ∥2Ḣ 1 + ∥u2 ∥2Ḣ 1 ,
K(t) = ∥u3 ∥66
and
M (t) = ∥∆u1 ∥22 + ∥∆u2 ∥22 ,
N (t) = ∥u33 ∥2Ḣ 1 .
Using the equation
⃗ u + ⃗g − ∇ϖ
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
with
1
⃗g = Pf⃗ and ϖ = −
∆
3 X
3
X
∂i ∂j (ui uj )
i=1 j=1
we get
d
I(t) =
dt
Z
Z
u53 ∂t u3 dx − J(t)2 ( −∂t u1 ∆u1 − ∂t u2 ∆u2 dx).
4/5
(11.31)
1/5
Indeed, we have ⃗u ∈ L10 L10 ((T0 , T ) × R3 ) (since ∥⃗u∥L10 L10 ≤ C∥⃗u∥L∞ Ḣ 1 ∥⃗u∥L2 Ḣ 2 ), so that
we easily check that ∂t ⃗u, ∆⃗u and |⃗u|5 belong to L2 L2 ((T0 , T ) × R3 ).
334
The Navier–Stokes Problem in the 21st Century (2nd edition)
This can be expanded into
Z
Z
Z
d
5
2
2
2
I(t) =ν u3 ∆u3 dx − νJ(t)
|∆u1 | dx − νJ(t)
|∆u2 |2 dx
dt
Z
Z
⃗ 3 dx + J(t)2 (∆u1 )⃗u · ∇u
⃗ 1 + (∆u2 )⃗u · ∇u
⃗ 2 dx
− u53 ⃗u · ∇u
Z
Z
+ u53 g3 dx − J(t)2 (g1 ∆u1 + g2 ∆u2 ) dx
Z
Z
− u53 ∂3 ϖ dx + J(t)2 (∆u1 )∂1 ϖ + (∆u2 )∂2 ϖ dx.
(11.32)
Next, we deal carefully with each term:
ˆ Integration by parts gives
Z
5 ⃗ 3 2
5
ν u53 ∆u3 dx = − ν∥∇(u
3 )∥2 = − νN (t)
9
9
(11.33)
ˆ We have obviously
Z
−ν
2
|∆u1 | dx − ν
Z
|∆u2 |2 dx = −νM (t)
ˆ Integration by parts gives (since div ⃗u = 0)
Z
Z
5
⃗
⃗ 5 ) dx = −5A = 0
A = −5 u3 ⃗u · ∇u3 dx = − u3 ⃗u · ∇(u
3
(11.34)
(11.35)
ˆ Writing ∆ = ∆2 + ∂32 , where ∆2 = ∂12 + ∂22 , we get
Z
⃗ 1 + (∆u2 )⃗u.∇u
⃗ 2 dx =
(∆u1 )⃗u · ∇u
Z
Z
⃗
⃗ 2 dx
− ∂3 u1 (∂3 ⃗u)∇u1 dx − ∂3 u2 (∂3 ⃗u) · ∇u
Z
Z
+ ∆2 u1 u3 ∂3 u1 dx + ∆2 u2 u3 ∂3 u2 dx
Z
+ ∆2 u1 (u1 ∂1 + u2 ∂2 )u1 dx
Z
+ ∆2 u2 (u1 ∂1 + u2 ∂2 )u2 dx
with
Z
∆2 u1 (u1 ∂1 + u2 ∂2 )u1 dx
Z
Z
= − ∂1 u1 ∂1 u1 ∂1 u1 dx − ∂1 u1 ∂1 u2 ∂2 u1 dx
Z
Z
− ∂2 u1 ∂2 u1 ∂1 u1 dx − ∂2 u1 ∂2 u2 ∂2 u1 dx
Z
− ∂1 u1 (u1 ∂1 + u2 ∂2 )∂1 u1 dx
Blow-up?
335
Z
−
∂2 u1 (u1 ∂1 + u2 ∂2 )∂2 u1 dx
Z
= − ∂1 u1 ∂1 u1 ∂1 u1 dx − ∂1 u1 ∂1 u2 ∂2 u1 dx
Z
Z
− ∂2 u1 ∂2 u1 ∂1 u1 dx − ∂2 u1 ∂2 u2 ∂2 u1 dx
Z
|∂1 u1 |2 + |∂2 u1 |2
− ∂3 u3
dx
2
Z
and similarly
Z
∆2 u2 (u1 ∂1 + u2 ∂2 )u2 dx
Z
Z
= − ∂1 u2 ∂1 u1 ∂1 u2 dx − ∂1 u2 ∂1 u2 ∂2 u2 dx
Z
Z
− ∂2 u2 ∂2 u1 ∂1 u2 dx − ∂2 u2 ∂2 u2 ∂2 u2 dx
Z
|∂1 u2 |2 + |∂2 u2 |2
− ∂3 u3
dx
2
In particular, we shall have to deal with the term
Z
Z
A(t) = − ∂1 u1 ∂1 u1 ∂1 u1 dx −
Z
Z
− ∂2 u1 ∂2 u1 ∂1 u1 dx −
Z
Z
− ∂1 u2 ∂1 u1 ∂1 u2 dx −
Z
Z
− ∂2 u2 ∂2 u1 ∂1 u2 dx −
∂1 u1 ∂1 u2 ∂2 u1 dx
∂2 u1 ∂2 u2 ∂2 u1 dx
∂1 u2 ∂1 u2 ∂2 u2 dx
∂2 u2 ∂2 u2 ∂2 u2 dx
which we rewrite as
Z
A(t) = −
(∂1 u1 + ∂2 u2 )(|∂1 u1 |2 + |∂2 u2 |2 ) dx
Z
+
(∂1 u1 + ∂2 u2 )∂1 u1 ∂2 u2 dx
Z
−
(∂1 u1 + ∂2 u2 )∂1 u2 ∂2 u1 dx
Z
−
Z
=
(∂1 u1 + ∂2 u2 )(|∂2 u1 |2 + |∂1 u2 |2 ) dx
∂3 u3 (|∂1 u1 |2 + |∂2 u2 |2 + |∂2 u1 |2 + |∂1 u2 |2 ) dx
Z
+ ∂3 u3 (∂1 u2 ∂2 u1 − ∂1 u1 ∂2 u2 ) dx
336
The Navier–Stokes Problem in the 21st Century (2nd edition)
This gives
Z
⃗ 1 + (∆u2 )⃗u · ∇u
⃗ 2 dx
(∆u1 )⃗u · ∇u
Z
Z
⃗ 1 dx − ∂3 u2 (∂3 ⃗u) · ∇u
⃗ 2 dx
= − ∂3 u1 (∂3 ⃗u)∇u
Z
Z
+ ∆2 u1 u3 ∂3 u1 dx + ∆2 u2 u3 ∂3 u2 dx
Z
|∂1 u1 |2 + |∂2 u1 |2 + |∂1 u2 |2 + |∂2 u2 |2
+ ∂3 u3
dx
2
Z
+ ∂3 u3 (∂1 u2 ∂2 u1 − ∂1 u1 ∂2 u2 ) dx
⃗ 1 ∥22r + 2∥∂3 ⃗u∥r ∥∇u
⃗ 2 ∥22r
≤ 2∥∂3 ⃗u∥r ∥∇u
r−1
r−1
2r ∥∂3 ⃗
2r ∥∂3 ⃗
+ ∥∆u1 ∥2 ∥u3 ∥ r−2
u∥r + ∥∆u2 ∥2 ∥u3 ∥ r−2
u∥r
2r
For 9/4 ≤ r ≤ 3, we have q1 = r−1
∈ [3, 18/5] ⊂ [2, 6]. As we look for a control of
∞
2
2
1
⃗ 1 and ∇u
⃗ 2 in L L ∩ L Ḣ ⊂ L∞ L2 ∩ L2 L6 , we may use interpolation and write,
∇u
for i = 1, 2
3
1
3
3
⃗ i ∥q ≤ ∥∇u
⃗ i ∥ q1 − 2 ∥∇u
⃗ i ∥ 2 − q1 ≤ C∥ui ∥1− 2r ∥∆ui ∥ 2r
∥∇u
1
2
6
2
Ḣ 1
3
3
2r
Similarly, we write that q2 = r−2
∈ [6, 18] and we look for a control of u33 in L∞ L2 ∩
2 1
∞ 2
2 6
L Ḣ ⊂ L L ∩ L L , hence of u3 in L∞ L6 ∩ L6 L18 . Thus we write
9
∥u3 ∥q2 ≤ ∥u3 ∥6q2
− 12
3
− q9
2
∥u3 ∥18
2
4− r9
≤ C∥u3 ∥6
3
−1
r
∥u33 ∥Ḣ
1
Thus, we get
Z
⃗ 1 + (∆u2 )⃗u · ∇u
⃗ 2 dx ≤
(∆u1 )⃗u · ∇u
3
(11.36)
3
C∥∂3 ⃗u∥r J(t)1− 2r M (t) 2r
1
3
1
2
3
+C∥∂3 ⃗u∥r M (t) 2 N (t) 2r − 2 K(t) 3 − 2r
ˆ We have
R
u53 g3 dx ≤ ∥u3 ∥510 ∥g3 ∥2 with
2
3
2
1
5
5
∥u3 ∥10 ≤ ∥u3 ∥65 ∥u3 ∥18
≤ C∥u3 ∥65 ∥u33 ∥Ḣ
.
1
and thus
Z
1
1
u53 g3 dx ≤ C∥⃗g ∥2 K(t) 3 N (t) 2
(11.37)
ˆ We have obviously
Z
−
(g1 ∆u1 + g2 ∆u2 ) dx ≤ ∥⃗g ∥2 M (t)1/2
ˆ We have
Z
−
u53 ∂3 ϖ dx = 5
Z
3r .
u43 ∂3 u3 ϖ dx ≤ 5∥∂3 ⃗u∥r ∥u3 ∥46r ∥ϖ∥ r−1
r−1
(11.38)
Blow-up?
337
3r
As the Riesz transforms are bounded on L r−1 , we find that
3r ≤ C
∥ϖ∥ r−1
3 X
3
X
′
3r . ≤ C
∥ui uj ∥ r−1
i=1 j=1
3
X
∥ui ∥26r .
r−1
i=1
3/2
Now, notice that L∞ Ḣ 1 ∩ L2 Ḣ 2 ⊂ L∞ Ḣ 1 ∩ L4 Ḃ2,1 ⊂ L∞ L6 ∩ L4 L∞ and that
6r
q3 = r−1
∈ [9, 54
5 ] ⊂ [6, 18], so that we may write, for i = 1, 2
6
1− q6
∥ui ∥q3 ≤ ∥ui ∥6q3 ∥ui ∥∞
and
3
1
+
3
1
2
q3
≤ C∥ui ∥Ḣ
∥∆ui ∥22
1
9
∥u3 ∥q3 ≤ ∥u3 ∥6q3
− 12
3
− q9
2
∥u3 ∥18
3
− q3
3
1−
1
1
= C∥ui ∥Ḣ 1 2r ∥∆ui ∥22r
3
1− 2r
≤ C∥u3 ∥6
1
2r
∥u33 ∥Ḣ
1
This gives
Z
− u53 ∂3 ϖ dx
23
1
1
3
3
3
3
≤ C∥∂3 ⃗u∥r (J(t)1− 2r M (t) 2r K(t)1− 2r N (t) 2r
+ K(t)1− 2r N (t) 2r ).
(11.39)
ˆ We write
(as div ⃗u = 0)
∆ϖ = −
3 X
3
X
∂i ∂j (ui uj ) = −
i=1 j=1
3 X
3
X
∂i uj ∂j ui
i=1 j=1
so that
Z
Z
(∆u1 )∂1 ϖ + (∆u2 )∂2 ϖ dx = −
(∂1 u1 + ∂2 u2 )∆ϖ dx
Z
=−
∂3 u3
3 X
3
X
∂i uj ∂j ui dx
i=1 j=1
Z
=−
∂3 u3
2 X
2
X
∂i uj ∂j ui dx
i=1 j=1
Z
+
−2
∂3 u3 (∂1 u1 + ∂2 u2 )2 dx
3 Z
X
∂i (∂1 u1 + ∂2 u2 )u3 ∂3 ui dx
i=1
and thus
Z
⃗ 1 ∥22r + ∥∇u
⃗ 1 ∥22r )
(∆u1 )∂1 ϖ + (∆u2 )∂2 ϖ dx ≤4∥∂3 ⃗u∥r (∥∇u
r−1
r−1
2r (∥∆u1 ∥2 + ∥∆u2 ∥2 )
+ 2∥∂3 ⃗u∥r ∥u3 ∥ r−2
3
3
≤C∥∂3 ⃗u∥r J(t)1− 2r M (t) 2r
1
3
1
2
3
+ C∥∂3 ⃗u∥r M (t) 2 N (t) 2r − 2 K(t) 3 − 2r
(11.40)
338
The Navier–Stokes Problem in the 21st Century (2nd edition)
Collecting together all those inequalities, we find that
d
5
I(t) ≤ − νN (t) − νJ(t)2 M (t)
dt
9
3
3
+ C∥∂3 ⃗u∥r J(t)2 J(t)1− 2r M (t) 2r
1
3
1
2
3
+ C∥∂3 ⃗u∥r J(t)2 M (t) 2 N (t) 2r − 2 K(t) 3 − 2r
1
1
+ C∥⃗g ∥2 K(t) 3 N (t) 2 + J(t)2 ∥⃗g ∥2 M (t)1/2
23
1
1
3
3
+ C∥∂3 ⃗u∥r J(t)1− 2r M (t) 2r K(t)1− 2r N (t) 2r
3
(11.41)
3
+ C∥∂3 ⃗u∥r K(t)1− 2r N (t) 2r
Using the fact that K(t) ≤ 6I(t), J(t) ≤ 21/3 I(t)1/3 and I(t) ≥ 1, we get
d
5
I(t) ≤ − νN (t) − νJ(t)2 M (t)
dt
9
3
3
+ C∥∂3 ⃗u∥r I(t)1− 2r J 2 (t)M (t) 2r
1
3
1
3
+ C∥∂3 ⃗u∥r J(t)2 M (t) 2 N (t) 2r − 2 I(t)1− 2r
1
2
1
2
1
2
(11.42)
1/2
J (t)M (t)
2
+ C∥⃗g ∥2 I(t) N (t) + I(t) ∥⃗g ∥2
1
3
2
+ C∥∂3 ⃗u∥r I(t)1− 2r J 2 (t)M (t) 2r N (t) 2r
3
3
+ C∥∂3 ⃗u∥r I(t)1− 2r N (t) 2r
Recalling that
1
p
=1−
3
2r ,
we find that
d
1
I(t) ≤ C ∥⃗g ∥22 + Cν 1−p ∥∂3 ⃗u∥p3 I(t)
dt
ν
(11.43)
If TMAX < +∞, I(t) must blow up when t → TMAX (as the L6 norm of ⃗u blows up,
RT
according to Serrin’s criterion), and thus, by Grönwall’s lemma, 0 MAX ∥∂3 ⃗u∥pr dt = +∞.
Proposition 11.7.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 1 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . If the maximal
existence time TMAX satisfies TMAX < +∞, then
RT
• 0 MAX ∥ϖ∥rp dt = +∞ for p2 + 3r = 2 and 1 < p < +∞ (and 3/2 < r < +∞).
• if f⃗ ∈ L2 H 1 ,
+∞).
R TMAX
0
p
⃗
∥∇ϖ∥
q dt = +∞ for
2
p
+
3
q
= 3 and
2
3
< p < +∞ (and 1 < q <
Proof. The first problem is, of course, to include the pressure ϖ in the estimates, and to
⃗ u as we have no good control on this term (except for its divergence,
exclude the term ⃗u · ∇⃗
⃗ = −∆ϖ).
as div(⃗u · ∇u)
It means that the proof will no longer be based on a Grönwall lemma applied to some
d
⃗ u and would not
norm ∥⃗u(t, .)∥2Ḣ α as the computations of dt
∥⃗u(t, .)∥2Ḣ α would involve ⃗u · ∇⃗
d
θ
⃗ u but,
depend on ϖ. Instead of that, we shall estimate dt ∥⃗u(t, .)∥θ : it will not involve ⃗u · ∇⃗
when θ ̸= 2, it will involve ϖ.
Blow-up?
339
Under the assumption that f⃗ ∈ L2 L2 , we shall estimate ∥⃗u∥44 . Under the assumption
f⃗ ∈ L2 H 1 , we shall estimate ∥⃗u∥θθ with θ > 4.
The quantity ∥⃗u∥44 :
As f⃗ ∈ L2 L2 , we know that, for every 0 < T0 < T < TMAX , ⃗u will belong to C([T0 , T ], H 1 ) ∩
L2 ((T0 , T ), H 2 ). Thus, ⃗u belongs to C([T0 , T ], Lθ (R3 )) for every θ ∈ (2, 6). Moreover, we
have
Z
Z
d
4
|⃗u| = 4 |⃗u|2 ∂t ⃗u · ⃗u dx
(11.44)
dt
This is obvious if ⃗u is regular enough; but if we write
⃗ u − ∇ϖ
⃗
∂t ⃗u = ν∆⃗u + Pf⃗ − ⃗u · ∇⃗
we find that ∂t ⃗u ∈ L2 L2 on (T0 , T1 ), while we know that ⃗u ∈ CL4 ∩ L6 L6 ; we may then
conclude by a density argument.
We have
3
X
∂j (|⃗u|2 ⃗u) = |⃗u|2 ∂j ⃗u + 2
∂j uk uk ⃗u
k=1
⃗ ⊗ ⃗u is bounded in L2 L6 on [T0 , T ] × R3 , we find that
and, as ⃗u is bounded in L∞ L6 and ∇
∂j (|⃗u|θ−2 ⃗u) ∈ L∞ L2 . Thus, we may write
Z
∆⃗u · |⃗u|2 ⃗u dx = −
3 Z
X
∂j ⃗u · ∂j (|⃗u|2 ⃗u) dx
j=1
Z
=−
3 Z
X
⃗ ⊗ ⃗u|2 dx − 2
|⃗u|2 |∇
2
|∂j ⃗u · ⃗u| dx
(11.45)
j=1
Z
=−
⃗ ⊗ ⃗u|2 dx − 1
|⃗u|2 |∇
2
Z
⃗ u|2 )|2 dx
|∇(|⃗
R
⃗ u) · |⃗u|2 ⃗u dx. As div ⃗u = 0, we have
The next term we study is I = (⃗u · ∇⃗
Z
⃗ u|2 ⃗u)) dx
I = − ⃗u · (⃗u · ∇(|⃗
=−I −2
3 X
3 Z
X
∂j uk uj uk |⃗u|2 dx
j=1 k=1
Z
=−I −
⃗ u|2 ) dx
|⃗u|2 (⃗u · ∇(|⃗
=−I
=0
so that
Z
⃗ u) · |⃗u|2 ⃗u dx = 0.
(⃗u · ∇⃗
Thus far, we have proven that
Z
Z
d
4
2 ⃗
2
2 2
(∥⃗u∥4 ) ≤ −4ν |⃗u| |∇ ⊗ ⃗u| dx − 2ν∥ |⃗u| ∥Ḣ 1 + 4 |⃗u|2 ⃗u · Pf⃗ dx
dt
Z
⃗ dx.
− 4 |⃗u|2 ⃗u · ∇ϖ
(11.46)
340
The Navier–Stokes Problem in the 21st Century (2nd edition)
We then write
Z
|4
|⃗u|2 ⃗u · Pf⃗ dx| ≤4∥|⃗u|2 ∥6 ∥⃗u∥3 ∥Pf⃗∥2
≤C∥ |⃗u|2 ∥Ḣ 1 ∥⃗u∥3 ∥f⃗∥2
(11.47)
ν
C2
≤ ∥ |⃗u|2 ∥2Ḣ 1 +
∥⃗u∥23 ∥f⃗∥22
2
2ν
2
ν
C
1
2
≤ ∥ |⃗u|2 ∥2Ḣ 1 +
( ∥⃗u∥44 + ∥⃗u∥2 )∥f⃗∥22 .
2
2ν 3
3
R 2
⃗ dx. We first integrate by
We are thus left with the task of estimating J = 4 |⃗u| ⃗u · ∇ϖ
parts and find (since div ⃗u = 0)
Z
⃗ u|2 dx
J = −4 ϖ⃗u · ∇|⃗
Thus, we get two different estimates for J:
Z
⃗
|J| ≤ 4 |∇ϖ||⃗
u|3 dx
and
Z
2
|J| ≤ 4
2
1/2 Z
|ϖ| |⃗u| dx
(11.48)
1/2
2 2
⃗
|∇(|⃗u| )| dx
(11.49)
Case p2 + 3r = 2 and 1 < p < +∞:
Let σ2 = 12 (1 − 1r ). We use estimate (11.49) to get
|J| ≤4∥ϖ⃗u∥2 ∥|⃗u|2 ∥Ḣ 1
1/2
≤4∥ϖ∥1/2
u∥σ ∥|⃗u|2 ∥Ḣ 1
r ∥ϖ∥σ/2 ∥⃗
1/2
As ϖ is given by Riesz transforms applied to ui uj and as σ/2 ∈ (1, +∞), we have ∥ϖ∥σ/2 ≤
C∥⃗u∥σ and thus
|J| ≤ C∥ϖ∥1/2
u∥2σ ∥|⃗u|2 ∥Ḣ 1
r ∥⃗
1 1
As 3/2 < r < +∞, we find that σ ∈ ( 12
, 4 ); writing
1
σ
1
= λ 14 + (1 − λ) 12
, we get
(1−λ)/2
∥⃗u∥σ ≤ ∥⃗u∥λ4 ∥⃗u∥1−λ
u∥λ4 ∥|⃗u|2 ∥2
12 ≤ C∥⃗
so that
|J| ≤C∥ϖ∥1/2
u∥2λ
u|2 ∥2−λ
r ∥⃗
4 ∥|⃗
Ḣ 1
ν
2 2
≤ ∥ |⃗u| ∥Ḣ 1 + Cν ∥ϖ∥1/λ
u∥44
r ∥⃗
2
Now, notice that we have λ =
6
σ
−
1
2
=
1
3
1− 2r
= p1 . It means that we finally get
d
(∥⃗u∥44 ) + ν∥ |⃗u|2 ∥2Ḣ 1 ≤ Cν (∥⃗u∥44 + ∥⃗u∥2 )∥f⃗∥22 + Cν ∥ϖ∥pr ∥⃗u∥44
dt
and we may conclude by using Grönwall’s lemma.
(11.50)
Blow-up?
Case
2
p
3
q
+
We have
and p2 +
341
= 3 and 1 < p < +∞:
2
3
p + q
3
r = 2.
⃗
= 3, thus 1 < q < 3. If 1r = 1q − 13 , then ∥ϖ∥r ≤ C∥∇ϖ∥
q , 3/2 < r < +∞
This case has thus already been dealt with.
The quantity ∥⃗u∥θθ , 4 < θ < +∞:
If f⃗ ∈ L2 H 1 , we know that, for every 0 < T0 < T < TMAX , ⃗u will belong to C([T0 , T ], H 2 ) ∩
L2 ((T0 , T ), H 3 ). Thus, ⃗u belongs to C([T0 , T ], Lθ (R3 )) for every θ ∈ (2, +∞). Moreover, we
have
Z
Z
d
θ
|⃗u| = θ |⃗u|θ−2 ∂t ⃗u · ⃗u dx.
(11.51)
dt
This is obvious if ⃗u is regular enough; but if we write
⃗ u − ∇ϖ,
⃗
∂t ⃗u = ν∆⃗u + Pf⃗ − ⃗u · ∇⃗
we find that ∂t ⃗u ∈ L2 L2 (and even in L2 H 1 ) on (T0 , T1 ), while we know that ⃗u ∈ CLθ ∩
L2(θ−1) L2(θ−1) ; we may then conclude by a density argument.
We have
3
X
∂j (|⃗u|θ−2 ⃗u) = |⃗u|θ−2 ∂j ⃗u + (θ − 2)
∂j uk uk |⃗u|θ−4 ⃗u
k=1
θ−2
and, as |⃗u|
write
Z
is bounded on [T0 , T ] × R , we find that ∂j (|⃗u|θ−2 ⃗u) ∈ L∞ L2 . Thus, we may
3
∆⃗u · |⃗u|θ−2 ⃗u dx = −
3 Z
X
∂j ⃗u · ∂j (|⃗u|θ−2 ⃗u) dx
j=1
Z
=−
⃗ ⊗ ⃗u|2 dx −
|⃗u|θ−2 |∇
3
X
Z
(θ − 2)
2
|∂j ⃗u · ⃗u| |⃗u|θ−4 dx
(11.52)
j=1
Z
=−
⃗ ⊗ ⃗u|2 dx −
|⃗u|θ−2 |∇
4
(θ − 2)
θ2
Z
⃗ u|θ/2 )|2 dx.
|∇(|⃗
R
⃗ u).|⃗u|θ−2 ⃗u dx. As div ⃗u = 0, we have
The next term we study is I = (⃗u · ∇⃗
Z
⃗ u|θ−2 ⃗u)) dx
I = − ⃗u · (⃗u · ∇(|⃗
= − I − (θ − 2)
3 X
3 Z
X
∂j uk uj uk |⃗u|θ−2 dx
j=1 k=1
2(θ − 2)
=−I −
θ
=−I
Z
⃗ u|θ/2 ) dx
|⃗u|θ/2 (⃗u · ∇(|⃗
=0
so that
Z
⃗ u) · |⃗u|θ−2 ⃗u dx = 0.
(⃗u · ∇⃗
(11.53)
Thus far, we have proven that
d
4(θ − 2)
(∥⃗u∥θθ ) ≤ −ν
∥ |⃗u|θ/2 ∥2Ḣ 1 + θ
dt
θ
Z
θ−2
|⃗u|
⃗u · Pf⃗ dx − θ
Z
⃗ dx
|⃗u|θ−2 ⃗u · ∇ϖ
342
The Navier–Stokes Problem in the 21st Century (2nd edition)
We write, as θ ≥ 2, |⃗u|θ−1 ≤ |⃗u|θ/2 + |⃗u|θ and thus
Z
|θ |⃗u|θ−2 ⃗u · Pf⃗ dx| ≤θ∥ |⃗u|θ/2 ∥2 ∥Pf⃗∥2 + θ∥ |⃗u|θ/2 ∥2 ∥ |⃗u|θ/2 ∥6 ∥Pf⃗∥3
θ/2
θ/2
≤θ∥⃗u∥θ ∥f⃗∥2 + Cθ∥⃗u∥θ ∥|⃗u|θ/2 ∥Ḣ 1 ∥f⃗∥H 1/2
2
(11.54)
3
θ−2
θ
θ 1 ⃗ 2
∥|⃗u|θ/2 ∥2Ḣ 1 + ∥f⃗∥2H 1 + ∥⃗u∥θθ (1 + C 2
∥f ∥H 1 )
θ
4
θ−2ν
R
⃗ dx.
We are thus left with the task of estimating J = θ |⃗u|θ−2 ⃗u · ∇ϖ
We first integrate by parts and find (since div ⃗u = 0)
Z
⃗ u|θ−2 dx
J = −θ ϖ⃗u · ∇|⃗
≤ν
⃗ u|θ−2 = 2(θ−2) |⃗u|θ/2−2 ∇(|⃗
⃗ u|θ/2 ). Thus, we get two different
For θ ≥ 4, we may write ∇|⃗
θ
estimates for J:
Z
⃗
|J| ≤ θ |∇ϖ||⃗
u|θ−1 dx
(11.55)
and
Z
|J| ≤ 2(θ − 2)
1/2 Z
1/2
⃗ u|θ/2 )|2 dx
|ϖ|2 |⃗u|θ−2 dx
|∇(|⃗
(11.56)
Case p2 + 3q = 3 and 2/3 < p ≤ 1:
In this case, we have q ∈ [3, +∞). We use (11.55) and (11.56) to get
⃗
|J| ≤ θ∥∇ϖ∥
u∥θ−1
q ∥⃗
(θ−1)
q
q−1
and (since the Riesz transforms are bounded on L(θ+2)/2 )
(θ+2)/2
|J| ≤ Cθ ∥⃗u∥θ+2
∥|⃗u|θ/2 ∥Ḣ 1 .
Recall that we seek a control in terms of ∥⃗u∥θ and ∥|⃗u|θ/2 ∥Ḣ 1 (hence in terms of ∥⃗u∥3θ ).
Thus we need to choose θ ≥ 4 such that
θ < (θ − 1)
q
< 3θ and θ < θ + 2 < 3θ.
q−1
The simplest choice would be
(θ − 1)
q
= θ + 2,
q−1
or equivalently θ = 3q − 2 ∈ [7, +∞). For this choice of θ, writing
θ−1
3
6
θ−1
θ(θ+2)
θ+2
∥v∥θ+2 ≤ ∥v∥θθ+2 ∥v∥3θ
≤ Cθ ∥v∥θθ+2 ∥|v|θ/2 ∥Ḣ
,
1
we find (since θ − 1 = 3(q − 1) =
2q
p )
(θ+2)(1− q1 )
⃗
|J| ≤ Cθ ∥∇ϖ∥
u∥θ+2
q ∥⃗
and
(θ+2) 12
|J| ≤ Cθ ∥⃗u∥θ+2
2(q−1)
p
⃗
≤ Cθ′ ∥∇ϖ∥
u∥θ
q ∥⃗
q
4
θp
∥|⃗u|θ/2 ∥Ḣ
1
1+ 3
∥|⃗u|θ/2 ∥Ḣ 1 ≤ Cθ′ ∥⃗u∥θp ∥|⃗u|θ/2 ∥Ḣ 1 θ
Blow-up?
343
Finally, writing |J| = |J|1/2 |J|1/2 , we find
1+ 3
θ
θ
1/2
⃗
|J| ≤ Cθ ∥∇ϖ∥
u∥θ2p ∥|⃗u|θ/2 ∥Ḣ21
q ∥⃗
2
+ θp
1+ 3
2
, is equal to 2p−1
Now, we check2 that the last exponent, 2 θ + θp
p : equivalently, we must
3
check that 32 − p1 = θ1 ( 32 + p2 ), or that 2q
= θ1 ( 92 − 3q ) and finally θ = 3q − 2.
Thus, we have
1
1 1− 2p
2p
p
⃗
∥|⃗u|θ/2 ∥Ḣ 1
(11.57)
|J| ≤ Cθ ∥∇ϖ∥
u∥θθ
q ∥⃗
From this and by Young’s inequality, we get
1
1
d
θ2
p
⃗
(∥⃗u∥θθ ) ≤ ∥f⃗∥2H 1 + ∥⃗u∥θθ (1 + Cθ ∥f⃗∥2H 1 + Cθ 2p−1 ∥∇ϖ∥
q)
dt
4
ν
ν
(11.58)
If TMAX < +∞, then, by Serrin’s criterion, ∥⃗u∥θθ must explode and thus, by Grönwall’s
RT
p
⃗
lemma, one must have 0 MAX ∥∇ϖ∥
q dx = +∞.
11.6
Vorticity
Vorticity has always played a prominent role in the study of turbulent flows. In his book
Vorticity and Turbulence [123], Chorin studied turbulence theory for incompressible flow
described in terms of the vorticity field. Similarly, in their book Vorticity and Incompressible
Flow [40], Bertozzi and Majda insist on vortex dynamics
which in lay terms refer to the interaction of local whirls or eddies in the fluid
The celebrated Beale–Kato–Majda criterion [27] expresses the link between blow-up and
high vorticity:
Beale–Kato–Majda criterion
Theorem 11.6.
Let ⃗u0 ∈ H 3 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 3 )∩L2 ((0, T ), H 4 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . Let ω
⃗ = curl ⃗u.
Then we have, for any T ∈ (0, TMAX )
2
1
⃗
2
R
C0 0T ∥⃗
ω ∥∞ dt
∥⃗u(T, .)∥2Ḣ 3 ≤ C0 e(∥⃗u0 ∥H 3 + ν ∥f ∥L2 H 2 )e
(11.59)
where the constant C0 does not depend on T .
In particular, if the maximal existence time TMAX satisfies TMAX < +∞, then
R TMAX
∥⃗
ω ∥∞ dt = +∞.
0
2 This
could be done by a simple scaling argument.
344
The Navier–Stokes Problem in the 21st Century (2nd edition)
As a matter of fact, the role of vorticity per se in this analytical criterion turned out not
to be as significant as it seemed. Indeed, the criterion could be expressed in a norm on ω
⃗
⃗ ⊗ ⃗u: the L∞ norm is not a good one
that is equivalent on a norm on the whole gradient ∇
⃗ ⊗ ⃗u
as it is unstable under Riesz transforms while one needs Riesz transforms to express ∇
in terms of ω
⃗:
⃗ ∧ω
⃗ ⊗ ⃗u = −∇
⃗ ⊗ 1 (∇
⃗ ).
∇
∆
Recall that we have seen in Theorem 11.3 that the L∞ norm may be replaced by the weaker
norm ∥⃗
ω ∥BM O (Kozono and Taniuchi [278]), or by the still weaker norm ∥⃗
ω ∥Ḃ 0
(Kozono,
∞,∞
Ogawa and Taniuchi [273]).
In order to highlight the role of vorticity, it is thus necessary to get into greater details
into the geometrical aspects of this role. Taylor [466] insisted on the role of vortex stretching
in the production (or dissipation) of vorticity. To explain this phenomenon, let us study the
ω |2 . If the flow is regular enough, we find
(local) enstrophy E = 21 |⃗
∂t E =⃗
ω · ∂t ω
⃗
⃗ω−ω
⃗ u) + ω
=ν⃗
ω · (∆⃗
ω) − ω
⃗ · (⃗u · ∇⃗
⃗ · ∇⃗
⃗ · curl f⃗
As div ⃗u = 0, we have
⃗ ω) =
ω
⃗ · (⃗u · ∇⃗
3
X
⃗
uj ω
⃗ .∂j ω
⃗ = ⃗u · ∇E.
j=1
On the other hand, we have
⃗ u) =
ω
⃗ · (⃗
ω · ∇⃗
3
3 X
X
i=1 j=1
where ϵ is the strain tensor
ϵ=
ωi ωj ∂j ui =
3 X
3
X
ϵi,j ωi ωj
i=1 j=1
1
Du + (Du)T .
2
Finally, we write
⃗ ⊗ω
∆E = ω
⃗ · ∆⃗
ω + |∇
⃗ |2
and we get the equation that expresses the material derivative enstrophy:
3 X
3
X
D
⃗ = ν∆E − ν|∇
⃗ ⊗ω
E = ∂t E + ⃗u · ∇E
⃗ |2 +
ϵi,j ωi ωj + ω
⃗ · curl f⃗
Dt
i=1 j=1
(11.60)
Thus,P
the inner
P3 production of enstrophy will be found in the regions where the quadratic
3
form i=1 j=1 ϵi,j ωi ωj is positive, i.e., where the vorticity ω
⃗ aligns with the eigenvectors
that correspond to positive eigenvalues of the tensor matrix (recall that the trace of ϵ is
equal to the divergence of ⃗u, hence is equal to 0, so that the eigenvalues cannot all be
negative). One can find discussions on this production of enstrophy through the interaction
between vorticity and strain and on its significance in the papers of Galanti, Gibbon and
Heritage [193] and of Tsinober [483].
In this section, we will focus on the result of Constantin and Fefferman [128], which
states that, whenever the direction of vorticity evolves regularly in the areas where the
vorticity is large, the solution cannot blow up. We will more precisely prove the following
generalization3 by Beirão da Vega and Berselli [30, 31]:
3 See
also the survey by Berselli [38] and the recent paper by Giga and Miura [211].
Blow-up?
345
Vorticity direction
Theorem 11.7.
Let ⃗u0 ∈ H 3 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 3 )∩L2 ((0, T ), H 4 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . Let ω
⃗ = curl ⃗u
⃗ x) = ω⃗ (t,x) .
and, for ω
⃗ (t, x) ̸= 0, ξ(t,
∥⃗
ω (t,x)|
Let R > 0. Then, if the maximal existence time TMAX satisfies TMAX < +∞, we
have
⃗ x) ∧ ξ(t,
⃗ y)|
|ξ(t,
lim sup
sup
= +∞.
1
−
|x − y| 2
|⃗
ω (t,x)|>R,|⃗
ω (t,y)|>R
t→TMAX
Proof. Let MR (t) = sup|⃗ω(t,x)|>R,|⃗ω(t,y)|>R
MR (t) ≤ C
⃗
⃗
|ξ(t,x)∧
ξ(t,y)|
1
|x−y| 2
. We have
1
1
∥⃗
ω ∥Ḃ 1/2 ≤ C ′ ∥∆⃗
ω ∥2 .
∞,∞
R
R
Thus, MR (t) is well defined as long as ⃗u remains controlled in the H 3 norm.
In order to show that ⃗u does not blow up in H 3 if MR (t) remains bounded, it is enough
to show that ⃗u does not blow up in H 1 ; as ⃗u is controlled in L∞ L2 ∩ L2 H 1 , we just have to
show that ω
⃗ does not blow up in L2 . From (11.60), we see that
d ∥⃗
ω ∥22
⃗ ⊗ω
(
) = −ν∥∇
⃗ ∥22 +
dt 2
Z
f⃗ · curl ω
⃗ dx +
Z X
3 X
3
ϵi,j ωi ωj dx
i=1 j=1
As ϵi,j is given by Riesz transforms of ω
⃗ , a direct estimate would give
d ∥⃗
ω ∥22
ν
1
(
) ≤ − ∥⃗
ω ∥2Ḣ 1 + C ∥f⃗∥2H 1 + C∥⃗
ω ∥33
dt 2
2
ν
ν
1
3/2
3/2
≤ − ∥⃗
ω ∥2Ḣ 1 + C ∥f⃗∥2H 1 + C ′ ∥⃗
ω ∥2 ∥⃗
ω ∥Ḣ 1
2
ν
1
1
≤ C ∥f⃗∥2H 1 + C ′′ 3 ∥⃗
ω ∥62
ν
ν
If ω
⃗ would belong to L4 L2 , we could control ∥⃗
ω ∥2 by Grönwall’s lemma; but we only know
that ω
⃗ belongs to L2 L2 . The information on MR (t) is then needed to lower the exponent of
∥⃗
ω ∥2 in the last inequality from 6 to 4, to allow the use of Grönwall’s lemma.
Write
Z X
Z
3 X
3
⃗ u) dx
ϵi,j ωi ωj dx = ω
⃗ · (⃗
ω · ∇⃗
i=1 j=1
with
⃗u = −
1 ⃗
(∇ ∧ ω
⃗ ).
∆
346
The Navier–Stokes Problem in the 21st Century (2nd edition)
If G is the Green function (the fundamental solution of −∆G = δ), we find
Z X
3 X
3
ϵi,j ωi ωj dx = −
i=1 j=1
ZZ X
3 X
3
ω
⃗ i (t, x)⃗
ωj (t, x)×
i=1 j=1
×(∂j ∂σ(i) G(y)⃗
ωσ2 (i) (t, x − y) − ∂j ∂σ2 (i) G(y)⃗
ωσ(i) (t, x − y))
ZZ X
3 X
3
=
|⃗
ω (t, x)| |⃗
ω (t, x)| |⃗
ω (t, x − y)|A(t, x, y, x − y) dx dy
i=1 j=1
where σ is the permutation 1 → 2 → 3 → 1,
A(t, x, y, z) =
−
3 X
3
X
ξi (t, x)ξj (t, x)(∂j ∂σ(i) G(y)ξσ2 (i) (t, z) − ∂j ∂σ2 (i) G(y)ξσ(i) (t, z))
i=1 j=1
and where the integrals are taken as principal values. The Fourier transform of A with
respect to the y variable gives
Â(t, x, η, z) =
3 X
3
X
ξi (t, x)ξj (t, x)ηj (ησ(i)
i=1 j=1
1
1
ξσ2 (i) (t, z) − ησ2 (i) 2 ξσ(i) (t, z))
|η|2
|η|
which we may rewrite as
3
1 X
⃗ x), ⃗η , ξ(t,
⃗ z)).
Â(t, x, η, z) = 2 (
ηj ξj (t, x)) Det (ξ(t,
|η| j=1
The main point is then the identity
Â(t, x, η, x) = 0
and thus
A(t, x, y, x) = 0.
⃗ where α
We then write ω
⃗ =α
⃗ + β,
⃗ = 1|⃗ω(t,x)|≤R ω
⃗ . We then have
Z X
3 X
3
1 ⃗
ϵi,j ωi ωj dx = −
(∇ ∧ ω
⃗ ) ) dx
∆
i=1 j=1
Z
1 ⃗
⃗
=− α
⃗ · (⃗
α·∇
(∇ ∧ ω
⃗ ) ) dx
∆
Z
1 ⃗
⃗
⃗
⃗
− β · (β · ∇
(∇ ∧ α
⃗ ) ) dx
∆
Z
1 ⃗ ⃗
⃗
⃗
⃗
− β · (β · ∇
(∇ ∧ β) ) dx
∆
Z
⃗
ω
⃗ · (⃗
ω·∇
with
Z
−
⃗
α
⃗ · (⃗
α·∇
1
⃗ ∧ω
∧ (∇
⃗ ) ) dx ≤C∥⃗
α∥23 ∥⃗
ω ∥3
∆
2/3
≤C∥⃗
ω ∥23 ∥⃗
ω ∥2 ∥⃗
α∥1/3
∞
5/3
≤C ′ ∥⃗
ω ∥2 ∥⃗
ω ∥Ḣ 1 R1/3
Z
−
⃗ · (β⃗
β
Blow-up?
1
⃗ 2 ∥⃗
⃗
∧ (∇ ∧ α
⃗ ) ) dx ≤C∥β∥
3 α ∥3
∆
347
5/3
≤C ′ ∥⃗
ω ∥2 ∥⃗
ω ∥Ḣ 1 R1/3
and
Z
−
⃗
β⃗ · (β⃗ · ∇
1
⃗ ) dx =
⃗ ∧ β)
∧ (∇
∆
3 X
3
X
ZZ
=
|⃗
ω (t, x)| |⃗
ω (t, x)| |⃗
ω (t, x−y)|A(t, x, y, x−y) dx dy
|⃗
ω (t,x)|>R, |⃗
ω (t,x)|>R i=1 j=1
In the last equality, as A(t, x, y, x) = 0, we may replace A(t, x, y, x − y) with A(t, x, y, x −
y) − A(t, x, y, x) or with A(t, x, y, x − y) + A(t, x, y, x). We have, for |⃗
ω (t, x)| > R and
|⃗
ω (t, x)| > R,
|A(t, x, y, x − y) − A(t, x, y, x)| ≤ C
1 ⃗
⃗
|ξ(x − y) − ξ(x)|
|y|3
|A(t, x, y, x − y) + A(t, x, y, x)| ≤ C
1 ⃗
⃗
|ξ(x − y) + ξ(x)|
|y|3
and
so that
1
⃗ − y) − ξ(x)|,
⃗
⃗ − y) + ξ(x)|)
⃗
min(|ξ(x
|ξ(x
|y|3
1
≤C ′ 5/2 MR (t)
|y|
|A(t, x, y, x − y)| ≤C
and thus
Z
−
⃗
β⃗ · (β⃗ · ∇
1
⃗ ) dx ≤C∥⃗
⃗ ∧ β)
∧ (∇
ω ∥23 ∥I1/2 ω
⃗ ∥3 MR (t)
∆
≤C ′ ∥⃗
ω ∥23 ∥⃗
ω ∥2 MR (t)
≤C ′′ ∥⃗
ω ∥22 ∥⃗
ω ∥Ḣ 1 MR (t).
Finally, we obtain
d ∥⃗
ω ∥22
ν
C
5/3
(
) ≤ − ∥⃗
ω ∥2Ḣ 1 + ∥f⃗∥2H 1 + C∥⃗
ω ∥2 ∥⃗
ω ∥Ḣ 1 R1/3 + C∥⃗
ω ∥22 ∥⃗
ω ∥Ḣ 1 MR (t)
dt 2
2
ν
1
1
4/3
≤C ∥f⃗∥2H 1 + C ′ (R2/3 ∥⃗
ω ∥2 + MR (t)2 ∥⃗
ω ∥22 )∥⃗
ω ∥22
ν
ν
and we conclude with Grönwall’s lemma.
RT
Remark: More generally, a similar proof gives that, if 2 ≤ r ≤ 3 and 0 MAX ∥⃗
ω ∥2r dt < +∞,
and if the maximal existence time TMAX satisfies TMAX < +∞, we have
sup0<t<TMAX
sup
|⃗
ω (t,x)|>R,|⃗
ω (t,y)|>R
⃗ x) ∧ ξ(t,
⃗ y)|
|ξ(t,
3
|x − y| r −1
= +∞.
348
The Navier–Stokes Problem in the 21st Century (2nd edition)
11.7
Squirts
Again, let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 ) ∩ L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 , where ⃗u0 ∈ H 1
with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), L2 ). Let us assume that the maximal existence time
TMAX satisfies TMAX < +∞.
We are going to discuss the behavior of ∥⃗u∥∞ .
ˆ As we have
∥⃗u∥∞ ≤ C∥⃗u∥Ḃ 3/2 ≤ C ′
2,1
we have that
Z
q
∥⃗u∥Ḣ 1 ∥⃗u∥Ḣ 2
T
∥⃗u∥4∞ dt < +∞
(11.61)
0
for all T < TMAX .
ˆ As TMAX < +∞, we have seen that
TMAX
Z
∥⃗u∥2∞ dx = +∞.
(11.62)
0
ˆ If f⃗ ∈ L2 H 1 , we have a more precise estimate [210]:
p
lim inf TMAX − t∥⃗u∥∞ > 0.
(11.63)
Indeed, if ⃗u(t0 , .) ∈ L∞ , we may use Picard’s algorithm to find a local solution of
Z
t
Wν(t−s) ∗ P(f⃗ − div(⃗v ⊗ ⃗v )) ds
⃗v = Wν(t−t0 ) ∗ ⃗u(t0 , .) +
t0
in L∞ ((t0 , t0 + T ), L∞ ). The existence time is estimated by the following inequality:
T ≥ Cν
1
(∥⃗u(t0 , .)∥∞ + ∥f⃗∥L2 H 1 )2
.
It is easy to check that ⃗u = ⃗v on (t0 , ∈ (TMAX , t0 + T ). In particular, we find that
T > TMAX − t0 . This gives
p
p
p
lim inf TMAX − t∥⃗u∥∞ ≥ lim inf Cν − TMAX − t∥f⃗∥L2 H 1
−
t→TMAX
−
t→TMAX
=
p
Cν > 0.
ˆ However, ∥⃗u(t, .)∥∞ cannot explode too fast, as we have
Z
TMAX
∥⃗u∥∞ dt < +∞.
0
(11.64)
Blow-up?
349
Indeed, we have seen that
∥⃗u(t, .)∥2 ≤ ∥⃗u0 ∥22 +
and
1
ν
∥f⃗∥2 ds
0
t
Z
0
t
Z
1
ν
⃗ ⊗ ⃗u∥22 dt ≤ (∥⃗u0 ∥22 +
∥∇
ν
t
Z
∥f⃗∥2 ds)2 .
0
Now, we write
t
Z
Wν(t−s) ∗ P(f⃗ − div(⃗u ⊗ ⃗u)) ds
⃗u = Wνt ∗ ⃗u(t0 , .) +
0
with
Z
TMAX
TMAX
Z
1
∥⃗u0 ∥H 1 dt < +∞,
(νt)1/4
∥Wνt ∗ ⃗u(t0 , .)∥∞ dt ≤ C
0
0
TMAX
Z
t
Z
p
Wν(t−s) ∗ Pf⃗ ds∥∞ dt ≤ TMAX
∥
0
s
Z
≤C
Wν(t−s) ∗ Pf⃗ ds∥2∞ dt)
∥
0
p
TMAXZ t
0
Z
TMAX
t
Z
∥
TMAX (
0
1/4
Wν(t−s) ∗ Pf⃗ ds∥2 ∥
0
Z
0
0
t
3/4
Wν(t−s) ∗ Pf⃗ ds∥Ḣ 2 dt)1/2
≤C ′ ∥f⃗∥L2 L2 (1 + TMAX ) < +∞
and
Z
TMAX
Z
∥
0
t
Wν(t−s) ∗ P div(⃗u ⊗ ⃗u) ds∥∞ dt
0
Z
≤C
TMAX
Z
∥
0
0
t
Wν(t−s) ∗P div(⃗u ⊗ ⃗u) ds∥Ḃ 3/2 dt
2,1
Z
TMAX
≤ Cν
∥ div(⃗u ⊗ ⃗u)∥Ḃ −1/2 dt
2,1
0
≤ Cν′
Z
TMAX
∥ div(⃗u ⊗ ⃗u)∥L3/2,1 dt
0
≤ Cν′′
Z
TMAX
∥⃗u∥2Ḣ 1 dt
0
≤
Cν′′′ (∥⃗u0 ∥22
+
1
ν
Z
TMAX
∥f⃗∥2 ds)2 < +∞
0
where we have used the inequalities between Sobolev, Besov and Lorentz norms
uv∥L3/2,1 ≤ C∥u∥2 ∥v∥L6,2 , ∥v∥L6,2 ≤ C∥v∥Ḣ 1 and ∥v∥Ḃ −1/2 ≤ C∥v∥L3/2,1 (which
2,1
are easily deduced from the classical Sobolev inequalities through real interpolation)
and the maximal regularity inequality for the heat kernel
Z t
∥
Wν(t−s) ∗ ∆v ds∥L1 Ḃ −1/2 ≤ C∥v∥L1 Ḃ −1/2
0
2,1
2,1
(see [313] for a proof).
In particular, inequality (11.64) precludes the possibility of squirt singularities at the
blow-up time, as was noted by Cordóba, Fefferman and De la Llave [131]. They introduced
350
The Navier–Stokes Problem in the 21st Century (2nd edition)
the notion of squirt singularities to give a unified treatment for various singularities that
had been studied for incompressible fluid mechanics, such as potato chip singularities, tube
collapse singularities, and saddle point singularities. Roughly speaking, a squirt corresponds
to a point x0 from which fluid particles will be expelled at higher and higher speed: there
exists a positive ϵ such that, for every t < TMAX , if a fluid particle lies in B(x0 , ϵ) at time
t, then there will be a time t′ ∈ (t, TMAX ) where the particle will be expelled from the ball
B(x0 , 2ϵ).
Of course, it means that we can follow the particle. The flow associated to the vector
field ⃗u will have path lines given by the characteristic equation
Ẋ(t) = ⃗u(t, X(t)).
This equation will be solvable if ⃗u ∈ L1t Lipx . If we assume that the forcing term f⃗ belongs
more precisely to L2 H 1 , then we know that, for 0 < T0 < T1 < TMAX , we have ⃗u ∈
C([T0 , T1 ], H 2 ) ∩ L2 ((T0 , T1 ), H 3 ), so that in particular
Z T1
⃗ ⊗ ⃗u∥4∞ dt < +∞.
∥∇
T0
Thus, we may follow the particles in the fluid.
Now, if we would have a squirt singularity at x0 , we would have for a particle lying in
B(x0 , ϵ) at time t and outside B(x0 , 2ϵ) at time t′
Z t′
Z TMAX
ϵ ≤ |X(t′ ) − X(t)| ≤
|⃗u(s, X(s))| ds ≤
∥⃗u∥∞ ds
t
and thus
Z
lim −inf
t→TMAX
t
TMAX
∥⃗u∥∞ ds ≥ ϵ > 0
t
But this is impossible due to inequality (11.64).
11.8
Eigenvalues of the Strain Matrix
Recall that the equation describing the evolution of the (local) enstrophy E = 12 |⃗
ω |2 is
3
3 X
X
D
⃗ = ν∆E − ν|∇
⃗ ⊗ω
E = ∂t E + ⃗u · ∇E
⃗ |2 +
ϵi,j ωi ωj + ω
⃗ · curl f⃗
Dt
i=1 j=1
where ϵ is the strain tensor
(11.65)
1
Du + (Du)T .
2
Enstrophy is increased in the regions where the vorticity ω
⃗ aligns with the eigenvectors that
correspond to positive eigenvalues of the tensor matrix ϵ (Galanti, Gibbon and Heritage
[193], Tsinober [483]). Recall that the eigenvalues λi (t, x) of ϵ satisfy λ1 +λ2 +λ3 = div ⃗u = 0,
so that, if λ1 ≤ λ2 ≤ λ3 we have λ1 ≤ 0 and λ3 ≥ 0. As a matter of fact, the sign of λ2
plays an important role. In 1987, numerical simulations by Ashurst, Kerstein, Kerr, and
Gibson [8] indicated that, when the fluid turns to turbulent, the vorticity aligns with the
eigenvector associated to λ2 .
A criterion for blow up involving the positive part of λ2 has recently been given by Miller
[361] (see [97] for a related result of Chae):
ϵ=
Blow-up?
351
Middle eigenvalue of the strain tensor
Theorem 11.8.
Let ⃗u0 ∈ H 1 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), L2 ). Let ⃗u be a solution of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u))
with ⃗u ∈ C([0, T ], H 1 )∩L2 ((0, T ), H 2 ) for all T < TMAX and ⃗u(0, .) = ⃗u0 . Let λ1 (t, x) ≤
λ2 (t, x) ≤ λ3 (t, x) be the eigenvalues of the strain tensor ϵ(t, x). Then if and p2 + 3q = 2
with 3/2 < q ≤ +∞, we have
∥⃗u(T, .)∥2Ḣ 1 ≤ (∥⃗u0 ∥2Ḣ 1 +
p
1 ⃗ 2
C ν 1− 2
∥f ∥L2 L2 )e 0
ν
RT
0
p
∥λ+
2 (t,x)∥Lq (dx) dt
(11.66)
where λ+
2 = max(0, λ2 ) and where the constant C0 does not depend on T .
In particular, if the maximal existence time TMAX satisfies TMAX < +∞, then
R TMAX +
∥λ2 (t, x)∥pLq (dx) dt = +∞.
0
⃗ u and ϵ = (
Proof. First, we remark that we have the identities, for ω
⃗ = ∇∧⃗
⃗ ⊗ ⃗u|2 =
|∇
3 X
3
X
j=1 k=1
=
∂j uk +∂k uj
)1≤j,k≤3
2
3 X
3
X
∂j uk + ∂k uj
∂j uk − ∂k uj
|∂j uk | =
+
2
2
j=1
2
2
k=1
3 X
3
X
|ϵj,k |2 +
j=1 k=1
3
1X
2
l=1
1 2
|ωl |2 = |ϵ|2 + |⃗
ω|
2
and
⃗ ∧ (∇
⃗ ∧ ⃗u) − ∇(div
⃗
−∆⃗u = ∇
⃗u)
so that, if ⃗u ∈ H 1 ,
Z
⃗ ⊗ ⃗u|2 dx =
|∇
Z
|⃗
ω |2 dx +
Z
| div ⃗u|2 dx.
Thus, if div ⃗u = 0, we have
∥⃗u∥2Ḣ 1 = ∥⃗
ω ∥22 = 2∥ϵ∥22 .
If ⃗u ∈ H 2 , we have
Z
|∆⃗u|2 dx =
Z
3
X X
1≤j≤3 k=1
∂j2 ⃗u · ∂k2 ⃗u dx =
Z
3
X X
|∂j ∂k ⃗u|2 dx.
1≤j≤3 k=1
⃗ ∧ (∂j ⃗u), we find as well, if ⃗u ∈ H 2 and div ⃗u = 0,
As ∂j ω
⃗ =∇
⃗ ⊗ ⃗u∥2 1 = ∥⃗
∥∆⃗u∥22 = ∥∇
ω ∥2Ḣ 1 = 2∥ϵ∥2Ḣ 1 .
Ḣ
In order to show that ∥⃗u∥Ḣ 1 remains bounded, we may equivalently show that ∥ϵ∥2
remains bounded. We may now control the evolution of ∥ϵ∥22 through the equalities:
352
The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ using ∥⃗
ω ∥22 = 2∥ϵ∥22 ,
Z
d
∥ϵ∥22 = ω
⃗ · ∂t ω
⃗ dx
dt
Z
Z
Z
2
⃗
⃗
⃗
= −ν |∇ ⊗ ω
⃗ | dx − ω
⃗ · (⃗u · ∇⃗
ω−ω
⃗ · ∇⃗u) dx + ω
⃗ · curl f⃗ dx
Z
Z
⃗ ⊗ ⃗u) dx − f⃗ · ∆⃗u dx
= − ν∥⃗
ω ∥2Ḣ 1 + (⃗
ω⊗ω
⃗ ) · (∇
with
⃗ ⊗ ⃗u) =
(⃗
ω⊗ω
⃗ ) · (∇
X X
ωj ωk ∂j uk = (⃗
ω⊗ω
⃗ ) · ϵ,
1≤j,k≤3
so that
d
∥ϵ∥22 = −2ν∥ϵ∥2Ḣ 1 +
dt
Z
Z
(⃗
ω⊗ω
⃗ ) · ϵ dx −
f⃗ · ∆⃗u dx
(11.67)
ˆ using ∥⃗u∥2Ḣ 1 = 2∥ϵ∥22 ,
d
∥ϵ∥22 = −
dt
Z
∆⃗u · ∂t ⃗u dx
Z
=−ν
|∆⃗u|2 dx −
Z X
3
⃗ u) dx
∂i ⃗u · ∂i (⃗u · ∇⃗
i=1
Z
+
⃗ dx −
∆⃗u · ∇p
= − ν∥∆⃗u∥22 −
Z
∆⃗u · f⃗ dx
Z X
3 X
3 X
3
Z
∂i uj ∂i uk ∂k uj dx −
f⃗ · ∆⃗u dx
i=1 j=1 k=1
with (writing Ω = 12 (∂i uj − ∂j ui )1≤i,j≤3 )
3 X
3 X
3
X
∂i uj ∂i uk ∂k uj =
i=1 j=1 k=1
=
3 X
3 X
3
X
3 X
3 X
3
X
ϵi,j ϵi,k ϵk,j +
i=1 j=1 k=1
+
3 X
3 X
3
X
(ϵi,j + Ωi,j )(ϵi,k + Ωi,k )(ϵk,j + Ωk,j )
i=1 j=1 k=1
3 X
3 X
3
X
ϵi,j ϵi,k Ωk,j + ϵi,j Ωi,k ϵk,j + Ωi,j ϵi,k ϵk,j
i=1 j=1 k=1
ϵi,j Ωi,k Ωk,j + Ωi,j ϵi,k Ωk,j + Ωi,j Ωi,k ϵk,j +
i=1 j=1 k=1
=
3 X
3 X
3
X
3 X
3 X
3
X
ϵi,j ϵi,k ϵk,j +
3 X
3 X
3
X
ϵi,j ϵi,k Ωk,j + ϵi,j Ωi,k ϵk,j + Ωi,j ϵi,k ϵk,j
i=1 j=1 k=1
ϵi,j Ωi,k Ωk,j + Ωi,j ϵi,k Ωk,j + Ωi,j Ωi,k ϵk,j +
i=1 j=1 k=1
=
Ωi,j Ωi,k Ωk,j
i=1 j=1 k=1
i=1 j=1 k=1
+
3 X
3 X
3
X
3 X
3 X
3
X
i=1 j=1 k=1
3 X
3 X
3
X
i=1 j=1 k=1
ϵj,i ϵi,k ϵk,j + 3
3 X
3 X
3
X
i=1 j=1 k=1
ϵi,j ϵi,k
(Ωk,j + Ωj,k )
2
Ωi,j Ωi,k Ωk,j
Blow-up?
+
3 X
3 X
3
X
353
ϵi,j (Ωi,k Ωk,j + Ωi,k Ωj,k + Ωk,i Ωk,j ) +
i=1 j=1 k=1
=
3 X
3 X
3
X
(Ωi,j + Ωj,i )
Ωi,k Ωk,j
2
i=1 j=1
k=1
3 X
3 X
3
X
ϵj,i ϵi,k ϵk,j −
i=1 j=1 k=1
1
4
3 X
3
X
ϵi,j ωi ωj
i=1 j=1
so that
1
d
∥ϵ∥22 = −2ν∥ϵ∥2Ḣ 1 +
dt
4
Z
Z
(⃗
ω⊗ω
⃗ ) · ϵ dx −
tr(ϵ3 ) dx −
Z
f⃗ · ∆⃗u dx.
(11.68)
R
Combining equations (11.67) and (11.68), we get rid of the term (⃗
ω⊗ω
⃗ ) · ϵ dx and find
Z
Z
d
4
tr(ϵ3 ) dx − f⃗ · ∆⃗u dx.
∥ϵ∥22 = −2ν∥ϵ∥2Ḣ 1 −
(11.69)
dt
3
Let λ1 ≤ λ2 ≤ λ3 be the eigenvalues of ϵ. Then λ1 + λ2 + λ3 = tr(ϵ) = div ⃗u = 0,
λ1 λ2 λ3 = det(ϵ), |ϵ|2 = tr(ϵ2 ) = λ21 + λ22 + λ23 ) and tr(ϵ3 ) = λ31 + λ32 + λ33 . Moreover, we have
(λ1 + λ2 + λ3 )3 = −2(λ31 + λ32 + λ33 + 3(λ1 + λ2 + λ3 )(λ21 + λ22 + λ23 ) + 6λ1 λ2 λ3
so that
tr(ϵ3 ) = 3 det(ϵ).
As λ1 λ3 ≤ 0, we have
− det(ϵ) ≤ λ+
2 (−λ1 λ3 ) ≤
1 + 2
λ |ϵ| .
2 2
Thus, we find:
d
∥ϵ∥22 ≤ −2ν∥ϵ∥2Ḣ 1 + 2
dt
We. then write, for 3/2 < q ≤ +∞, r =
Z
2
q
q−1
Z
2
λ+
2 |ϵ| dx −
∈ [1, 3) and
Z
1
2r
f⃗ · ∆⃗u dx.
(11.70)
= (1 − σ) 21 + σ 16 (σ ∈ [0, 1))
2(1−σ)
+
+
2
2
λ+
2 |ϵ| dx ≤2∥λ2 ∥q ∥ϵ∥r/2 ≤ 2∥λ2 ∥q ∥ϵ∥2
∥ϵ∥2σ
6
1
σ
1−σ
≤ν∥ϵ∥2Ḣ 1 + Cν − 1−σ ∥λ+
∥ϵ∥22
2 ∥q
and
Z
−
1 ⃗ 2
1 ⃗ 2
ν
∥f ∥2 = ν∥ϵ∥2Ḣ 1 +
∥f ∥2
f⃗ · ∆⃗u dx ≤ ∥∆⃗u∥22 +
2
2ν
2ν
and we get
2
σ
1 ⃗ 2
d
1−σ
∥ϵ∥22 ≤ Cν − 1−σ ∥λ+
∥ϵ∥22 +
∥f ∥2 .
2 ∥q
dt
2ν
As ∥⃗u∥2Ḣ 1 = 2∥ϵ∥22 and 1 − σ =
3
2r
−
1
2
=1−
3
2q
= p2 , we get
σ
1
d
p
∥⃗u∥2Ḣ 1 ≤ Cν − 1−σ ∥λ+
u∥2Ḣ 1 + ∥f⃗∥22 .
2 ∥q ∥⃗
dt
ν
and we conclude by Grönwall’s lemma.
Remark: We can use the inequality of Lemarié-Rieusset in [316]:
∥f g∥2 ≤ Cr ∥f ∥
(11.71)
3
Ṁ 2, r
∥g∥Ḃ r
2,1
(11.72)
354
The Navier–Stokes Problem in the 21st Century (2nd edition)
for 0 < r < 1, and by duality
∥f h∥Ḃ −r ≤ Cr ∥f ∥
In particular, writing k =
∥h∥2 .
p kp
|k| |k| |k|, we get
p
∥kg∥Ḃ −r ≤ Cr2 ∥ |k|∥2
3
Ṁ 2, r
2,∞
and thus, for 3/2 < q < +∞ and
Z
3
Ṁ 2, r
2,∞
+
2
2
λ+
2 |ϵ| dx ≤ C∥λ2 ∥Ṁ 1,q ∥ϵ∥
3
2q
Ḃ2,1
2
p
+
3
q
∥g∥Ḃ r = Cr2 ∥k∥2
3
Ṁ 1, 2r
2,1
∥g∥Ḃ r
2,1
= 2,
2− q3
≤ C∥λ+
2 ∥Ṁ 1,q ∥ϵ∥2
3
2
1
2(1− p
)
q
p
∥ϵ∥Ḣ
= C∥λ+
1
2 ∥Ṁ 1,q ∥ϵ∥2 ∥ϵ∥Ḣ 1
.
Thus, we find that a necessary condition for blow up in finite time is that
R TMAX + p
∥λ2 ∥Ṁ 1,q dt = +∞.
0
Chapter 12
Leray’s Weak Solutions
12.1
The Rellich Lemma
Existence of weak solutions relies on an energy estimate (Leray energy inequality) and
on a compactness lemma in (L2t L2x )loc which goes back to the Rellich lemma.
Rellich’s lemma was published in 1930 [409].
qR In modern terms, it states that the set of
1,2
d
⃗ |2 dx ≤ 1 and f = 0 for |x| > 1
|f |2 + |∇f
functions f ∈ W (R ) such that ∥f ∥W 1,2 =
is a compact subset of L2 (Rd ). This can be easily generalized by replacing the W 1,2 norm
with a H s norm with positive s. (Another generalization was proved by Kondrashov in
1945, replacing L2 norms by Lp norms [271]; Rellich’s lemma is often quoted therefore as
the Rellich–Kondrashov lemma).
When considering an evolution equation, it is often very useful to consider a variant of
the (generalized) Rellich lemma that has been highlighted by Lions in 1961 [336]. (Again,
this theorem has been extended from the context of Hilbert spaces to more general (reflexive)
Banach spaces by Aubin [9] and Lions [337]; the extended theorem is known as the Aubin–
Lions lemma).
Theorem 12.1 (Rellich–Lions theorem).
Let I be an open interval of R and Ω an open subset of Rd . Let (un )n∈N be a sequence of
measurable functions on I × Ω such that, for every φ ∈ D(I × Ω), we have:
• for some positive α > 0,
sup ∥φun ∥L2 (I,H α (Rd )) < +∞
n∈N
• for some negative β < 0 and some p ∈ (1, 2],
sup ∥φ∂t un ∥Lp (I,H β (Rd )) < +∞
n∈N
Then, there exists a subsequence (unk )k∈N which converges strongly to a limit u in (L2t L2x )loc :
for every φ ∈ D(I × Ω),
ZZ
lim
k→+∞
φ2 |unk − u|2 dt dx = 0.
Proof. First, we consider a fixed φ. Let vn = φun . We have supn∈N ∥vn ∥L2 (I,H α (Rd )) < +∞
and supn∈N ∥∂t vn ∥Lp (I,H β (Rd )) < +∞. In particular, if θr = cr |t|r−1 is the inverse Fourier
transform of |τ |−r (0 < r < 1) and if r = p1 − 12 , we find that supn∈N ∥θr ∗ ∂t vn ∥L2t H β < +∞.
If Vn is the Fourier transform in t and x (i.e., on Rd+1 ), we find by the Plancherel equality
that
ZZ
sup
|Vn (τ, ξ)|2 (1 + |ξ|2 )α dτ dξ < +∞
n∈N
DOI: 10.1201/9781003042594-12
355
356
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
ZZ
sup
|τ |2(1−r) |Vn (τ, ξ)|2 (1 + |ξ|2 )β dτ dξ < +∞
n∈N
As β < 0 < α, we may write as well
ZZ
sup
(1 + |τ |2 )1−r |Vn (τ, ξ)|2 (1 + |ξ|2 )β dτ dξ < +∞
n∈N
(This inequality is valid as well in the case p = 2, r = 0). We now write for 0 < γ < 1 − r,
γ = γ1 + γ2 ,
ZZ
ZZ
(1 + |τ |2 + |ξ|2 )γ |Vn (τ, ξ)|2 dτ dξ ≤
(1 + |τ |2 )γ (1 + |ξ|2 )γ |Vn (τ, ξ)|2 dτ dξ
ZZ
(1−r)γ1
γ
≤(
(1 + |τ |2 )1−r |Vn (τ, ξ)|2 (1 + |ξ|2 ) γ dτ dξ) 1−r ×
ZZ
(1−r)γ2
γ
×(
|Vn (τ, ξ)|2 (1 + |ξ|2 ) 1−r−γ dτ dξ))1− 1−r
The choice γ =
β
1−r−β
γ1 = 1−r+α−β
α and γ2 = 1−r+α−β
α gives then
ZZ
sup
(1 + |τ |2 + |ξ|2 )γ |Vn (τ, ξ)|2 dτ dξ < +∞
1−r
1−r+α−β α,
n∈N
Thus, the sequence vn is bounded in H γ (R × Rd ) (with γ > 0), and the support of vn is
contained in a fixed compact set (the support of φ); thus, we may apply Rellich’s lemma
and find a subsequence that is strongly convergent in L2t L2x .
To finish the proof, it is then enough to consider an exhaustion of I × Ω by compact
sets (Kl )l∈N 1 , test functions φl ∈ D(I × Ω) such that φl = 1 on Kl and to use the Cantor
diagonal process.
12.2
Leray’s Weak Solutions
In 1934, Leray [328] exhibited global weak solutions for the Cauchy problem for the
Navier–Stokes equations with a divergence-free initial value ⃗u0 ∈ L2 and a forcing term
f⃗ ∈ L2 ((0, T ), H −1 (R3 )). He first recalled Oseen’s formula for regular solutions:
Z t
⃗ ⃗u) ds
⃗u = Wνt ∗ ⃗u0 +
Wν(t−s) ∗ P(f⃗ − ⃗u · ∇
(12.1)
0
However, L2 is not a good space for looking for solutions of this equation by Picard’s
2
2 1
iterative algorithm: if ⃗u0 ∈ L2 , then Wνt ∗ ⃗u0 ∈ L∞
g ∈ L2t Hx−1 ,
t Lx ∩ Lt Ḣx ; conversely, if ⃗
Rt
∞ 2
2 1
then 0 Wν(t−s) ∗ P⃗g ds belongs to Lt Lx ∩ Lt Ḣx on [0, T ] (T < +∞) or on every bounded
subinterval of [0, +∞) (T = +∞). But the problem is that, for ⃗u ∈ L∞ L2 ∩ L2 Ḣ 1 with
⃗ u ∈ L2 H −1 but only ⃗u · ∇⃗
⃗ u ∈ L2 H −3/2 .
div ⃗u = 0, we do not have ⃗u · ∇⃗
⃗ u with (θϵ ∗ ⃗u) · ∇⃗
⃗u
Leray’s idea was then to alleviate the non-linearity
by replacing ⃗u · ∇⃗
R
1
3
in the equation, where θ is fixed in D(R ) with θ dx = 1, θ ≥ 0, ϵ > 0 and θϵ = ϵ3 θ( xϵ ).
In modern terms, θϵ is called a mollifier (a term coined by Friedrichs in 1944 [183]).
1K
l
is compact, Kl is contained in the interior of Kl+1 and ∪l∈N Kl = I × Ω.
Leray’s Weak Solutions
357
The strategy of proof then follows three steps:
ˆ use the Picard algorithm to solve on a small interval of time the equation ∂t ⃗u + P((⃗u ∗
⃗ u) = ν∆⃗u + Pf⃗, with a time of existence controlled by the size of ∥⃗u0 ∥2
θϵ ) · ∇⃗
ˆ establish an energy estimate on this solution to get a control on its size in L2 and
thus be able to extend it globally
ˆ use the Rellich–Lions theorem to relax the mollification and get a solution to the
⃗ u) = ν∆⃗u + Pf⃗
Navier–Stokes equations ∂t ⃗u + P(⃗u · ∇⃗
The solutions we will obtain by this method satisfy the Leray energy inequality (12.2) and
will be called Leray weak solutions:
Definition 12.1.
⃗ u) =
Let ⃗u0 ∈ L2 with div ⃗u0 = 0 and f⃗ ∈ L2t Hx−1 . A weak solution ⃗u of equations ∂t ⃗u+P(⃗u·∇⃗
3
⃗
ν∆⃗u + Pf on (0, T ) × R with initial value ⃗u0 is called a Leray weak solution if it satisfies
2
2 1
• ⃗u ∈ L∞
t Lx ∩ Lt Ḣx
• for every t ∈ (0, T ),
∥⃗u(t, .)∥2 ≤ ∥⃗u0 ∥22 − 2ν
Z
t
⃗ ⊗ ⃗u∥22 ds + 2
∥∇
t
Z
0
⟨⃗u|f⃗⟩H 1 ,H −1 ds.
(12.2)
0
Leray’s mollification
Theorem 12.2.
Let ⃗u0 ∈ L2 , with div ⃗u0 = 0, and f⃗ ∈ L2t Hx−1 on (0, T ) × R3 . Then
• for ϵ > 0, the problem associated to the mollifier θϵ
⃗ u) = ν∆⃗u + Pf⃗
∂t ⃗u + P((⃗u ∗ θϵ ) · ∇⃗
(12.3)
with initial value ⃗u(0, .) = ⃗u0 has a unique solution ⃗uϵ such that ⃗uϵ ∈
on every bounded subinterval of [0, T ].
Moreover we have the following inequality:
∥⃗uϵ (t, .)∥22 + ν
Z
t
⃗ ⊗ ⃗uϵ (s, .)∥22 ≤ (∥⃗u(0, .)∥22 +
∥∇
0
1
ν
Z
0
2
2 1
L∞
t Lx ∩Lt Hx
T
∥f⃗∥2H −1 ds)eνt .
(12.4)
x
2
2 1
• there exists a sequence ϵk → 0 and a function ⃗u ∈ L∞
t Lx ∩ Lt Hx (on every bounded subinterval of [0, T ]) such that ⃗u(ϵk ) is weakly convergent to ⃗u.
Moreover ⃗u is a Leray weak solution of the Navier–Stokes problem
⃗ u) = ν∆⃗u + Pf⃗,
∂t ⃗u + P(⃗u · ∇⃗
⃗u(0, .) = ⃗u0 .
Proof.
• First step: Local existence of ⃗uϵ .
We start from the obvious inequality
∥⃗u ∗ θϵ ∥∞ ≤ ϵ−3/2 ∥⃗u∥2 ∥θ∥2 .
(12.5)
358
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, for ⃗u and ⃗v in L∞ L2 ∩ L2 H 1 with div ⃗u = 0, we have for every 0 < T0 < T ,
⃗ v∥ 2
∥(⃗u ∗ θϵ ) · ∇⃗
u ∗ θϵ ) ⊗ ⃗v ) ∥L2 Ḣ −1
L ((0,T0 ),Ḣ −1 ) =∥ div ((⃗
≤C∥(⃗u ∗ θϵ ) ⊗ ⃗v ∥L2 L2
p
≤C ′ T0 ϵ−3/2 ∥⃗u∥L∞ L2 ∥⃗v ∥L∞ L2
Let ∥⃗u∥ν,T0 = ∥⃗u∥L∞ ((0,T0 ),L2 ) +
Z
∥Wνt ∗ ⃗u0 +
0
t
√
ν∥⃗u∥L2 (0,T0 ),Ḣ 1 ) . We have
p
1
Wν(t−s) ∗ Pf⃗ ds∥ν,T0 ≤ C0 (∥⃗u0 ∥2 + √ (1 + T0 ν)∥f⃗∥L2 H −1 )
ν
and
Z
∥
0
t
p
⃗ v ) ds∥ν,T ≤ C0 √1
Wν(t−s) ∗ P((⃗u ∗ θϵ ) · ∇⃗
T0 ϵ−3/2 ∥⃗u∥L∞ L2 ∥⃗v ∥L∞ L2 .
0
ν
Thus, we find existence (and uniqueness) of a solution ⃗u = ⃗uϵ of the equation
Z
⃗u = Wνt ∗ ⃗u0 +
t
⃗ ⃗u) ds
Wν(t−s) ∗ P(f⃗ − (⃗u ∗ θϵ ) · ∇
(12.6)
0
for T0 small enough to ensure
1+
p
T0 ν ≤ 2
and
T0 ≤ ϵ3 ν
1
16 C04 (∥⃗u0 ∥2 +
⃗∥L2 H −1 )2
√2 ∥f
ν
.
(12.7)
• Second step: Energy estimates and global existence of ⃗uϵ .
To show the existence of a global solution to (12.3), it is then enough to show that the L2
norm of ⃗uϵ remains bounded (as the existence time T0 is controlled by the L2 norm of the
Cauchy data by (12.7)).
Since div (⃗uϵ ∗ θϵ ) = θϵ ∗ div ⃗uϵ = 0, we have
Z
⃗ uϵ dx = 0
⃗uϵ . (⃗uϵ ∗ θϵ ) · ∇⃗
hence
d
∥⃗uϵ ∥22 =2
dt
Z
∂t ⃗uϵ · ⃗uϵ dx
= − 2ν∥⃗uϵ ∥2Ḣ 1 + 2⟨f⃗|⃗uϵ ⟩H −1 ,H 1
1
≤ − ν∥⃗uϵ ∥2Ḣ 1 + ν∥⃗uϵ ∥22 + ∥f⃗∥2H −1
ν
(12.8)
so that
∥⃗uϵ (t, .)∥22
Z
+ν
0
t
∥⃗uϵ ∥2Ḣ 1
ds ≤
∥⃗u0 ∥22
1
+ ∥f⃗∥2L2 H −1 + ν
ν
Z
t
∥⃗uϵ (s, .)∥22 ds
0
We thus get (by Grönwall’s lemma) the energy estimate (12.4) and, therefore, the global
existence of ⃗uϵ .
Leray’s Weak Solutions
359
• Third step: Weak convergence.
From the energy estimate (12.4), we know that ⃗uϵ remains bounded in L∞ L2 ∩ L2 Ḣ 1 .
Moreover, since ∥θ∥1 = 1, we have ∥⃗uϵ ∗ θϵ ∥2 ≤ ∥⃗uϵ ∥2 . As we have
∂t ⃗uϵ = ν∆⃗uϵ + P(f⃗ − div(⃗uϵ ∗ θϵ ) ⊗ ⃗uϵ ))
we can see that ∂t ⃗uϵ remains bounded in L2 H −3/2 (on every bounded subinterval of [0, T ]).
We may then use the Rellich–Lions theorem (Theorem 12.1) for the set of functions ⃗vϵ
defined on (−T, T ) by ⃗vϵ (t, x) = ⃗uϵ (t, x) for t > 0 and ⃗vϵ (t, x) = ⃗uϵ (−t, x) for t < 02 and
find a sequence ϵn → 0 and a function ⃗u such that:
• on every bounded subinterval of [0, T ], ⃗u(ϵn ) is *-weakly convergent to ⃗u in L∞ L2 and
in L2 Ḣ 1
• ⃗u(ϵn ) is strongly convergent to ⃗u in L2loc ([0, T ) × R3 ): for every compact subset K of
RT R
R3 and every T0 < T , limn→+∞ 0 0 K |⃗uϵn − ⃗u|2 dx dt = 0.
Since ⃗uϵ is bounded in L∞ L2 , we get that ⃗u(ϵn ) ∗θϵn strongly converges ⃗u in L2loc ((0, T )×R3 )
⃗ ⊗ ⃗uϵ *-weakly converges to
(and even in (Lpt L2x )loc ((0, T ) × R3 ) for every p < +∞); as ∇
n
2
⃗
⃗
⃗ u in
∇ ⊗ ⃗u in Lloc we get that the sequence (⃗uϵn ∗ θϵn ) · ∇ ⃗uϵn is *-weakly convergent to ⃗u · ∇⃗
q −3/2
(Lt Hx
)loc for every 1 < q < 2; as the sequence is bounded in L2 H −3/2 , we have *-weak
convergence
in L2 H −3/2 as well; as P is bounded on H −3/2 , we get the *-weak convergence
⃗ ⃗uϵ to P(⃗u · ∇⃗
⃗ u) in L2 H −3/2 .
of P (⃗uϵn ∗ θϵn ) · ∇
n
Thus, the weak limit ⃗u satisfies
⃗ ⃗u).
∂t ⃗u = ν∆⃗u + P(f⃗ − ⃗u · ∇
• Fourth step: Global energy estimates for the weak limit.
We remark that, for all T < +∞, ⃗uϵ belongs to L2 ((0, T ), H 1 ) and ∂t ⃗uϵ belongs to
L2 ((0, T ), H −3/2 ), so that (from Lemma (6.1)), we can represent ⃗uϵ as
Z
⃗uϵ (t, .) = ⃗u0 +
t
∂t ⃗uϵ (s, .) ds
0
(so that ⃗uϵ ∈ C([0, T ], H −3/2 )). Moreover, ∂t ⃗uϵ is bounded in L2 ((0, T ), H −3/2 ), hence the
weak convergence of ⃗uϵn to ⃗u gives the weak convergence of ∂t ⃗uϵn to ∂t ⃗u in L2 ((0, T ), H −3/2 ),
and then the weak convergence of ⃗uϵn (t, .) to ⃗u(t, .) in H −3/2 , and in L2 as ⃗uϵ (t, .) is bounded
in L2 .
For fixed t, we thus have the weak convergence of (⃗uϵn (t, .), 10<s<t ⃗uϵn (s, .)) in L2 ×
L2 ((0, t), Ḣ 1 ) and thus
∥⃗u(t, .)∥22 + 2ν∥⃗u∥2L2 ((0,t),Ḣ 1 ) ≤ lim inf ∥⃗uϵn (t, .)∥22 + 2ν∥⃗uϵn ∥2L2 ((0,t),Ḣ 1 )
n→+∞
Z t
2
≤ lim ∥⃗u0 ∥2 + 2
⟨⃗uϵn |f⃗⟩H 1 ,H −1 ds
n→+∞
0
Z t
=∥⃗u0 ∥22 + 2
⟨⃗u|f⃗⟩H 1 ,H −1 ds
0
2 We
check easily that ∂t⃗vϵ is the distribution defined by ∂t ⃗
uϵ (t, x) for t < 0 and −∂t ⃗
uϵ (−t, x) for t < 0.
360
The Navier–Stokes Problem in the 21st Century (2nd edition)
As a matter of fact, we have a better convergence of ⃗uϵn to ⃗u than just in L2loc :
Lemma 12.1.
RT
If T0 ≤ T and T0 < +∞, then limϵn →0 0 0 ∥⃗uϵn − ⃗u∥22 dt = 0.
Proof. From estimate (12.4), we know that, if T0 ≤ T and T0 < +∞,
√
MT0 = sup ∥⃗uϵ ∥L∞ ((0,T0 ),L2 ) + ν∥⃗u∥ϵ ∥L2 ((0,T0 ),Ḣ 1 ) < +∞.
ϵ>0
We write
⃗ uϵ = (⃗uϵ ∗ θϵ ) · ∇⃗
⃗ uϵ + ∇p
⃗ ϵ.
P (⃗uϵ ∗ θϵ ) · ∇⃗
x
Let ϕ ∈ D(R3 ) be equal to 1 on the ball B(0, 1) and, for R > 1, let ϕR (x) = ϕ( R
) and
⃗uR,ϵ = (1 − ϕR (x))⃗uϵ . We have, for t < T0 ,
∥⃗uϵ,R (t, .)∥22 − ∥(1 − ϕR )⃗u0 ∥22 = 2
Z tZ
(1 − ϕR )2 ⃗uϵ · ∂t ⃗uϵ dx ds
0
=−2
3 Z tZ
X
k=1
−2
3 Z tZ
X
k=1
Z tZ
⃗ ⊗ ⃗uϵ |2 dx ds
(1 − ϕR )2 |∇
0
(∂k (1 − ϕR )2 )⃗uϵ · ∂k ⃗uϵ dx ds
0
Z tZ
2
⃗
⃗ − ϕR )2 dx ds
+
|⃗uϵ | (⃗uϵ ∗ θϵ ) · ∇(1 − ϕR ) dx ds + 2
pϵ ⃗uϵ · ∇(1
0
0
Z tZ
+2
(1 − ϕR )2 ⃗uϵ · Pf⃗ dx ds
0
Z
C t
≤
∥⃗uϵ ∥2 ∥⃗uϵ ∥Ḣ −1 + ∥⃗uϵ ∥33 + ∥⃗uϵ ∥3 ∥pϵ ∥3/2 dt
R 0
Z t
+2
(∥⃗uϵ ∥2 + ∥⃗uϵ ∥Ḣ 1 )∥(1 − ϕR )2 Pf⃗∥H −1 ds
0
!
r
p
C′
T0 2
≤
MT0 + + C ′ ( T0 + ν −1/2 )MT0 ∥(1 − ϕR )2 Pf⃗∥L2 ((0,T0 ),H −1 ) .
R
ν
2
Thus, we have
sup sup ∥(1 − ϕR )⃗uϵ ∥22
ϵ>0 0<t<T0
≤ C0 (
1
+ ∥(1 − ϕR )⃗u0 ∥22 + ∥(1 − ϕR )2 Pf⃗∥L2 ((0,T0 ),H −1 ) )
R
(12.9)
where C0 does not depend on R. As (1−ϕR )⃗uϵn (t, .) is weakly convergent in L2 to (1−ϕR )⃗u,
this estimate remains valid for (1 − ϕR )⃗u.
On the other hand, we have limn→+∞ ∥ϕR (⃗uϵn −⃗u)∥L2 ((0,T0 ),L2 ) = 0. Thus, we find that,
for every R > 1, we have
lim sup ∥⃗uϵn − ⃗uϵ ∥L2 ((0,T0 ),L2 )
n→+∞
≤ 2C0 T0 (
1
+∥(1 − ϕR )⃗u0 ∥22 + ∥(1 − ϕR )2 Pf⃗∥L2 ((0,T0 ),H −1 ) ).
R
Leray’s Weak Solutions
361
To prove the Lemma, we just have to let R go to +∞: if we write Pf⃗ = f⃗0 +
avec f⃗0 , . . . , f⃗3 ∈ L2 ((0, T0 ), L2 ), we have
∥(1 − ϕR ) Pf⃗∥L2 ((0,T0 ),H −1 ) ≤
2
3
X
P3
k=1
∂k f⃗k
3
∥(1 −
ϕR )2 f⃗i ∥L2 L2
i=0
1 X ⃗
+C
∥fi ∥L2 L2 = o(1).
R i=1
Proposition 12.1 (Strong Leray energy inequality).
The solution ⃗u constructed in Theorem 12.2 satisfies the strong Leray energy inequality: for
almost every t0 in (0, T ) and for every t ∈ (t0 , T ), we have
∥⃗u(t, .)∥22
Z
t
+ 2ν
t0
∥⃗u∥2Ḣ 1
ds ≤
∥⃗u(t0 )∥22
Z
t
⟨f⃗|⃗u⟩H −1 ,H 1 ds
+2
(12.10)
t0
Proof. 3
Let t1 > 0. For fixed t1 > t0 ≥ 0, we have the weak convergence of
(⃗uϵn (t1 , .), 1t0 <s<t1 ⃗uϵn (s, .)) in L2 × L2 ((t0 , t1 ), Ḣ 1 ) and thus
Z
∥⃗u(t1 , .)∥22 + 2ν∥⃗u∥2L2 ((t0 ,t1 ),Ḣ 1 ) ≤ lim inf ∥⃗uϵn (t1 , .)∥22 + 2ν
n→+∞
≤ lim inf ∥⃗uϵn (t0 , .)∥22 + 2
n→+∞
Z
t1
t0
t1
∥⃗uϵn ∥2Ḣ 1 ds
⟨⃗u|f⃗⟩H 1 ,H −1 ds.
t0
The problem is now to estimate lim inf n→+∞ ∥⃗uϵn (t0 , .)∥22 . By Lemma 12.1, we know that
∥⃗uϵn (t, .) − ⃗u(t, .)∥2 converges to 0 in L2 norm on (0, T0 ) for every finite T0 , hence almost
everywhere. Thus, for almost every t0 , we have limn→+∞ ∥⃗uϵn (t0 , .)∥2 = ∥⃗u(t0 , .)∥2 .
Proposition 12.2.
Let ⃗u be a Leray weak solution that satisfies the strong Leray energy inequality for almost
every t0 in (0, T ): for every t ∈ (t0 , T ), we have
∥⃗u(t, .)∥22
Z
t
+ 2ν
t0
∥⃗u∥2Ḣ 1
ds ≤
∥⃗u(t0 )∥22
Z
t
⟨f⃗|⃗u⟩H −1 ,H 1 ds.
+2
(12.11)
t0
Then it satisfies inequality (12.11) for every Lebesgue point t0 of the map t 7→ ∥⃗u(t, .)∥22 .
Proof. Let t0 < t and ϵ < t − t0 . For almost every t1 ∈ (t0 , t0 + ϵ), we have
∥⃗u(t, .)∥22 + 2ν
Z
t
t1
∥⃗u∥2Ḣ 1 ds ≤ ∥⃗u(t1 )∥22 + 2
Z
t
⟨f⃗|⃗u⟩H −1 ,H 1 ds.
t1
Integrating in t1 , we get
ZZ
1
+ 2ν
∥⃗u(s, .)∥2Ḣ 1 ds dt1
ϵ
t0 ≤t1 ≤s≤t
Z
ZZ
1 t0 +ϵ
1
∥⃗u(t1 )∥22 dt1 + 2
≤
⟨f⃗|⃗u⟩H −1 ,H 1 ds dt1 .
ϵ t0
ϵ
t0 ≤t1 ≤s≤t
∥⃗u(t, .)∥22
3 Thanks
to T. Tao’s students who noticed that the proof given in the first edition was incorrect.
362
The Navier–Stokes Problem in the 21st Century (2nd edition)
We let ϵ go to 0 and get
∥⃗u(t, .)∥22 + 2ν
Z
∥⃗u(s, .)∥2Ḣ 1 ds
t0 ≤s≤t
Z t0 +ϵ
1
≤ lim inf
ϵ→0 ϵ
∥⃗u(t1 )∥22
Z
dt1 + 2
⟨f⃗|⃗u⟩H −1 ,H 1 ds.
t0 ≤s≤t
t0
If t0 is a Lebesgue point of the map t 7→ ∥⃗u(t, .)∥22 , then
Z
1 t0 +ϵ
lim
∥⃗u(t1 )∥22 dt1 = ∥⃗u(t0 , .)∥22
ϵ→0 ϵ t
0
12.3
Weak-Strong Uniqueness: The Prodi–Serrin Criterion
Theorem 12.2 shows global existence of weak Leray solutions (when ⃗u0 ∈ L2 and f⃗ ∈
but gives no clue on whether those solutions are unique or not. If ⃗u0 belongs more
precisely to H 1 and f⃗ to L2t L2x , then Theorem 7.1 gives the local existence of a unique mild
1
2 2
solution ⃗u ∈ L∞
t H ∩ Lt H . It is easy to check that, as long as this mild solution is defined,
the Leray weak solutions coincide with this solution (and thus we have uniqueness in the
class of Leray solutions). Such a result is called weak-strong uniqueness.
L2t Hx−1 )
Weak-strong uniqueness
Theorem 12.3.
Let ⃗u0 ∈ H 1 , with div ⃗u0 = 0, and f⃗ ∈ L2t L2x on (0, T ) × R3 . Assume that the Navier–
Stokes problem
⃗ u) = ν∆⃗u + Pf⃗, ⃗u(0, .) = ⃗u0 .
∂t ⃗u + P(⃗u · ∇⃗
(12.12)
1
2 2
has a solution ⃗u1 on (0, T ) × R3 such that ⃗u1 ∈ L∞
u2 is a Leray weak
t H ∩ Lt H . If ⃗
solution of the same Navier–Stokes problem, then ⃗u2 = ⃗u1 .
Proof. We have ⃗u1 ∈ L2 H 2 with ∂t ⃗u1 ∈ L2 H −1 while ⃗u2 ∈ L2 H 1 with ∂t ⃗u2 ∈ L2 H −2 . This
is enough to get that
Z
Z t
2
⃗u1 (t, x) · ⃗u2 (t, x) dx = ∥⃗u0 ∥2 +
⟨⃗u1 |∂t ⃗u2 ⟩H 2 ,H −2 + ⟨∂t ⃗u1 |⃗u2 ⟩H −1 ,H 1 ds.
0
If w
⃗ = ⃗u2 − ⃗u1 , we write
∥w(t,
⃗ .)∥22 =∥⃗u2 (t, .)∥22 + ∥⃗u1 (t, .)∥22 − 2⟨⃗u1 (t, .)|⃗u2 (t, .)⟩L2 ,L2
=∥⃗u2 (t, .)∥22 − ∥⃗u1 (t, .)∥22 − 2⟨⃗u1 (t, .)|w(t,
⃗ .)⟩L2 ,L2
Z t
=∥⃗u2 (t, .)∥22 − ∥⃗u1 (t, .)∥22 − 2
⟨⃗u1 |∂t w⟩
⃗ H 2 ,H −2 + ⟨∂t ⃗u1 |w⟩
⃗ H −1 ,H 1 ds
0
where
ˆ the Leray energy inequality gives
∥⃗u2 (t, .)∥22 ≤ ∥⃗u0 ∥22 + 2
Z
0
t
⟨⃗u2 |f⃗⟩H 1 ,H −1 ds − 2ν
Z
0
t
⃗ ⊗ ⃗u2 ∥2 ds
∥∇
2
Leray’s Weak Solutions
RtR
⃗ u1 dx ds = 0 and thus
ˆ the regularity of ⃗u1 gives 0 ⃗u1 · (⃗u1 · ∇)⃗
∥⃗u1 (t, .)∥22
=
∥⃗u0 ∥22
Z
+2
t
⟨⃗u1 |f⃗⟩H 1 ,H −1 ds − 2ν
0
Z
363
t
⃗ ⊗ ⃗u1 ∥22 ds
∥∇
0
ˆ using the Navier–Stokes equations on ⃗u1 , we get
Z t
Z t
Z t
⃗ ⊗ ⃗u1 |∇
⃗ ⊗ w⟩
⟨∂t ⃗u1 |w⟩
⃗ H −1 ,H 1 ds = − ν
⟨∇
⃗ L2 ,L2 ds +
⟨f⃗|w⟩
⃗ H −1 ,H 1 ds
0
0
0
Z t
⃗ u1 |w⟩
−
⟨⃗u1 · ∇⃗
⃗ H −1 ,H 1 ds
0
⃗ u2 − ⃗u1 · ∇⃗
⃗ u1 ), we get
ˆ using ∂t w
⃗ = ν∆w
⃗ − P(⃗u2 · ∇⃗
Z
0
t
Z t
⃗ ⊗ ⃗u1 |∇
⃗ ⊗ w⟩
⟨⃗u1 |∂t w⟩
⃗ H 2 ,H −2 , ds = − ν
⟨∇
⃗ L2 ,L2 ds
0
Z t
⃗ u2 − ⃗u1 · ∇⃗
⃗ u1 ⟩H 2 ,H −2 ds
−
⟨⃗u1 |⃗u2 · ∇⃗
0
ˆ Moreover, we have (since div ⃗u1 = div ⃗u2 = 0)
⃗ u1 ⟩H 2 ,H −2 = ⟨⃗u1 |⃗u2 · ∇⃗
⃗ u1 ⟩H 2 ,H −2 = 0
⟨⃗u1 |⃗u1 · ∇⃗
and
⃗ u1 |w⟩
⃗ w⟩
⟨⃗u1 · ∇⃗
⃗ H −1 ,H 1 = −⟨⃗u1 |⃗u1 · ∇
⃗ H 2 ,H −2
so that
⃗ u1 |w⟩
⃗ u2 − ⃗u1 · ∇⃗
⃗ u1 ⟩H 2 ,H −2 = ⟨⃗u1 |w
⃗ w⟩
⟨⃗u1 · ∇⃗
⃗ H −1 ,H 1 + ⟨⃗u1 |⃗u2 · ∇⃗
⃗ ·∇
⃗ H 2 ,H −2
This gives finally that
Z t
Z t
Z t
1
⃗ w⟩
∥w(t,
⃗ .)∥22 ≤ −2ν
∥w∥
⃗ 2Ḣ 1 ds + 2
⟨⃗u1 |w
⃗ ·∇
⃗ H 2 ,H −2 ds ≤
∥⃗u1 ∥2∞ ∥w∥
⃗ 22 ds.
2ν
0
0
0
We then conclude by Grönwall’s lemma, as
RT
0
∥⃗u1 ∥2∞ ds ≤ C
p
∥⃗u1 ∥L2 Ḣ 1 ∥⃗u1 ∥L2 Ḣ 2 .
Corollary 12.1.
Let ⃗u0 ∈ L2 and f⃗ ∈ L2 ((0, +∞), L2 ) ∩ L2 ((0, +∞), Ḣ −1 ). Then the Navier–Stokes problem
⃗ u) = ν∆⃗u + Pf⃗,
∂t ⃗u + P(⃗u · ∇⃗
⃗u(0, .) = ⃗u0 .
(12.13)
2
2 1
has a weak solution ⃗u on (0, +∞) × R3 such that ⃗u ∈ L∞
u satisfies the
t L ∩ Lt Ḣ and ⃗
strong Leray inequality.
Moreover, we have
lim ∥⃗u(t, .)∥2 = 0.
t→+∞
Proof. We construct ⃗u by Leray’s mollification (see Theorem 12.2). The global control of ⃗u
2
2 1
in L∞
t L ∩ Lt Ḣ is provided by the inequality
∥⃗u(t, .)∥22 + ν
Z
0
t
⃗ ⊗ ⃗u∥2 ds ≤ ∥⃗u0 ∥2 +
∥∇
2
2
1
ν
Z
0
t
∥f⃗∥2Ḣ −1 ds.
364
The Navier–Stokes Problem in the 21st Century (2nd edition)
Recall that we have seen in Theorem 7.3 that if ⃗u(t0 , .) ∈ H 1 , f⃗ ∈ L2 ((t0 , +∞), L2 ) ∩
R +∞
L2 ((t0 , +∞), Ḣ −1/2 ) and if moreover ∥⃗u(t0 , .)∥Ḣ 1/2 < ϵ0 ν and t0 ∥f⃗(s, .)∥2 − 1 ds < ϵ20 ν 3 ,
Ḣ 2
then the Navier–Stokes problem with forcing term f⃗ and value ⃗u(t0 , .) at time t = t0 has a
global solution ⃗v on (t0 , +∞) which belongs to C([t0 , +∞), H 1 ) ∩ L2 (t0 , +∞), Ḣ 2 ).
2
2 1
4 1/2
As ⃗u belongs to L∞
. Thus, the set of times t such that
t L ∩ Lt Ḣ , it belongs to Lt Ḣ
R +∞
∥⃗u(t, .)∥Ḣ 1/2 ≥ ϵ0 ν is of finite measure. As we have limt→+∞ t
∥f⃗(s, .)∥2 − 1 ds = 0 and
Ḣ
2
as the set of Lebesgue points of t 7→ ∥⃗u(t, .)∥22 has a complement of null measure, we may
find a time t0 such that
ˆ ∥⃗u(t0 , .)∥Ḣ 1/2 < ϵ0 ν
R +∞
ˆ t0 ∥f⃗(s, .)∥2 − 1 ds < ϵ20 ν 3
Ḣ
2
ˆ ⃗u is a weak Leray solution on (t0 , +∞): for every t ∈ (t0 , +∞), we have
Z t
Z t
2
2
2
∥⃗u(t, .)∥2 + 2ν
∥⃗u∥Ḣ 1 ds ≤ ∥⃗u(t0 )∥2 + 2
⟨f⃗|⃗u⟩H −1 ,H 1 ds
t0
t0
Then, by the weak-strong uniqueness theorem of Serrin, we find that ⃗u coincides on (t0 , +∞)
with the mild solution ⃗v ∈ C([t0 , +∞), H 1 ) ∩ L2 ((t0 , +∞), Ḣ 2 ). Thus, we shall prove the
corollary if we prove that limt→+∞ ∥⃗v (t, .)∥2 = 0. If t0 < τ < t, we find that
Z t
⃗v (t, .) = Wν(t−τ ) ∗ ⃗v (τ, .) +
Wν(t−s) ∗ P(f⃗ − div(⃗v ⊗ ⃗v )) ds
τ
so that
Z t
∥⃗v (t, .)∥2 ≤ ∥Wν(t−τ ) ∗ ⃗v (τ, .)∥2 + C(
∥Pf⃗∥2Ḣ −1 + ∥P div(⃗v ⊗ ⃗v )∥2Ḣ −1 ds)1/2
τ
which gives
lim sup ∥⃗v (t, .)∥2 ≤
Z
C (
τ
t→+∞
+∞
Z
2
1/2
⃗
∥f ∥Ḣ − 1 ds)
+ sup ∥⃗v (t, .)∥Ḣ 1/2 (
t>t0
τ
+∞
∥⃗v ∥2Ḣ 1
1/2
ds)
.
Letting τ go to +∞, we get
lim ∥⃗v (t, .)∥2 = 0.
t→+∞
Weak-strong uniqueness has been proved under many various assumptions, in a generalization of the proof of Theorem 12.3. The idea is to consider two solutions ⃗u1 and ⃗u2 of
the same Navier–Stokes problem associated to ⃗u0 ∈ L2 and f⃗ ∈ L2 H −1 such that ⃗u1 and ⃗u2
belong to L2 H 1 ∩ L∞ L2 (hence ∂t ⃗u1 and ∂t ⃗u2 belong to L2 H −3/2 ), and with assumptions
that ⃗u2 is a Leray solution and that ⃗u1 satisfies ⃗u1 ∈ X for some well-chosen space X, and
to try to find a control on ∥w(t,
⃗ .)∥2 for w
⃗ = ⃗u2 − ⃗u1 .
The first step is to write a convenient representation for ∥w∥
⃗ 22 . We write again
∥w(t,
⃗ .)∥22 = ∥⃗u2 (t, .)∥22 − ⟨⃗u1 |⃗u1 + 2w⟩
⃗ L2 ,L2 .
We then use a mollifier θϵ and write
∥w(t,
⃗ .)∥22 = ∥⃗u2 (t, .)∥22 − lim+ ⟨⃗u1 ∗ θϵ |⃗u1 + 2w⟩
⃗ L2 ,L2 .
ϵ→0
Leray’s Weak Solutions
365
We have ⃗u1 ∗θϵ ∈ L2 H 3/2 and ∂t (⃗u1 ∗θϵ ) ∈ L2 H −1 , while ⃗u1 +2w
⃗ ∈ L2 H 1 and ∂t (⃗u1 +2w)
⃗ ∈
2 −3/2
L H
. Thus, we may write
⟨⃗u1 ∗ θϵ |⃗u1 + 2w⟩
⃗ L2 ,L2 =⟨⃗u0 ∗ θϵ |⃗u0 ⟩L2 ,L2
t
Z
⟨∂t (⃗u1 ∗ θϵ )|⃗u1 + 2w⟩
⃗ H −1 ,H 1 + ⟨⃗u1 ∗ θϵ |∂t (⃗u1 + 2w)⟩
⃗ H 3/2 ,H −3/2 ds
+
0
and thus
⟨⃗u1 |⃗u1 + 2w⟩
⃗
L2 ,L2
=
∥⃗u0 ∥22
Z
t
⃗ ⊗ ⃗u1 |∇(⃗
⃗ u1 + 2w)⟩
− 2ν
⟨∇
⃗ L2 ,L2 ds
0
Z t
+2
⟨f⃗|⃗u1 + w⟩
⃗ H −1 ,H 1 ds − lim+ Jϵ
ϵ→0
0
with
t
Z
Jϵ =
⃗ u1 ) ∗ θϵ |⃗u1 ⟩H −1 ,H 1 + ⟨⃗u1 ∗ θϵ |⃗u1 · ∇⃗
⃗ u1 ⟩H 3/2 ,H −3/2 ds
⟨(⃗u1 · ∇⃗
Z t
⃗ u1 ) ∗ θϵ |w⟩
⃗ u1 + ⃗u2 · ∇⃗
⃗ u2 ⟩H 3/2 ,H −3/2 ds.
+2
⟨(⃗u1 · ∇⃗
⃗ H −1 ,H 1 + ⟨⃗u1 ∗ θϵ | − ⃗u1 · ∇⃗
0
0
Recalling now that ⃗u2 satisfies the Leray energy inequality, we get
Z t
2
⃗ ⊗ w∥
∥w(t,
⃗ .)∥2 ≤ − 2ν
∥∇
⃗ 22 ds + lim+ Jϵ .
(12.14)
ϵ→0
0
Inequality (12.14) has been established for any solution ⃗u1 in L2 H 1 ∩L∞ L2 . The problem
is now to see for which spaces X the condition ⃗u1 ∈ X allows one to express the limit in
(12.14) and to get w
⃗ = 0.
The Prodi–Serrin uniqueness criterion
Theorem 12.4.
Let ⃗u0 ∈ L2 , with div ⃗u0 = 0, and f⃗ ∈ L2t Hx−1 on (0, T ) × R3 . Assume that the Navier–
Stokes problem
⃗ u) = ν∆⃗u + Pf⃗, ⃗u(0, .) = ⃗u0 .
∂t ⃗u + P(⃗u · ∇⃗
(12.15)
(0)
2
2 1
has a solution ⃗u1 on (0, T ) × R3 such that ⃗u1 ∈ L∞
t L ∩ Lt H ∩ XT , where
2
2 1
• XT is the space of pointwise multipliers on (0, T ) × R3 from L∞
t L ∩ Lt Ḣ to
2 2
Lt Lx , normed with ∥u∥XT = sup∥v∥L∞ L2 +∥v∥L2 Ḣ 1 ≤1 ∥uv∥L2 L2 ;
t
t
(0)
• XT is the space of multipliers u in XT such that, for every t0 ∈ [0, T ),
limt1 →t+ ∥1(t0 ,t1 ) (t)u(t, x)∥XT = 0.
0
If ⃗u2 is a Leray weak solution of the same Navier–Stokes problem, then ⃗u2 = ⃗u1 .
2
2 1
Proof. If ⃗u1 ∈ L∞
u1 = 0 and ⃗v ∈ L2 H 1 , we write
t L ∩ Lt H ∩ XT with div ⃗
Z
0
t
⃗ u1 ) ∗ θϵ |⃗v ⟩H −1 ,H 1 ds = −
⟨(⃗u1 · ∇⃗
Z
0
t
⃗ ⊗ ⃗v ⟩L2 ,L2 ;
⟨(⃗u1 ⊗ ⃗u1 ) ∗ θϵ |∇
366
The Navier–Stokes Problem in the 21st Century (2nd edition)
since ⃗u1 ⊗ ⃗u1 ∈ L2 L2 , we have
Z t
Z t
⃗ u1 ) ∗ θϵ |⃗v ⟩H −1 ,H 1 ds = −
⃗ ⊗ ⃗v ⟩L2 ,L2 .
lim
⟨(⃗u1 ∇⃗
⟨⃗u1 ⊗ ⃗u1 |∇
ϵ→0+
0
2
L∞
t L
0
2
L∞
t L
L2t H 1
2
If ⃗u1 ∈
∩
∩ XT and ⃗v ∈
∩ L H 1 , we have that limϵ→0+ ∥⃗u1 ∗ θϵ −
⃗u1 ∥L2 L6 = 0, so that limϵ→0+ ∥(⃗u1 ∗ θϵ − ⃗u1 ) ⊗ ⃗v ∥L2 L3/2 = 0. Moreover
Z
∥(⃗u1 ∗ θϵ ) ⊗ ⃗v ∥L2 L2 ≤ ∥⃗u1 (t, x − y) ⊗ ⃗v (x)∥L2 L2 θϵ (y) dy
Z
= ∥⃗u1 (t, x) ⊗ ⃗v (x + y)∥L2 L2 θϵ (y) dy
Z
≤∥⃗u1 ∥XT
∥⃗v (t, x + y)∥L∞ L2 ∩L2 Ḣ 1 θϵ (y) dy
=∥⃗u1 ∥XT ∥⃗v ∥L∞ L2 ∩L2 Ḣ 1 .
This gives that (⃗u1 ∗ θϵ ) ⊗ ⃗v is weakly convergent in L2 L2 to ⃗u1 ⊗ ⃗v and thus
t
Z
⃗ v ⟩H 3/2 ,H −3/2 ds =
⟨⃗u1 ∗ θϵ |⃗v · ∇⃗
lim
ϵ→0+
0
Z tZ
⃗ v dx ds.
⃗u1 · (⃗v · ∇)⃗
0
2
2 1
Moreover, we have, for ⃗v1 and ⃗v2 in ⃗v ∈ L∞
u1 = 0 and (⃗u1 ∗ θϵ ) ⊗⃗vi
t L ∩ L Ḣ (since div ⃗
2 2
is weakly convergent in L L to ⃗u1 ⊗ ⃗vi )
Z tZ
0
⃗ v2 dx ds = lim
⃗v1 · (⃗u1 · ∇)⃗
+
Z tZ
⃗ v2 dx ds
⃗v1 · ((⃗u1 ∗ θϵ ) · ∇)⃗
Z tZ
⃗ v1 dx ds
= − lim+
⃗v2 · ((⃗u1 ∗ θϵ ) · ∇)⃗
ϵ→0
0
Z tZ
⃗ v1 dx ds
=−
⃗v2 · (⃗u1 · ∇)⃗
ϵ→0
0
0
Thus,
RtR
0
⃗ u1 dx ds = 0 and we may transform inequality (12.14) into
⃗u1 · (⃗u1 · ∇)⃗
∥w(t,
⃗ .)∥22 ≤ − 2ν
Z
t
⃗ ⊗ w∥
∥∇
⃗ 22 ds + 2
0
Z
Z tZ
⃗ u1 ) · w
⃗ w)
(⃗u1 · ∇⃗
⃗ + ⃗u1 · (⃗u2 · ∇
⃗ dx ds
0
t
= − 2ν
⃗ ⊗ w∥
∥∇
⃗ 22 ds + 2
0
Z tZ
(12.16)
⃗ w)
⃗u1 · (w
⃗ ·∇
⃗ dx ds
0
If 0 ≤ t0 < t1 < T are such that w
⃗ = 0 on [0, t0 ], we find that, on [0, t1 ],
∥w(t,
⃗ .)∥22 + 2ν
Z
t
⃗ ⊗ w∥
∥∇
⃗ 22 ds ≤
(12.17)
0
2
2(∥w∥
⃗ L2 ((0,t1 ),Ḣ 1 ) + ∥w∥
⃗ L∞ ((0,t1 ),L2 ) ) ∥1(t0 ,t1 ) ⃗u1 ∥XT
(0)
1
If 4(1 + 2ν
)∥1(t0 ,t1 ) ⃗u1 ∥XT < 1, we obtain w
⃗ = 0 on [0, t1 ]. Thus, if ⃗u1 ∈ XT , we find that
w
⃗ = 0 on (0, T ) and ⃗u2 = ⃗u1 .
Leray’s Weak Solutions
367
(0)
Remark: Of course, the assumption ⃗u1 ∈ XT is very restrictive on ⃗u0 : if f⃗ = 0, we
may apply the theory developed in Chapter 5. Indeed, we have obviously that 10<t<T ⃗u1 ∈
V 2,1 (R × R3 ). As the bilinear operator
Z
t
Wν(t−s) ∗ P div(⃗u ⊗ ⃗v ) ds
B(⃗u, ⃗v ) = 1t>0
0
is bounded on V 2,1 (R × R3 ), we find that 10<t<T Wνt ∗ ⃗u0 ∈ V 2,1 (R × R3 ). Moreover, since
(0)
⃗u1 ∈ XT , we find that limt0 →0+ ∥10<t<t0 Wνt ∗ ⃗u0 ∥V 2,1 (R×R3 ) = 0. This gives that ⃗u1 must
be the mild solution associated to ⃗u0 through Picard’s algorithm.
Proposition 12.3.
(0)
Let XT be the space described in Theorem 12.4. Then
(0)
• for p2 + 3q = 1 and 2 ≤ p < +∞, we have Lpt Lqx ⊂ XT (this is the original Prodi–Serrin
criterion [406, 435] )
(0)
• for p2 + 3q = 1 and 2 ≤ p < +∞, we have Lpt M(Ḣ 3/q 7→ L2 ) ⊂ XT (Lemarié-Rieusset
[313])
• for
2
p
+
3
q
(0)
= 1 and 2 ≤ p < +∞, we have Lpt Ṁ 2,q ⊂ XT (Lemarié–Rieusset [316])
(0)
• C([0, T ], L3 ) ⊂ XT (Von Wahl [494])
(0)
• if V01 is the closure of L3 in M(Ḣ 1 7→ L2 ), then C([0, T ], V01 ) ⊂ XT (Lemarié–Rieusset
[313])
Proof. For q > 3/2, we have Lq ⊂ M(Ḣ 3/q 7→ L2 ) ⊂ Ṁ 2,q . Moreover, we have Ṁ 2,q =
3/q
M(Ḃ2,1 7→ L2 ). (A simple proof of this statement, using a decomposition on a wavelet
basis, is given in Lemarié-Rieusset [316]).
3/q
When 2 < p < ∞, we consider Lp Ṁ 2,q . We have Ḃ2,1 = [L2 , Ḣ 1 ]3/q,1 , hence ∥v∥Ḃ 3/q ≤
2,1
1−3/q
C∥v∥2
3/q
3/q
∥v∥Ḣ 1 and thus L∞ L2 ∩ L2 Ḣ 1 ⊂ Lr Ḃ2,1 with
1
r
=
3
2q
=
1
2
− p1 . Thus, Lp Ṁ 2,q ⊂
(0)
XT .
When p = 2, we have obviously that L2 L∞ is a pointwise multiplier from L∞ L2 to L2 L2
(0)
so that L2 L∞ ⊂ XT . (Remark: L∞ = M(L2 7→ L2 ) = Ṁ 2,∞ .)
When p = ∞, we have obviously that L∞ V 1 is a pointwise multiplier from L2 Ḣ 1 to
2 2
L L . Moreover, if ϵ > 0, we may split u ∈ C([0, T ], V01 ) into u = v + w, where v ∈
C([0, T ], V01 ) ∩ L∞ L∞ and ∥w∥L∞ V 1 < ϵ. Thus, if t0 < t1 , we find
√
∥1(t0 ,t1 ) u∥XT ≤ ∥1(t0 ,t1 ) v∥XT + ∥1(t0 ,t1 ) w∥XT ≤ t1 − t0 ∥v∥L∞ L∞ + ∥w∥L∞ V 1
so that
lim sup ∥1(t0 ,t1 ) u∥XT ≤ ϵ
t1 →t+
0
(0)
As ϵ is arbitrary, we find that C([0, T ], V01 ) ⊂ XT .
The endpoints of the Prodi–Serrin criterion have been slightly extended by Kozono and
Taniuchi [277] when p = 2 and Kozono and Sohr [275] (extending a result of Masuda [350])
when p = ∞.
368
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proposition 12.4.
Let ⃗u0 ∈ L2 , with div ⃗u0 = 0, and f⃗ ∈ L2t Hx−1 on (0, T )×R3 . Assume that the Navier–Stokes
problem
⃗ u) = ν∆⃗u + Pf⃗, ⃗u(0, .) = ⃗u0 .
∂t ⃗u + P(⃗u · ∇⃗
(12.18)
2
2 1
has two solutions ⃗u1 and ⃗u2 on (0, T ) × R3 such that ⃗u1 and ⃗u2 belong to L∞
t L ∩ Lt H
and that ⃗u2 is a weak Leray solution. Then
• If ⃗u1 belongs to L2t BM O, then ⃗u2 = ⃗u1 .
3
∞ p
⃗
• If ⃗u1 belongs to L∞
u2 = ⃗u1 .
t L and f belongs to Lt L for some p ∈ (1, 3), then ⃗
Proof.
(a) Case ⃗u1 ∈ L2t BM O: we start from inequality (12.14):
∥w(t,
⃗ .∥22 ≤ − 2ν
Z
t
⃗ ⊗ w∥
∥∇
⃗ 22 ds − lim Jϵ .
ϵ→0+
0
The div-curl lemma of Coifman, Lions, Meyer, and Semmes [124, 313] gives that ⃗u1 ·
⃗ u1 and ⃗u2 ·∇⃗
⃗ u2 belong to L2 H1 (where H1 is the Hardy space, the pre-dual of BM O).
∇⃗
⃗ ϵ ∗ w)
Similarly, ⃗u1 · ∇(θ
⃗ belongs to L2 H1 and is controlled by ∥⃗u1 ∥L∞ L2 ∥θϵ ∗ w∥
⃗ L2 Ḣ 1 .
2 1
Thus, as we have the strong convergence of θϵ ∗ w
⃗ to w
⃗ in L Ḣ , we find that
Z t
⃗ ⊗ w∥
∥w(t,
⃗ .)∥22 ≤ − 2ν
∥∇
⃗ 22 ds
0
Z t
⃗ u1 |⃗u1 ⟩H1 ,BM O + ⟨⃗u1 |⃗u1 · ∇⃗
⃗ u1 ⟩BM O,H1 ds
+
⟨⃗u1 · ∇⃗
0
Z t
⃗ w⟩
⃗ u1 + ⃗u2 · ∇⃗
⃗ u2 ⟩BM O,H1 ds
+2
− ⟨⃗u1 |⃗u1 · ∇
⃗ BM O,,H1 + ⟨⃗u1 | − ⃗u1 · ∇⃗
0
Moreover, we have, for j = 1, 2,
Z t
Z t
⃗ u1 ⟩BM O,H1 ds = −
⃗ ϵ ∗ ⃗u1 )⟩BM O,H1 ds
⟨⃗u1 ∗ θϵ |⃗uj · ∇⃗
⟨⃗u1 |⃗uj · ∇(θ
0
0
so that
Z
t
⃗ u1 ⟩BM O,H1 ds = −
⟨⃗u1 |⃗uj · ∇⃗
t
Z
0
⃗ u1 ⟩BM O,H1 ds = 0.
⟨⃗u1 |⃗uj · ∇⃗
0
This gives finally
∥w(t,
⃗ .)∥22 ≤ − 2ν
Z
0
t
⃗ ⊗ w∥
∥∇
⃗ 22 ds + 2
Z
t
⃗ w⟩
⟨⃗u1 |w
⃗ ·∇
⃗ BM O,,H1 ds
0
Z t
Z t
⃗ ⊗ w∥
≤ − 2ν
∥∇
⃗ 22 ds + 2C
∥⃗u1 ∥BM O ∥w∥
⃗ 2 ∥w∥
⃗ Ḣ 1 ds
0
0
Z
C2 t
∥⃗u1 ∥2BM O ∥w∥
⃗ 22 ds
≤
2ν 0
(12.19)
and we conclude w
⃗ = 0 by Grönwall’s lemma.
3
(b) Case ⃗u1 ∈ L∞
u1 and ∂t ⃗u2 belong to L2 H −3/2 , ⃗u1 and ⃗u2 are continuous from
t L : as ∂t ⃗
−3/2
[0, T ) to H
, thus the set of times t such that ⃗u1 = ⃗u2 is closed. If ⃗u1 =
̸ ⃗u2 , let T ∗
∗
be the maximal time such that ⃗u1 = ⃗u2 on [0, T ]. As ⃗u1 is bounded in L3 and in L2
Leray’s Weak Solutions
369
and continuous in H −3/2 , we find that it is weakly continuous from [0, T ) to L3 ∩ L2 ;
in particular, ⃗u1 (T ∗ ) ∈ L3 ∩ L2 . It is easy to check that, following Theorem 7.5, we
may construct a solution ⃗u3 on a small interval [T ∗ , T ∗ + ϵ] such that ⃗u3 belongs to
2
2 1
∗
∗
3
L∞
u1 belongs to XT , so that ∂t ⃗u1 ∈ L2 H −1
t L ∩ Lt H ∩ C([T , T + ϵ], L ). But ⃗
and ⃗u1 satisfies the Leray energy equality on [0, T ), while ⃗u2 satisfies the same Leray
energy equality on [0, T ∗ ]; thus, ⃗u1 and ⃗u2 are weak Leray solutions on [T ∗ , T ∗ + ϵ] and
applying Proposition 12.3 to ⃗u3 , we find that ⃗u3 = ⃗u1 and ⃗u3 = ⃗u2 , so that ⃗u1 = ⃗u2
on [0, T ∗ + ϵ], which contradicts the definition of T ∗ .
Of course, L3 does not play a specific role in the Kozono–Sohr theorem. A general result
is the following one:
Proposition 12.5.
Let V01 be the closure of M(Ḣ 1 7→ L2 ). Let E be a Banach space such that
• E ⊂ V01 (continuous embedding)
• E is the dual of a Banach space E0 such that D is dense in E0 .
Let ⃗u0 ∈ L2 , with div ⃗u0 = 0. Assume that the Navier–Stokes problem
⃗ u) = ν∆⃗u,
∂t ⃗u + P(⃗u · ∇⃗
⃗u(0, .) = ⃗u0 .
(12.20)
2
2 1
has two solutions ⃗u1 and ⃗u2 on (0, T ) × R3 such that ⃗u1 and ⃗u2 belong to L∞
t L ∩ Lt H
∞
and that ⃗u2 is a weak Leray solution. If ⃗u1 belongs to Lt E, then ⃗u2 = ⃗u1 .
Proof. The proof is similar to the proof of Kozono and Sohr’s theorem. As ∂t ⃗u1 and ∂t ⃗u2
belong to L2 H −3/2 , ⃗u1 and ⃗u2 are continuous from [0, T ) to H −3/2 , thus the set of times
t such that ⃗u1 = ⃗u2 is closed. If ⃗u1 ̸= ⃗u2 , let T ∗ be the maximal time such that ⃗u1 = ⃗u2
on [0, T ∗ ]. As ⃗u1 is bounded in E and in L2 and continuous in H −3/2 , we find that it is
weakly continuous from [0, T ) to E ∩ L2 ; in particular, ⃗u1 (T ∗ ) ∈ E ∩ L2 . It is easy to check
that, following Theorem 8.2, we may construct a solution ⃗u3 on a small interval [T ∗ , T ∗ + ϵ]
such that ⃗u3 belongs to C([T ∗ , T ∗ + ϵ], V01 ). Moreover, it is easy to check that this solution
2
2 1
belongs to L∞
t L ∩ Lt H .
⃗u1 belongs to XT , so that ∂t ⃗u1 ∈ L2 H −1 and ⃗u1 satisfies the Leray energy equality on
[0, T ), while ⃗u2 (which is equal to ⃗u1 on [0, T ∗ ]) satisfies the same Leray energy equality on
[0, T ∗ ]; thus, ⃗u1 and ⃗u2 are weak Leray solutions on [T ∗ , T ∗ + ϵ] and applying Proposition
12.3 to ⃗u3 , we find that ⃗u3 = ⃗u1 and ⃗u3 = ⃗u2 , so that ⃗u1 = ⃗u2 on [0, T ∗ + ϵ], which
contradicts the definition of T ∗ .
Example: mixed-norm Lebesgue spaces.
Obvious examples of spaces E that fulfill the hypotheses of Proposition 12.5 are the
Lorentz spaces L3,q with 1 ≤ q < +∞. But we may find other examples, such as the case
of mixed-norm Lebesgue spaces that has been recently considered by Phan and Robertson
[396]:
Definition 12.2.
L(p1 ,p2 ,p3 ) = Lpx33 Lpx22 Lpx11 is the space of measurable functions f on R3 such that

Z
∥f ∥L(p1 ,p2 ,p3 )
=
Z Z
p1
|f (x1 , x2 , x3 )|
! pp3
pp2
2
1
dx1
dx2
 p1
3
dx3 
< +∞.
370
The Navier–Stokes Problem in the 21st Century (2nd edition)
Phan and Robertson’s result states that the weak-strong uniqueness result stated in
Proposition 12.5 holds for E = L(p1 ,p2 ,p3 ) , where p1 , p2 , p3 ∈ [2, +∞), p11 + p12 + p13 = 1
and p3 > 2. To prove this, we only need to check that L(p1 ,p2 ,p3 ) ⊂ M(Ḣ 1 7→ L2 ).
As L(p1 ,p2 ,p3 ) ⊂ Ṁ min(p1 ,p2 ,p3 ),3 for p11 + p12 + p13 = 1 and as Ṁ p,3 ⊂ M(Ḣ 1 7→ L2 )
for 2 < p ≤ 3, the result is obvious for p1 ̸= 2 and p2 =
̸ 2. In order to prove the
result for the general case (including the cases where p1 or p2 is equal to 2), Phan and
Robertson use a Sobolev embedding theorem in mixed-norm spaces they found in the book
by Besov, Il’in and Nikol’skiı̆ [41]: for 2 ≤ q1 , q2 , q3 ≤ +∞ with q11 + q12 + q13 = 12 and
2 < q3 < +∞, we have the continuous embedding Ḣ 1 ⊂ Lq1 ,q2 ,q3 .
The proof of Kozono and Tanyuchi suggested to many authors a further extension of the
Prodi-Serrin criterion for 1 ≤ p < 2, using paradifferential calculus (Ribaud [412], Gallagher
and Planchon [201], Germain [204]). However, their results were generalized by Chen, Miao,
and Zhang [115] in a very simple way that does not use para-differential calculus.
Proposition 12.6 (Chen, Miao, and Zhang).
Let ⃗u0 ∈ L2 , with div ⃗u0 = 0, and f⃗ ∈ L2t Hx−1 on (0, T )×R3 . Assume that the Navier–Stokes
problem
⃗ u) = ν∆⃗u + Pf⃗, ⃗u(0, .) = ⃗u0
∂t ⃗u + P(⃗u · ∇⃗
(12.21)
2
2 1
has two solutions ⃗u1 and ⃗u2 on (0, T ) × R3 such that ⃗u1 and ⃗u2 belong to L∞
t L ∩ Lt H
2 ∞
∞ 2
2 1
∞ 2
and that ⃗u2 is a weak Leray solution. If ⃗u1 belongs to (Lt L ∩ Lt H ∩ L L ) + (Lt L ∩
L2t H 1 ∩ L1 Ẇ 1,∞ ), then ⃗u2 = ⃗u1 .
r
with 1 < p < 2 and p2 = 1 + r, then ⃗u2 = ⃗u1 .
In particular, if ⃗u1 ∈ Lp Ḃ∞,∞
r
(0 < r < 1) (the spaces of Hölderian functions of
Proof. First, let us remark that Ḃ∞,∞
1,∞
Hölder exponent r) and Ẇ
(the space of Lipschitzian functions) are defined modulo the
r
or L2 ∩ Ẇ 1,∞ , the constants are fixed.
constants; however, on L2 ∩ Ḃ∞,∞
2
2
2 1
2 ∞
∞ 2
2 1
∞ 2
1+r Ḃ r
We now check that L∞
t L ∩ Lt H ∩ L
∞,∞ ⊂ (Lt L ∩ Lt H ∩ L L ) + (Lt L ∩
L2t H 1 ∩ L1 Ẇ 1,∞ ). Indeed, we use a mollifier θϵ and write
u(t, x) = u ∗ θϵ(t) + (u − u ∗ θϵ(t) ) = U + V.
2 1
∞ 2
2
2 1
⃗
We have, of course, U ∈ L∞
t L ∩ Lt H and V ∈ Lt L ∩ Lt H . Moreover, ∥∇U (t, .)∥∞ ≤
−1+r
C∥u(t, .)∥Ḃ r ϵ(t)
while ∥V (t, .)∥∞ ≤ C∥u(t, .)∥Ḃ r ϵ(t)r . The choice ϵ(t) =
∞,∞
−p/2
∥u(t, .)∥Ḃ r
∞,∞
gives U ∈ L1 Ẇ 1,∞ and V ∈ L2 L∞ .
∞,∞
We now prove the general case. We start again from inequality (12.14):
∥w(t,
⃗ .∥22
Z
≤ − 2ν
0
t
⃗ ⊗ w∥
∥∇
⃗ 22 ds + lim+ Jϵ
ϵ→0
with
Z
Jϵ =
t
⃗ u1 ) ∗ θϵ |⃗u1 ⟩H −1 ,H 1 + ⟨⃗u1 ∗ θϵ |⃗u1 · ∇⃗
⃗ u1 ⟩H 3/2 ,H −3/2 ds
⟨(⃗u1 · ∇⃗
Z t
⃗ u1 ) ∗ θϵ |w⟩
⃗ u1 + ⃗u2 · ∇⃗
⃗ u2 ⟩H 3/2 ,H −3/2 ds
+2
⟨(⃗u1 · ∇⃗
⃗ H −1 ,H 1 + ⟨⃗u1 ∗ θϵ | − ⃗u1 · ∇⃗
0
0
Leray’s Weak Solutions
371
2
2 1
1
1,∞
2
2 1
2 ∞
⃗ +V
⃗ , where U
⃗ ∈ (L∞
⃗ ∈ (L∞
We write ⃗u1 = U
) and V
t L ∩Lt H ∩L Ẇ
t L ∩Lt H ∩L L ).
Then, for j = 1, 2, we have
Z t
Z t
⃗ u1 ) ∗ θϵ |⃗uj ⟩H −1 ,H 1 ds =
⃗U
⃗ ) ∗ θϵ |⃗uj ⟩L2 ,L2 ds
⟨(⃗u1 · ∇⃗
⟨(⃗u1 · ∇
0
0
Z t
⃗ ) ∗ θ ϵ |∇
⃗ ⊗ ⃗uj ⟩L2 ,L2 , ds
−
⟨(⃗u1 ⊗ V
0
and
t
Z
⃗ uj ⟩H 3/2 ,H −3/2 ds =
⟨⃗u1 ∗ θϵ |⃗uj · ∇⃗
0
Z
t
⃗ ∗ θϵ |⃗uj · ∇⃗
⃗ uj ⟩L∞ ,L1 ds
⟨V
Z t
⃗ ⊗U
⃗ ) ∗ θϵ |⃗uj ⊗ ⃗uj ⟩L∞ ,L1 ds
−
⟨(∇
0
0
By *-weak convergence in the space variable and then dominated convergence in the time
variable, we find that
Z t
2
⃗ ⊗ w∥
∥w(t,
⃗ .)∥2 ≤ − 2ν
∥∇
⃗ 22 ds
0
Z tZ
Z tZ
⃗
⃗
⃗ · (⃗u1 · ∇
⃗ w)
+2
(⃗u1 · ∇U ) · w
⃗ dx ds − 2
V
⃗ dx ds
0
0
Z tZ
⃗ · (⃗u2 · ∇⃗
⃗ u2 − ⃗u1 · ∇u
⃗ 1 ) dx ds
+2
V
0
Z tZ
⃗U
⃗ ) − ⃗u1 · (⃗u1 · ∇
⃗U
⃗ ) dx ds
−2
⃗u2 · (⃗u2 · ∇
0
Z t
⃗ ⊗ w∥
= − 2ν
∥∇
⃗ 22 ds
0
Z tZ
Z tZ
⃗U
⃗ ) dx ds + 2
⃗ · (w
⃗ w)
−2
w
⃗ · (w
⃗ ·∇
V
⃗ ·∇
⃗ dx ds
0
0
Z tZ
⃗ · (w
⃗ u1 ) − ⃗u1 · (w
⃗U
⃗ ) dx ds.
+2
V
⃗ · ∇⃗
⃗ ·∇
0
A similar proof by mollification shows that
Z tZ
Z tZ
⃗ · (w
⃗V
⃗ ) dx ds =
⃗ · (w
⃗U
⃗ ) dx ds = 0
V
⃗ ·∇
U
⃗ ·∇
0
0
so that finally we get
Z t
⃗ ⊗ w∥
∥w(t,
⃗ .∥22 ≤ − 2ν
∥∇
⃗ 22 ds
0
Z tZ
Z tZ
⃗U
⃗ ) dx ds + 2
⃗ · (w
⃗ w)
−2
w
⃗ · (w
⃗ ·∇
V
⃗ ·∇
⃗ dx ds
0
0
Z t
⃗ ⊗ w∥
≤ − 2ν
∥∇
⃗ 22 ds
0
Z t
Z t
⃗ ⊗U
⃗ ∥∞ ds + 2
⃗ ∥∞ ∥w∥
⃗ ⊗ w∥
∥w∥
⃗ 22 ∥∇
∥V
⃗ 2 ∥∇
⃗ 2 ds
+2
0
0
Z t
⃗ ⊗U
⃗ ∥∞ + 1 ∥V
⃗ ∥2∞ )∥w∥
(∥∇
≤2
⃗ 22 ds
4ν
0
and we conclude by Grönwall’s lemma.
372
The Navier–Stokes Problem in the 21st Century (2nd edition)
r
The case of ⃗u1 ∈ Lp Ḃ∞,∞
with p2 = r + 1 and 0 < r < 1 could have been treated directly
by elementary interpolation arguments:
Lemma 12.2.
1−r
r
If ⃗u ∈ Ḃ∞,∞
, ⃗v ∈ L2 with div ⃗v = 0 and w
⃗ ∈ Ḃ2,1
, then
Z
|
⃗u. div(⃗v ⊗ w)
⃗ dx| ≤ C∥⃗u∥Ḃ r
∞,∞
∥⃗v ∥2 ∥w∥
⃗ Ḃ 1−r .
2,1
Proof. Let T be the operator (⃗v , w)
⃗ 7→ T (⃗v , w)
⃗ = div((P⃗v ) ⊗ w).
⃗ If ⃗v ∈ L2 and w
⃗ ∈ L2 , then
−1
0
⃗w
⃗v ⊗w
⃗ ∈ L1 ⊂ Ḃ1,∞
, hence T (⃗v , w)
⃗ ∈ Ḃ1,∞ . If ⃗v ∈ L2 and w
⃗ ∈ Ḣ 1 , then T (⃗v , w)
⃗ = (P⃗v )·∇
⃗∈
−1
−0
1
0
2
2
1
L ⊂ Ḃ1,∞ . By interpolation, T is bounded from L × [L , Ḣ ]1−r,1 to [Ḃ1,∞ , Ḃ1,∞ ]1−r,1 ,
1−r
−r
hence from L2 × Ḃ2,1
to Ḃ1,1
.
Let us make a final remark. For 1 ≤ p < +∞, the results of Propositions 12.3, 12.4, and
12.6, may (partially) be unified in a single statement: let L2,λ be the Morrey–Campanato
space of locally square integrable functions u such that
s
Z
1
∥u∥L2,λ =
sup
|u(x) − mB(x0 ,r) u|2 dx < +∞
3+2λ
x0 ∈R3 , r>0 r
B(x0 ,r)
This space is defined modulo the constants.
⃗ ∈ L∞ .
ˆ For λ = 1, we have L2,λ = Ẇ 1,∞ : u ∈ Ẇ 1,∞ ⇔ ∇u
λ
ˆ For 0 < λ < 1, we have L2,λ = Ḃ∞,∞
(where the homogeneous Besov space is
λ
λ−1
λ−1
⃗ ∈ Ḃ∞,∞
defined by u ∈ Ḃ∞,∞
⇔ ∇u
: as λ − 1 < 0, we have already defined Ḃ∞,∞
unambiguously). This equality can be checked by using a decomposition on a wavelet
basis, for instance.
ˆ For λ = 0, we have L2,λ = BM O.
ˆ For −3/2 < λ < 0, we can see that, for u ∈ L2,λ , L(u) = limR→+∞
2,q
1
q
exists; moreover, u − L(u) belongs to Ṁ , where
+
2
2,λ
∥u − L(u)∥Ṁ 2,q . Of course, if u ∈ L ∩ L , then L(u) = 0.
λ
3
1
|B(0,R)
R
B(0,R)
u(x) dx
= 0 and ∥u∥L2,λ ≈
Thus, we get:
The generalized Prodi–Serrin uniqueness criterion
Theorem 12.5.
Let ⃗u0 ∈ L2 , with div ⃗u0 = 0, and f⃗ ∈ L2t Hx−1 on (0, T ) × R3 . Assume that the Navier–
Stokes problem
⃗ u) = ν∆⃗u + Pf⃗, ⃗u(0, .) = ⃗u0
∂t ⃗u + P(⃗u · ∇⃗
(12.22)
2
2 1
p 2,λ
has a solution ⃗u1 on (0, T ) × R3 such that ⃗u1 ∈ L∞
, where 1 ≤ p <
t L ∩ Lt H ∩ L L
2
+∞ and p = 1 + λ.
If ⃗u2 is a Leray weak solution of the same Navier–Stokes problem, then ⃗u2 = ⃗u1 .
Leray’s Weak Solutions
12.4
373
Weak-Strong Uniqueness and Morrey Spaces on the Product
Space R × R3
In the preceding section, we have considered the inequality
Z t
Z tZ
2
2
⃗ w)
⃗u1 · (w
⃗ ·∇
⃗ dx ds
∥w(t,
⃗ .)∥2 ≤ −2ν
∥w∥
⃗ Ḣ −1 ds + 2
0
0
RtR
⃗ w)
and we have estimated the term 0 ⃗u1 · (w
⃗ ·∇
⃗ dx ds using only size estimates on w
⃗ with
∞ 2
respect to the time variable: w
⃗ ∈ Lt Lx ∩ L2t Ḣx1 . But we know that we have some time
regularity on w:
⃗ ∂t w
⃗ ∈ L2 ((0, T ), H −3/2 ).
This suggests some generalizations of Theorem 12.4. We begin first with some lemmas
on the Sobolev spaces Ḣ r (R) with 0 < r < 1/2.
Lemma 12.3.
If I is an interval of R, then the pointwise multiplication by 1I is bounded on Ḣ r (R) with
0 < r < 1/2:
∥1I f ∥Ḣ r ≤ C∥f ∥Ḣ r
where C does not depend on I.
Proof. It is enough to prove the theorem for I = (0, +∞) as
1(a,b) (t) = 1(0,+∞) (t − a)(1 − 1(0,∞) (t − b))
(for t ̸= b). But for I = (0, +∞), the boundedness of the multiplier 1I on Ḣ r is equivalent
to the boundedness of the Hilbert transform on L2 (|τ |2r dτ ) (by the Plancherel equality for
the Fourier transform); as |τ |2r is a Muckenhoupt weight in A2 for 0 < r < 1/2 [448], the
Hilbert transform is actually bounded on L2 (|τ |2r dτ ).
From this lemma, we see that we can define Ḣ r (I) as the space of functions f in Ḣ r (R)
which are equal to 0 outside I. We then have the following important lemma:
Lemma 12.4.
Let I be a bounded interval (a, b) of R, w be a function defined on I × R3 such that w ∈
−3/2
) with w(a, .) = 0, then w ∈ L2x Ḣ 2/5 (I) and
L2t (I, Hx1 ) and ∂t w ∈ L2t (I, Hx
3/5
2/5
∥w∥L2 Ḣ 2/5 (I) ≤ C∥w∥L2 H 1 ∥∂t w∥L2 H −3/2
(12.23)
x
where C does not depend on I.
Proof. If I = (a, b), we define W on (0, 1) × R3 as
√
W (t, x) = b − a w(a + t(b − a), x).
−3/2
We have that W ∈ L2t ((0, 1), Hx1 ) and ∂t W ∈ L2t ((0, 1), Hx
) and that
∥W ∥L2t Hx1 = ∥w∥L2t Hx1 , ∥∂t W ∥L2 H −3/2 = (b − a)∥∂t w∥L2 H −3/2 ,
t
x
t
while
∥W ∥L2 Ḣ 2/5 ((0,1)) = (b − a)2/5 ∥w∥L2 Ḣ 2/5 (I) .
x
Thus, we may only consider the case I = (0, 1).
x
x
374
The Navier–Stokes Problem in the 21st Century (2nd edition)
If I = (0, 1) we define ω on R × R3 as ω(t, x) = w(t, x) when 0 < t < 1, ω(t, x) = w(2 −
t, x) when 1 < t < 2, and ω(t, x) = 0 when t < 0 or t > 2. Then ω ∈ L2t (R, Hx1 ) and ∂t ω ∈
−3/2
L2t (I, Hx
); in Fourier variables, we find ω̂(1+|ξ|2 )1/2 ∈ L2τ L2ξ and |τ |ω̂(1+|ξ|)−3/2 ∈ L2τ L2ξ
2/5
so that |τ |2/5 ω̂ ∈ L2τ L2ξ . Thus, ω ∈ L2x Ḣt
, and w = 1(0,1) ω ∈ L2x Ḣ 2/5 ((0, 1)).
2
2 1
2 2
We thus caan modify Theorem 12.4 by replacing multipliers from L∞
t Lx ∩ Lt Ḣx to L L
∞ 2
2 1
2 2/5
2 2
by multipliers from Lt Lx ∩ Lt Ḣx ∩ Lx Ḣt to L L :
Extension of the Prodi–Serrin uniqueness criterion
Theorem 12.6.
Let ⃗u0 ∈ L2 , with div ⃗u0 = 0, and f⃗ ∈ L2t Hx−1 on (0, T ) × R3 . Assume that the Navier–
Stokes problem
⃗ u) = ν∆⃗u + Pf⃗, ⃗u(0, .) = ⃗u0 .
∂t ⃗u + P(⃗u · ∇⃗
(12.24)
(0)
2
2 1
has a solution ⃗u1 on (0, T ) × R3 such that ⃗u1 ∈ L∞
t L ∩ Lt H ∩ YT , where
2
2 1
• YT is the space of pointwise multipliers on (0, T ) × R3 from L∞
t L ∩ Lt H ∩
2 2/5
2 2
Lx Ḣt to Lt Lx , normed with
∥u∥YT =
∥uv∥L2 L2 ;
sup
∥v∥L∞ L2 +∥v∥L2 H 1 +∥v∥
t
t
2/5 ≤1
L2
x Ḣt
(0)
• YT is the space of multipliers u in YT such that, for every t0 ∈ [0, T ),
lim ∥1(t0 ,t1 ) (t)u(t, x)∥YT = 0.
t1 →t+
0
If ⃗u2 is a Leray weak solution of the same Navier–Stokes problem, then ⃗u2 = ⃗u1 .
2
2 1
Proof. We follow the proof of Theorem 12.4. If ⃗u1 ∈ L∞
u1 = 0
t L ∩ Lt H ∩ YT with div ⃗
2 1
and ⃗v ∈ L H , we write
Z t
Z t
⃗ u1 ) ∗ θϵ |⃗v ⟩H −1 ,H 1 ds = −
⃗ ⊗ ⃗v ⟩L2 ,L2
⟨(⃗u1 · ∇⃗
⟨(⃗u1 ⊗ ⃗u1 ) ∗ θϵ |∇
0
0
and find
Z
lim+
ϵ→0
0
t
⃗ u1 ) ∗ θϵ |⃗v ⟩H −1 ,H 1 ds = −
⟨(⃗u1 · ∇⃗
t
Z
⃗ ⊗ ⃗v ⟩L2 ,L2 .
⟨⃗u1 ⊗ ⃗u1 |∇
0
2
2 1
2
2 1
If ⃗u1 ∈ L∞
v ∈ L∞
u1 ∗ θϵ −
t L ∩ Lt H ∩ YT and ⃗
t L ∩ L H , we have that limϵ→0+ ∥⃗
⃗u1 ∥L2 L6 = 0, so that limϵ→0+ ∥(⃗u1 ∗ θϵ − ⃗u1 ) ⊗ ⃗v ∥L2 L3/2 = 0. Moreover
Z
∥(⃗u1 ∗ θϵ ) ⊗ ⃗v ∥L2 L2 ≤ ∥⃗u1 (t, x − y) ⊗ ⃗v (x)∥L2 L2 θϵ (y) dy
Z
= ∥⃗u1 (t, x) ⊗ ⃗v (x + y)∥L2 L2 θϵ (y) dy
Z
≤∥⃗u1 ∥YT
∥⃗v (t, x + y)∥L∞ L2 ∩L2 H 1 ∩L2 Ḣ 2/5 θϵ (y) dy
x
=∥⃗u1 ∥YT ∥⃗v ∥L∞ L2 ∩L2 H 1 ∩L2 Ḣ 2/5 .
x
t
t
Leray’s Weak Solutions
375
This gives that (⃗u1 ∗ θϵ ) ⊗ ⃗v is weakly convergent in L2 L2 to ⃗u1 ⊗ ⃗v and thus
Z t
Z tZ
⃗ v ⟩H 3/2 ,H −3/2 ds =
⃗ v dx ds.
lim
⟨⃗u1 ∗ θϵ |⃗v · ∇⃗
⃗u1 · (⃗v · ∇)⃗
ϵ→0+
0
0
2
2 1
L∞
t L ∩L H
Moreover, we have, for ⃗v1 and ⃗v2 in ⃗v ∈
(since div ⃗u1 = 0 and (⃗u1 ∗ θϵ ) ⊗⃗vi
is weakly convergent in L2 L2 to ⃗u1 ⊗ ⃗vi )
Z tZ
Z tZ
⃗ v2 dx ds = lim
⃗ v2 dx ds
⃗v1 · (⃗u1 · ∇)⃗
⃗v1 · ((⃗u1 ∗ θϵ ) · ∇)⃗
ϵ→0+ 0
0
Z tZ
⃗ v1 dx ds
= − lim+
⃗v2 · ((⃗u1 ∗ θϵ ) · ∇)⃗
ϵ→0
0
Z tZ
⃗ v1 dx ds
=−
⃗v2 · (⃗u1 · ∇)⃗
0
Thus,
RtR
0
⃗ u1 dx ds = 0 and we may transform inequality (12.14) into
⃗u1 · (⃗u1 · ∇)⃗
∥w(t,
⃗ .)∥22 ≤ − 2ν
Z
t
⃗ ⊗ w∥
∥∇
⃗ 22 ds + 2
0
Z
= − 2ν
Z tZ
⃗ u1 ) · w
⃗ w)
(⃗u1 · ∇⃗
⃗ + ⃗u1 · (⃗u2 · ∇
⃗ dx ds
0
t
⃗ ⊗ w∥
∥∇
⃗ 22 ds + 2
0
Z tZ
(12.25)
⃗ w)
⃗u1 · (w
⃗ ·∇
⃗ dx ds
0
If 0 ≤ t0 < t1 < T are such that w
⃗ = 0 on [0, t0 ], we find that, on [0, t1 ],
Z t
⃗ ⊗ w∥
∥w(t,
⃗ .)∥22 + 2ν
∥∇
⃗ 22 ds ≤
(12.26)
0
2(∥w∥
⃗ L2 ((0,t1 ),H 1 ) + ∥w∥
⃗ L∞ ((0,t1 ),L2 ) + ∥w∥
⃗ L2 Ḣ 2/5 ((0,t1 )) )2 ∥1(t0 ,t1 ) ⃗u1 ∥YT
x
We have ∂t w
⃗ = ∆w
⃗ − P div(⃗u2 ⊗ w
⃗ +w
⃗ ⊗ ⃗u1 ), so that
∥w∥
⃗ L2 Ḣ 2/5 ((0,t1 )) ≤C0 (∥w∥
⃗ L2 ((t0 ,t1 ),Ḣ 1 ) + ∥∂t w∥
⃗ L2 ((t0 ,t1 ),Ḣ −3/2) )
x
≤C0 ∥w∥
⃗ L2 ((t0 ,t1 ),Ḣ 1 ) + C1 (1 + ∥⃗u0 ∥2 )∥w∥
⃗ L2 ((t0 ,t1 ),Ḣ 1 )
If 4(1 + C0 +
⃗u1 ∈
(0)
YT ,
1+C1 (1+∥⃗
u0 ∥2 )
)∥1(t0 ,t1 ) ⃗u1 ∥YT
2ν
< 1, we obtain w
⃗ = 0 on [0, t1 ]. Thus, if
we find that w
⃗ = 0 on (0, T ) and ⃗u2 = ⃗u1 .
As the space XT of Theorem 12.4 is obviously embedded into YT , we obtain a larger
class of weak-strong uniqueness. For instance, we have:
Proposition 12.7.
p,11/2
For 1 < p ≤ 11/2, let M5/2 be the Morrey space on R × R3 defined by ∥u∥Mp,11/2 < +∞,
5/2
where
∥u∥Mp,11/2 =
5/2
sup
x0 ∈R3 ,t0 ∈R,R>0
1
R11/2−p
!1/p
ZZ
p
|u(t, x)| dt dx
.
|t−t0 |<R5/2 ,|x−x0 |<R
p,11/2
Let Zp,T be the space of the functions u defined on (0, T )×R3 such that 10<t<T u ∈ M5/2
Then, for 2 < p ≤ 11/2, we have:
.
376
The Navier–Stokes Problem in the 21st Century (2nd edition)
• Zp,T ⊂ YT
(0)
11/2
• the closure Zp,T of Lt
11/2
Lx
(0)
(0)
in Zp,T satisfies Zp,T ⊂ YT
Proof. Let X be the space of homogenous type (R × R3 , δ5/2 , µ), where δ5/2 is the parabolic
(quasi)-distance
δ5/2 ((t, x), (s, y)) = |t − s|2/5 + |x − y|
(12.27)
and µ is the Lebesgue measure dµ = dt dx.
p,11/2
Then the homogeneous dimension Q of X is equal to 11/2 and M5/2 is the Morrey
space Ṁ p,Q (X). Then we may apply Theorem 5.3 and Proposition 5.4 on Riesz potentials to
see that the elements of Ṁ p,Q (X) with 2 < pRR≤ Q are pointwise multipliers from W 1 (X) to
1
v(s, y) ds dy with v ∈ L2t L2x .
L2t L2x , where u ∈ W 1 (X) if and only if u =
δ5/2 (t−s,x−y)Q−1
2/5
Moreover, by Proposition 5.6, we have W 1 (X) = W 5/2,3/2 (R × R3 ) = L2t Ḣx1 ∩ L2x Ḣt
Thus, we find that Zp,T ⊂ YT .
p,11/2
We easily can check that, locally in space and time, M5/2
scaling of mild solutions: recall that the space
where
∥u∥Mp,5 =
2
sup
R5−p
satisfies the parabolic
2 < p ≤ 5, defined by ∥u∥Mp,5 < +∞,
2
!1/p
ZZ
1
x0 ∈R3 ,t0 ∈R,R>0
Mp,5
2 ,
.
p
|u(t, x)| dt dx
,
|t−t0 |<R2 ,|x−x0 |<R
was discussed on page 98 in Chapter 5. If K is a compact subset of R × R3 and
p,11/2
u belongs to M5/2 , then 1K (t, x)u belongs to Mp,5
2 . Indeed, if R ≥ 1, we have
RR
RR
p
5−p
p
|u(t,
x)|
dt
dx
≤
R
|u(t,
x)|
dt dx; if R < 1, we may split the
|t−t0 |<R2 ,|x−x0 |<R
K
interval [t0 − R2 , t0 + R2 ] into an union of O(R−1/2 ) intervals [t0,i − R5/2 , t0,i + R5/2 ], so
that
ZZ
p
O(R−1/2 )Z Z
|u(t, x)| dt dx =
|t−t0 |<R2 ,|x−x0 |<R
X
i=1
|u(t, x)|p dt dx
|t−t0,i |<R5/2 ,|x−x0 |<R
≤C R−1/2 ∥u∥p
p,11/2
M5/2
12.5
R11/2−p .
Almost Strong Solutions
Uniqueness remains an open problem in the class of Leray solutions. There have been
many papers dealing with uniqueness in some subclasses of Leray solutions: one studies
uniqueness in a class L∞ L2 ∩ L2 Ḣ 1 ∩ X. Chemin proved uniqueness of solutions in L∞ L2 ∩
−1
−1
L2 Ḣ 1 ∩C([0, T ], Ḃ∞,∞
) (which implies that ⃗u0 not only belongs to L2 but belongs to Ḃ∞,∞
),
−1+ 3
under a further assumption: ⃗u0 belongs to the closure of test functions in some space Ḃp,∞ p
with p < +∞ [106]. Lemarié-Rieusset removed the assumption on ⃗u0 and proved uniqueness
2
−1
r
in L∞ L2 ∩L2 Ḣ 1 ∩C([0, T ], Ḃ∞,∞
) [317]. Then uniqueness was proved in L r+1 ((0, T ), Ḃ∞,∞
)∩
∞ 2
2 1
L L ∩L Ḣ for −1 < r ≤ 1: the case r < 0 was proved by May in an extension of LemariéRieusset’s method [355], while the case r > −1/2 was proved by Chen, Miao and Zhang
[115].
Leray’s Weak Solutions
377
Definition 12.3.
Let ⃗u0 ∈ L2 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), L2 ). An almost strong solution ⃗u of the
Navier–Stokes equations on (0, T )
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u)),
⃗u(0, .) = ⃗u0
is a solution ⃗u such that:
• ⃗u ∈ L∞ ((0, T ), L2 ) ∩ L2 ((0, T ), H 1 )
• limt→0+ ∥⃗u(t, .) − ⃗u0 ∥2 = 0
• ⃗u ∈ C((0, T ], H 1 ).
Note that such a solution satisfies Leray’s energy equality: for all 0 ≤ t0 < t1 ≤ T ,
Z t1
Z t1 Z
2
2
⃗
∥⃗u(t1 , .)∥2 = ∥⃗u(t0 , .)∥2 − 2
∥∇ ⊗ ⃗u∥2 ds + 2
⃗u(s, x) · f⃗(s, x) dx ds.
t0
t0
Of course, if ⃗u0 ∈ H 1 , then we may use Serrin’s weak-strong uniqueness theorem to ensure
that an almost strong solution ⃗uAS and the classical strong solution ⃗uCL ∈ C([0, T ], H 1 )
coincide: ⃗uAS = ⃗uCL .
Proposition 12.8.
Let ⃗u0 ∈ L2 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), L2 ). Assume that ⃗u is a solution of the
Navier–Stokes equations on (0, T )
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u)),
⃗u(0, .) = ⃗u0
such that ⃗u ∈ L∞ ((0, T ), L2 ) ∩ L2 ((0, T ), H 1 ) and assume moreover that, for some r ∈
(−1, 1), we have
2
r
⃗u ∈ L r+1 ((0, T ), Ḃ∞,∞
).
Then ⃗u is an almost strong solution.
Proof. First step: right-continuity of t ∈ [0, T ) 7→ ⃗u(t, .) ∈ L2 :
Case r > 0:
2
r
When r > 0, it is easy to check that, if ⃗u ∈ L r+1 ((0, T ), Ḃ∞,∞
) ∩ L∞ L2 ∩ L2 Ḣ 1
⃗ u ∈ L2 Ḣ −1 + L1 L2 . Just recall that we have seen that
with div ⃗u = 0, then ⃗u · ∇⃗
2
r
1
2 ∞
1+r
L Ḃ∞,∞ ⊂ L Lip + L L (see the proof of Proposition 12.6). Writing ⃗u = ⃗v + w
⃗
with ⃗v ∈ L1 Lip and w
⃗ ∈ L2 L∞ , we just write
⃗ u = ⃗u · ∇⃗
⃗v+∇
⃗ · (⃗u ⊗ w)
⃗u · ∇⃗
⃗
and get
⃗ u∥ 1 2 2 −1 ≤ C∥⃗u∥L∞ L2 (∥w∥L
∥⃗u · ∇⃗
⃗ 1 Lip + ∥⃗v ∥L2 L∞ ) ≤ C ′ ∥⃗u∥L∞ L2 ∥⃗u∥
L L +L Ḣ
2
r
L 1+r Ḃ∞,∞
.
⃗ u ∈ L1 L2 + Ḣ −1 , we get that B(⃗u, ⃗u) ∈ C([0, T ], L2 ). As ⃗u = Wνt ∗ ⃗u0 −
From ⃗u · ∇⃗
B(⃗u, ⃗u), we see that ⃗u ∈ C([0, T ], L2 ).
378
The Navier–Stokes Problem in the 21st Century (2nd edition)
Case r < 0:
2
r
When r < 0, it is easy as well to check that, if ⃗u ∈ L r+1 ((0, T ), Ḃ∞,∞
) ∩ L∞ L2 ∩
⃗ u ∈ L2 Ḣ −1 + L1 L2 . We write (using the Littlewood–
L2 Ḣ 1 with div ⃗u = 0, then ⃗u · ∇⃗
Paley decomposition)
∥∆j ⃗u ⊗ ∆k ⃗u∥2 ≤ ∥∆min(j,k) ⃗u∥∞ ∥∆max(j,k) ⃗u∥2
which gives
∥(⃗u ⊗ ⃗u)(t, .)∥Ḣ 1+r ≤ ∥⃗u(t, .)∥Ḃ r
∞,∞
∥⃗u(t, .)∥Ḣ 1
and
⃗
∥⃗u · ∇u∥
2
L 2+r Ḣ r
≤ ∥⃗u(t, .)∥
2
r
L 1+r Ḃ∞,∞
∥⃗u(t, .)∥L2 Ḣ 1
2
and we end by using the embedding L 2+r Ḣ r ⊂ L1 L2 + L2 Ḣ −1 . Thus, again we get
that ⃗u ∈ C([0, T ], L2 ).
Case r = 0:
The case r = 0 is more delicate. We use a Littlewood–Paley decomposition and use
the norm
X
∥⃗u∥LP = (
∥∆j ⃗u∥22 )1/2
j∈Z
which is a Hilbertian norm equivalent to the L2 norm. As t 7→ ⃗u(t, .) is continuous from
[0, T ] to D′ and is bounded from [0, T ] to L2 , we find that it is *-weakly continuous
from [0, T ] to L2 . In particular, we shall have that
lim ∥⃗u(t, .) − ⃗u(t0 , .)∥2 = 0 ⇔ lim+ ∥⃗u(t, .) − ⃗u(t0 , .)∥LP = 0
t→t+
0
t→t0
⇔ lim sup ∥⃗u(t, .)∥2LP − ∥⃗u(t0 , .)∥2LP ≤ 0.
t→t+
0
We have
Z
d
∥∆j ⃗u(t, .)∥22 =2 ∆j ⃗u(t, x) · ∆j (∂t ⃗u)(t, x) dx
dt
Z
⃗ u) dx
=2 ∆j ⃗u · (ν∆∆j ⃗u + ∆j f⃗ − ∆j (⃗u · ∇⃗
Z
2
2
2
⃗
⃗
⃗ u) dx
≤ −2ν∥∇⊗∆
⃗
u
∥
+
∥∆
⃗
u
∥
+
∥∆
f
∥
−
2
∆j ⃗u · ∆j (⃗u · ∇⃗
j
j
j
2
2
2
R
⃗ l ⃗u) dx = 0, so that
If k ≥ j + 5 and |l − k| ≥ 4, then ∆j ⃗u · ∆j (∆k ⃗u · ∇∆
Z
Z
⃗ u) dx = ∆j ⃗u · ∆j (Sj+5 ⃗u · ∇⃗
⃗ u) dx
∆j ⃗u · ∆j (⃗u · ∇⃗
Z
X X
⃗ l ⃗u) dx
+
∆j ⃗u · ∆j (∆k ⃗u · ∇∆
k≥j+5 |k−l|≤3
We have, for k ≥ j + 5 and |l − k| ≤ 3,
Z
⃗ l ⃗u) dx
Aj,k,l = − ∆j ⃗u · ∆j (∆k ⃗u · ∇∆
=
3 Z
X
∆k ui (∆l ⃗u.∂i ∆∗j ∆j ⃗u) dx
i=1
≤C∥∆k ⃗u∥∞ 2j ∥∆j ⃗u∥2 ∥∆l ⃗u∥2
Leray’s Weak Solutions
379
so that
X X
X
X
Aj,k,l ≤C∥⃗u∥Ḃ 0
∞,∞
j∈Z k≥j+5 |l−k|≤3
∥∆j ⃗u∥2 (
j∈Z
≤C
′
X
2j−l 2l ∥∆⃗ul ∥2 )
l≥j+2
⃗
∥⃗u∥Ḃ 0 ∥⃗u∥2LP ∥∇
∞,∞
⊗ ⃗u∥2LP
R
⃗ u) dx, we note that
For the term Bj = − ∆j ⃗u · ∆j (Sj+5 ⃗u · ∇⃗
Z
⃗ j ⃗u) dx = 0
∆j ⃗u.(Sj+5 ⃗u · ∇∆
and we write
Bj = −
3 Z
X
∆j ⃗u.[∆j , Sj+5 ui ]∂i ⃗u dx
(12.28)
i=1
and
Bj =
3
X
∂i ∆j ⃗u.[∆j , Sj+5 ui ]⃗u dx.
(12.29)
i=1
We have (writing ∆j h = ψ( 2Dj )h and ψ = Ψ̂)
Z
Z
[∆j , Sj+5 ui ]h = 23j (Sj+5 ui (y) − Sj+5 ui (x))Ψ(2j (x − y))h(y) dy.
⃗ i ∈ Ḃ −1 , so that ∥∇S
⃗ j+5 ui ∥∞ ≤ C2j and
We have ∇u
0,∞
Z
|[∆j , Sj+5 ui ]h| ≤C∥⃗u∥Ḃ 0
∞,∞
≤C ′ ∥⃗u∥Ḃ 0
∞,∞
2
3j
Z
2j |x − y||Ψ(2j (x − y))||h(y)| dy
(12.30)
Mh (x).
From (12.28) and (12.30), we get
Bj ≤ C∥⃗u∥Ḃ 0
∞,∞
⃗ ⊗ ⃗u∥2 ∥∆j ⃗u∥2 ,
∥∇
while we get, from (12.29) and (12.30),
Bj ≤ C∥⃗u∥Ḃ 0
∞,∞
∥⃗u∥2 2j ∥∆j ⃗u∥2 .
and thus
X
q
Bj ≤C∥⃗u∥Ḃ 0
∞,∞
⃗ ⊗ ⃗u∥2
∥⃗u∥2 ∥∇
j∈Z
X
2j/2 ∥∆j ⃗u∥2 .
j∈Z
≤C ′ ∥⃗u∥Ḃ 0
q
∞,∞
′′
≤C ∥⃗u∥Ḃ 0
∞,∞
⃗ ⊗ ⃗u∥2 ∥⃗u∥ 1/2
∥⃗u∥2 ∥∇
Ḃ
2,1
⃗ ⊗ ⃗u∥2 .
∥⃗u∥2 ∥∇
Collecting all those estimates, we find
d
⃗ ⊗ ⃗u∥2LP + ∥⃗u∥2LP + ∥f⃗∥2LP + C∥⃗u∥ 0 ∥⃗u∥2LP ∥∇
⃗ ⊗ ⃗u∥2LP
∥⃗u∥2LP ≤ −2ν∥∇
Ḃ∞,∞
dt
(12.31)
380
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, for 0 ≤ t0 < t1 ,
∥⃗u(t, .)∥2LP ≤ ∥⃗u(t0 , .)∥2LP +
Z
t
(∥⃗u∥2LP + ∥f⃗∥2LP ) ds + C ′ ∥⃗u∥2L∞ L2
Z
t0
t
t0
∥⃗u∥2Ḃ 0
ds
∞,∞
and we get
lim sup ∥⃗u(t, .)∥2LP ≤ ∥⃗u(t0 , .)∥2LP .
t→t+
0
Thus, t 7→ ⃗u(t, .) is strongly right-continuous from [0, T ) to L2 .
Second step: energy equality.
By interpolation, we find that, for ρ =
1/3
∥⃗u∥L3 Ḃ 1/3 ≤ C∥⃗u∥
3,3
2
L 1+r
1−r
2
2/3
∥⃗u∥
r
Ḃ∞,∞
2
Lρ
ρ
Ḃ2,2
1/3
∈ (0, 1),
≤ C ′ ∥⃗u∥
1−ρ
1/3
2
L 1+r
r
Ḃ∞,∞
ρ
3
∥⃗u∥L∞
u∥L3 2 Ḣ 1 .
L2 ∥⃗
1/3
1/3
In particular, ⃗u belongs to L3t b3,∞ , where b3,∞ is the closure of test functions in Ḃ3,∞ .
Then, we may use Duchon and Robert’s theorem [159] (see Theorem 13.7 below) and
conclude that ⃗u satisfies the local energy equality:
∂t (
2
2
|⃗u|2
⃗ ⊗ ⃗u|2 = ν∆( |⃗u| ) + 2⃗u · f⃗ − div(( |⃗u| + p)⃗u).
) + ν|∇
2
2
2
Using the right-continuity of t ∈ [0, T ) 7→ ⃗u(t, .) ∈ L2 , we conclude that ⃗u satisfies the
Leray energy equality: for all 0 ≤ t0 ≤ t1 ≤ T ,
Z t1
Z t1 Z
⃗ ⊗ ⃗u∥22 ds + 2
∥⃗u(t1 , .)∥22 = ∥⃗u(t0 , .)∥22 − 2
∥∇
⃗u · f⃗ dx ds
t0
t0
(see the discussion on page 119).
Third step: continuity in H 1 norm.
As ⃗u ∈ L2 H 1 , we know that ⃗u(t, .) belongs to H 1 for almost every time. If ⃗u(t0 , .)
belongs to H 1 , then we know that there will be a solution ⃗v of the Navier–Stokes
equations on some small interval [t0 , t0 + δ] such that ⃗v ∈ C([t0 , t0 + δ], H 1 ) ∩ L2 H 2 .
By Serrin’s weak-strong uniqueness theorem (Theorem 12.3), we know that ⃗v = ⃗u.
Moreover, if we look at the maximal existence time of ⃗u as a solution in C([t0 , T ∗ ), H 1 ),
2
r
we know that T ∗ = T as ⃗u ∈ L r+1 Ḃ∞,∞
(see Theorem 11.3).
Thus, we find that ⃗u ∈ C((0, T ), H 1 ) so that ⃗u is an almost strong solution.
We may now state the uniqueness theorem of May [355] and Chen, Miao and Zhang
[115]:
Uniqueness for almost strong solutions
Theorem 12.7.
Let ⃗u0 ∈ L2 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 1 ). Assume that ⃗u and ⃗v are two
solutions of the Navier–Stokes equations on (0, T )
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u)),
⃗u(0, .) = ⃗u0
Leray’s Weak Solutions
381
such that ⃗u and ⃗v belong to L∞ ((0, T ), L2 ) ∩ L2 ((0, T ), H 1 ). Assume moreover that, for
some r1 , r2 ∈ (−1, 1), we have
2
2
r1
r2
⃗u ∈ L r1 +1 ((0, T ), Ḃ∞,∞
) and ⃗v ∈ L r2 +1 ((0, T ), Ḃ∞,∞
).
Then ⃗u = ⃗v .
Proof. We may, of course, assume that r2 ≤ r1 .
Case r1 > 0:
In that case, we know that we have weak-strong uniqueness (Proposition 12.6). As ⃗v satisfies
2
r1
the Leray energy (in)equality and ⃗u ∈ L r1 +1 ((0, T ), Ḃ∞,∞
) with r1 > 0, we know that ⃗u = ⃗v .
Case r1 < 0:
As f⃗ ∈ L2 H 1 , we find that we may enhance the regularity of ⃗u and ⃗v to C((0, T ), H 2 ).
Hence, they belong to C((0, T ), L∞ ). In particular, we√have that, for every t0 ∈ (0, T ), the
function ηt0 : τ ∈ [0, T − t0 ) 7→ ηt0 (τ ) = sup0<θ<τ θ∥⃗u(t0 + θ, .)∥∞ is continuous and
satisfies ηt0 (0) = 0. Moreover, we have, for β ∈ (θ/4, θ/2)
Z θ
⃗u(t0 + θ, .) = Wν(θ−β) ∗ ⃗u(t0 + β, .) −
Wν(θ−s) ∗ P div(⃗u(t0 + s, .) ⊗ ⃗u(t0 + s)) ds
β
so that
√
θ∥⃗u(t0 + θ, .)∥∞ ≤ Cθ
1+r1
2
r1
∥⃗u(t0 + β, .)∥Ḃ∞,∞
+ Cθ sup ∥⃗u(t0 + s, .)∥2∞
β<s<θ
Averaging over (θ/4, θ/2), we find
√
θ∥⃗u(t0 + θ, .)∥∞ ≤Cθ
1+r1
2
R θ/2
θ/4
r1
∥⃗u(t0 + β, .)∥Ḃ∞,∞
dβ
θ
≤C0 (∥⃗u∥
2
r
1
L 1+r1 (t0 ,t0 +θ),Ḃ∞,∞
)
+ Cηt0 (θ)2
+ ηt0 (θ)2 )
and thus
ηt0 (τ ) ≤ C0 (∥⃗u∥
2
r
1
L 1+r1 (t0 ,t0 +τ ),Ḃ∞,∞
)
+ ηt0 (τ )2 ).
If τ0 is small enough to grant that
∥⃗u∥
sup
0<t0 <T /2
2
r
1
L 1+r1 (t0 ,t0 +τ0 ),Ḃ∞,∞
)
<
1
4C02
we get that, for 0 < t0 < T /2 and 0 ≤ τ < τ0
ηt0 (τ ) ≤ 2C0 ∥⃗u∥
2
r
1
L 1+r1 (t0 ,t0 +τ ),Ḃ∞,∞
Letting t0 go to 0, we find that, for 0 < t < T0 ,
√
t∥⃗u(t, .)∥∞ ≤ 2C0 ∥⃗u∥
2
r
.
1
L 1+r1 (0,t),Ḃ∞,∞
.
A similar estimate holds for ⃗v . Now, if w
⃗ = ⃗u − ⃗v , we write
Z t
w
⃗ =−
Wν(t−s) ∗ P div(⃗u ⊗ w
⃗ +w
⃗ ⊗ ⃗v ) ds
0
(12.32)
382
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
Z
t
∥w(t,
⃗ .)∥2 ≤ C
1
p
0
ν(t − s)
∥w(s,
⃗ .)∥2 (∥⃗u(s, .)∥∞ + ∥⃗v (s, .)∥∞ ) ds
so that
sup ∥w(t,
⃗ .)∥2 ≤ C sup ∥w(t,
⃗ .)∥2 (∥⃗u∥
0<t<t0
0<t<t0
2
r
1
L 1+r1 (0,t0 ),Ḃ∞,∞
+ ∥⃗v ∥
2
r
2
L 1+r2 (0,t0 ),Ḃ∞,∞
).
If t0 is so small that
C(∥⃗u∥
2
r
1
L 1+r1 (0,t0 ),Ḃ∞,∞
+ ∥⃗v ∥
2
r
2
L 1+r2 (0,t0 ),Ḃ∞,∞
) < 1,
we find w
⃗ = 0 on (0, t0 ], hence local uniqueness. This uniqueness propogates to (0, T ), as
we have uniqueness in C([t0 , T ], H 1 ).
Case r1 = 0:
Let w
⃗ = ⃗u − ⃗v . One more time, we use a Littlewood–Paley decomposition but we do not
use the norm
X
∥w∥
⃗ LP = (
∥∆j w∥
⃗ 22 )1/2
j∈Z
which is a Hilbertian norm equivalent to the L2 norm. Instead of it, we shall use the norm
X
∥w∥
⃗ LP,σ = (∥S0 w∥
⃗ 22 +
2−2jσ ∥∆j w∥
⃗ 22 )1/2
j∈N
which is a Hilbertian norm equivalent to the H −σ norm. If σ > 0, we have the embedding
L2 ⊂ H −σ , so that the map t ∈ [0, T ) 7→ w(t,
⃗ .) ∈ H −σ is (strongly) continuous. We write
Z
d
∥∆j w(t,
⃗ .)∥22 =2 ∆j w(t,
⃗ x) · ∆j (∂t w)(t,
⃗
x) dx
dt
Z
⃗ u − ⃗v ∇⃗
⃗ v )) dx
=2 ∆j w
⃗ · (ν∆∆j w
⃗ − ∆j (⃗u · ∇⃗
Z
2
⃗
⃗ u − ⃗v · ∇⃗
⃗ v )) dx
= −2ν∥∇ ⊗ ∆j w∥
⃗ 2 − 2 ∆j w
⃗ · ∆j (⃗u · ∇⃗
If k ≥ j + 5 and |l − k| ≥ 4, then
Z
Z
⃗ l ⃗u) dx = ∆j w
⃗ l⃗v ) dx = 0,
∆j w
⃗ · ∆j (∆k ⃗u · ∇∆
⃗ · ∆j (∆k⃗v · ∇∆
so that
Z
⃗ u − ⃗v · ∇⃗
⃗ v ) dx =
∆j w
⃗ · ∆j (⃗u · ∇⃗
+
X
Z
X Z
⃗ u − Sj+5⃗v · ∇⃗
⃗ v )) dx
∆j w
⃗ · ∆j (Sj+5 ⃗u · ∇⃗
⃗ l ⃗u − ∆k⃗v · ∇∆
⃗ l⃗v ) dx
∆j w
⃗ · ∆j (∆k ⃗u · ∇∆
k≥j+5 |k−l|≤3
We have, for k ≥ j + 5 and |l − k| ≤ 3,
Z
⃗ l ⃗u − ∆k⃗v · ∇∆
⃗ l⃗v ) dx
Aj,k,l = − ∆j w
⃗ · ∆j (∆k ⃗u · ∇∆
Z
⃗ l ⃗u + ∆k⃗v · ∇∆
⃗ l w)
= − ∆j w
⃗ · ∆j (∆k w
⃗ · ∇∆
⃗ dx
Z
Z
⃗ ∗j ∆j w)
⃗ ∗j ∆j w)
= ∆l ⃗u.(∆k w.
⃗ ∇∆
⃗ dx + ∆l w(∆
⃗ k⃗v .∇∆
⃗ dx
r2
≤C2j ∥∆j w∥
⃗ 2 (2−kr2 ∥⃗v ∥Ḃ∞,∞
∥∆l w∥
⃗ 2 + ∥⃗u∥Ḃ 0
∞,∞
∥∆k w∥
⃗ 2 ).
Leray’s Weak Solutions
Thus, we get
X
X
2−2jσ
j∈N
X
383
Aj,k,l
k≥j+5 |k−l|≤3
≤C
X
j∈N
=C
X
X
2−2jσ 2j ∥∆j w∥
⃗ 2
r2
(2−kr2 ∥⃗v ∥Ḃ∞,∞
+ ∥⃗u∥Ḃ 0
∞,∞
X
r2
2−jσ 2j ∥⃗v ∥Ḃ∞,∞
∥∆j w∥
⃗ 2
j∈N
+C
2−jσ ∥⃗u∥Ḃ 0
∞,∞
j∈N
X
2−(j−k)σ 2−k(σ+r2 )) ∥∆k w∥
⃗ 2
k≥j+2
X
2j ∥∆j w∥
⃗ 2
j∈N
As σ > 0, we get
X
X
2−2jσ
)∥∆k w∥
⃗ 2
k≥j+2
X
2−(j−k)−σ 2−kσ ∥∆k w∥
⃗ 2
k≥j+2
Aj,k,l ≤
k≥j+5 |k−l|≤3


r2
C∥⃗v ∥Ḃ∞,∞
X
1/2 
1/2
X
⃗ 22  
22j(1−σ) ∥∆k w∥
⃗ 22 
2−2j(σ+r2 ) ∥∆j w∥
j∈N
+ C∥⃗u∥Ḃ 0
∞,∞
j∈N

1/2 
1/2
X
X

2−2jσ ∥∆j w∥
⃗ 22  
22j(1−σ) ∥∆k w∥
⃗ 22 
j∈N
r2
≤C∥⃗v ∥Ḃ∞,∞
j∈N


X

2−2jσ ∥∆j w∥
⃗ 22 
1+r2
2
2

 1−r
2
X

22j(1−σ) ∥∆k w∥
⃗ 22 
j∈N
+ C∥⃗u∥Ḃ 0
∞,∞
j∈N

1/2 
1/2
X
X

2−2jσ ∥∆j w∥
⃗ 22  
22j(1−σ) ∥∆k w∥
⃗ 22 
j∈N
j∈N
(as 0 ≤ −r2 ≤ 1).
R
⃗ jw
Now, we write (as ∆j w.(S
⃗ j+5⃗v · ∇∆
⃗ dx = 0)
Z
⃗ u − Sj+5⃗v · ∇⃗
⃗ v )) dx
B j = − ∆j w
⃗ · ∆j (Sj+5 ⃗u · ∇⃗
Z
=−
Z
⃗ u) dx −
∆j w
⃗ · ∆j (Sj+5 w
⃗ · ∇⃗
Z
⃗ j+8 ⃗u) dx −
∆j w
⃗ · ∆j (Sj+5 w
⃗ · ∇S
=
=
⃗ u + Sj+5⃗v · ∇
⃗ w))
∆j w
⃗ · ∆j (Sj+5 w
⃗ · ∇⃗
⃗ dx
≤C∥∆j w∥
⃗ 2 ∥Sj+5 w∥
⃗ 2 2j ∥⃗u∥Ḃ 0
∞,∞
Z
⃗ w)
∆j w
⃗ · ([∆j , Sj+5⃗v ].∇
⃗ dx
Z
⃗ j+8 w)
∆j w
⃗ · ([∆j , Sj+5⃗v ].∇S
⃗ dx
⃗ j+8 w∥
r2
+ C∥∆j w∥
⃗ 2 2−jr2 ∥⃗v ∥Ḃ∞,∞
∥∇S
⃗ 2.
384
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, we get
X
2−2jσ Bj ≤C∥⃗u∥Ḃ 0
X
∞,∞
j∈N
j∈N
r2
+C∥⃗v ∥Ḃ∞,∞
X
X
2−2jσ 2j ∥∆j w∥
⃗ 2 (∥S0 w∥
⃗ 2+
= C∥⃗u∥Ḃ 0
∞,∞
j(1−σ)
2
X
2k ∥∆k w∥
⃗ 2)
0≤k≤j+4
∥∆j w∥
⃗ 2 (2
−jσ
∥S0 w∥
⃗ 2+
j∈N
r2
+C∥⃗v ∥Ḃ∞,∞
X
2−2jσ 2−jr2 ∥∆j w∥
⃗ 2 (∥S0 w∥
⃗ 2+
j∈N
X
∥∆k w∥
⃗ 2)
k≤j+4
X
2
−(j−k)σ −kσ
2
∥∆k w∥
⃗ 2)
k≤j+4
X
2−j(σ+r2 ) ∥∆j w∥
⃗ 2 (2−jσ ∥S0 w∥
⃗ 2+
j∈N
2−(j−k)σ 2k(1−σ) ∥∆k w∥
⃗ 2)
k≤j+4
As σ > 0 and 0 ≤ −r2 ≤ 1, we find
X
2−2jσ Bj
j∈N
≤ C∥⃗u∥Ḃ 0
∞,∞

1/2 
1/2
X
X

22j(1−σ) ∥∆j w∥
⃗ 22  ∥S0 w∥
⃗ 22 +
2−2jσ ∥∆j w∥
⃗ 22 
j∈N
r2
+C∥⃗v ∥Ḃ∞,∞
j∈N

1/2 
1/2
X
X

2−2j(σ+r2 ) ∥∆j w∥
⃗ 22 ) ∥S0 w∥
⃗ 22 +
22j(1−σ) ∥∆j w∥
⃗ 22 
j∈N
≤ C∥⃗u∥Ḃ 0
∞,∞
j∈N

1/2 
1/2
X
X

22j(1−σ) ∥∆j w∥
⃗ 22  ∥S0 w∥
⃗ 22 +
2−2jσ ∥∆j w∥
⃗ 22 
j∈N
j∈N
2 
2

 1+r
 1−r
2
2
X
X

∥S0 w∥
r2
+C∥⃗v ∥Ḃ∞,∞
2−2jσ ∥∆j w∥
⃗ 22 
⃗ 22 +
22j(1−σ) ∥∆j w∥
⃗ 22 
j∈N
j∈N
We can perform similar estimates when dealing with
d
⃗ 22 ,
dt ∥S0 w∥
and we obtain finally
d
∥w∥
⃗ 2LP,σ ≤
dt
1/2

⃗ ⊗ w∥
−ν∥∇
⃗ 2LP,σ +C∥⃗u∥Ḃ 0
∞,∞
∥S0 w∥
⃗ 22 +
X
22j(1−σ) ∥∆j w∥
⃗ 22 
∥w∥
⃗ LP,σ
j∈N
2
 1−r
2

1+r2
2
r2
+C∥⃗v ∥Ḃ∞,∞
∥w∥
⃗ LP,σ ∥S0 w∥
⃗ 22 +
X
22j(1−σ) ∥∆j w∥
⃗ 22 
j∈N
We then use Bernstein’s inequality (for j ≥ 0)
⃗ ⊗ w∥
∥∆j ∇
⃗ 22 ≥ η∥∆j w∥
⃗ 22
(for a positive constant η) and Young’s inequality
caγ b1−γ ≤ γ
c γ1
ϵ
1
a + (1 − γ)ϵ 1−γ b
.
Leray’s Weak Solutions
385
for 0 < γ < 1 and positive a, b, c, ϵ, for the values γ = 12 and γ =
enough to ensure that
2
1 2 1 − r2 1−r
ϵ +
ϵ 2 <η
2
2
and we get
1+r2
2
, and for ϵ small
2
d
2
)∥w∥
⃗ 2LP,σ .
∥w∥
⃗ 2LP,σ ≤ C∥S0 w∥
+ ∥⃗v ∥Ḃ1+r
⃗ 22 + C(∥⃗u∥2Ḃ 0
r2
∞,∞
∞,∞
dt
As ∥S0 w∥
⃗ 2 ≤ ∥w∥
⃗ LP,σ , we may use the Grönwall lemma and get that w
⃗ = 0, hence ⃗u =
⃗v .
Theorem 12.7 may be generalized to the limit values r1 = 1 or r2 = −1 in the following
way:
Theorem 12.8.
Let XrT be defined, for −1 ≤ r ≤ 1 as
• X1T = L1 ((0, T ), Lip)
2
r
• for −1 < r < 1, XrT = L 1+r ((0, T ), Ḃ∞,∞
)
−1
• X−1
T = C([0, T ], Ḃ∞,∞ ).
Let ⃗u0 ∈ L2 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 1 ). Assume that ⃗u and ⃗v are
two solutions of the Navier–Stokes equations on (0, T )
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u)),
⃗u(0, .) = ⃗u0
such that ⃗u and ⃗v belong to L∞ ((0, T ), L2 ) ∩ L2 ((0, T ), H 1 ). Assume moreover that, for
some r1 , r2 ∈ (−1, 1), we have
⃗u ∈ XrT1 and ⃗v ∈ XrT2 .
Then ⃗u = ⃗v .
Proof. Step 1: almost strong solutions.
The first step is to check that a solution ⃗u ∈ L∞ L2 ∩ L2 H 1 ∩ XrT (for some r ∈ [−1, 1])
is indeed an almost strong solution. This has already been proved for |r| < 1. The
proof for r = 1 is exactly the same as the one for 0 < r < 1.
The proof for r = −1 is similar to the one for −1 < r < 0 but more delicate. We begin
by the interpolation inequality
1/3
2/3
∥⃗u∥L3 Ḃ 1/3 ≤ ∥⃗u∥L∞ Ḃ −1 ∥⃗u∥L2 Ḃ 1 .
∞,∞
3,3
1/3
L3 b3,∞
2,2
1/3
b3,∞
1/3
(where
is the closure of test functions in Ḃ3,∞ ).
In particular, ⃗u belongs to
Then, we may again use Duchon and Robert’s theorem [159] (see Theorem 13.7 below)
and conclude that ⃗u satisfies the local energy equality:
∂t (
2
2
|⃗u|2
⃗ ⊗ ⃗u|2 = ν∆( |⃗u| ) + 2⃗u · f⃗ − div(( |⃗u| + p)⃗u).
) + ν|∇
2
2
2
386
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, for every Lebesgue point t0 of the map t 7→ ∥⃗u(t, .)∥2 , we find that ⃗u satisfies
the Leray energy equality: for all t0 ≤ t1 ≤ T ,
Z t1
Z t1 Z
⃗ ⊗ ⃗u∥22 ds + 2
∥⃗u(t1 , .)∥22 = ∥⃗u(t0 , .)∥22 − 2
∥∇
⃗u · f⃗ dx ds
t0
t0
(see the discussion on page 119).
As ⃗u ∈ L2 H 1 , we know that ⃗u(t, .) belongs to H 1 for almost every time. If t0 is a
Lebesgue point of the map t 7→ ∥⃗u(t, .)∥2 such that ⃗u(t0 , .) belongs to H 1 , then we
know that there will be a solution ⃗v of the Navier–Stokes equations on some small
interval [t0 , t0 + δ] such that ⃗v ∈ C([t0 , t0 + δ], H 1 ) ∩ L2 H 2 . Moreover, by Serrin’s
weak-strong uniqueness theorem (Theorem 12.3), we know that ⃗v = ⃗u. Moreover,
if we look at the maximal existence time of ⃗u as a solution in C([t0 , T ∗ ), H 1 ), we
−1
know that T ∗ = T as ⃗u ∈ C([t0 , T ], Ḃ∞,∞)
(see Theorem 11.3). Hence, we get that
1
⃗u ∈ C((0, T ]), H ).
As f⃗ ∈ L2 H 1 , we find that we may enhance the regularity of ⃗u to C((0, T ), H 2 ).
Hence, it belongs to C((0, T ), L∞ ). In particular,
for every t0 ∈ (0, T ), the function
√
ηt0 : τ ∈ [0, T − t0 ) 7→ ηt0 (τ ) = sup0<θ<τ θ∥⃗u(t0 + θ, .)∥∞ is continuous and satisfies
ηt0 (0) = 0. Moreover, we have
Z θ
⃗u(t0 + θ, .) = Wν(θ/2) ∗ ⃗u(t0 + θ/2, .) −
Wν(θ−s) ∗ P div(⃗u(t0 + s, .) ⊗ ⃗u(t0 + s)) ds
θ/2
so that
√
√
θ∥⃗u(t0 + θ, .)∥∞ ≤ θ∥Wν(θ/2) ∗ ⃗u(t0 + θ/2, .)∥∞ + Cθ
sup
∥⃗u(t0 + s, .)∥2∞
θ/2<s<θ
−1
Let b−1
u ∈ L4 Ḣ 1/2 , we see that
∞,∞ be the closure of the Schwartz class S in Ḃ∞,∞ . As ⃗
−1
⃗u(t, .) ∈ H 1/2 for almost every t; as S is dense in Ḣ 1/2 and H 1/2 ⊂ Ḃ∞,∞
, we get
−1
−1
that ⃗u(t, .) ∈ b∞,∞ for almost every t; by continuity of t 7→ ⃗u(t, .) in Ḃ∞,∞ norm, we
see that ⃗u(t, .) ∈ b−1
∞,∞ for every t. Thus, for t ∈ [0, T ] and ϵ > 0, there exists M (t, ϵ),
⃗
⃗t , ∥⃗
−1
α
⃗ t and βt so that ⃗u(t, .) = α
⃗t + β
αt ∥Ḃ∞,∞
< ϵ and ∥β⃗t ∥∞ < M (t, ϵ). As [0, T ] is
−1
compact and thus t ∈ [0, T ] 7→ ⃗u(t, .) ∈ Ḃ∞,∞
is uniformly continuous, we can choose
M (t, ϵ) independently from t. We thus obtain
√
√
θ∥⃗u(t0 + θ, .)∥∞ ≤ θM (ϵ) + C0 ϵ + C0 θ sup ∥⃗u(t0 + s, .)∥2∞
θ/2<s<θ
and thus
ηt0 (τ ) ≤
For ϵ <
1
8C02
and T (ϵ) =
√
1
(8C0 M (ϵ))2 ,
τ M (ϵ) + C0 (ϵ + ηt0 (τ )2 ).
(12.33)
we get that, for 0 < t0 < T and 0 ≤ τ < T (ϵ)
√
ηt0 (τ ) ≤ 2( τ M (ϵ) + C0 ϵ.
Letting t0 go to 0, we find that, for 0 < t < T (ϵ),
√
√
t∥⃗u(t, .)∥∞ ≤ 2 tM (ϵ) + C0 ϵ.
√
Thus, we get that sup0<t<T t∥⃗u(t, .)∥∞ < +∞ and
√
lim+ t∥⃗u(t, .)∥∞ = 0.
t→0
(12.34)
Leray’s Weak Solutions
387
This gives that ⃗u is an almost strong solution, as
∥⃗u(t, .) − ⃗u0 ∥2 ≤∥Wνt ∗ ⃗u0 − ⃗u0 ∥2 + ∥B(⃗u, ⃗u)(t, .)∥2
≤∥Wνt ∗ ⃗u0 − ⃗u0 ∥2 + C∥⃗u∥L∞ L2 sup
√
s∥⃗u(s, .)∥∞
0<s<t
so that
lim ∥⃗u(t, .) − ⃗u0 ∥2 = 0.
t→0+
Step 2: uniqueness.
Let ⃗u ∈ L∞ L2 ∩ L2 H 1 ∩ XrT1 and ⃗v ∈ L∞ L2 ∩ L2 H 1 ∩ XrT2 , with −1 ≤ r2 ≤ r1 ≤ 1,
be two solutions of the same Navier–Stokes equations. We have already proved that
⃗u = ⃗v , in the case −1 < r2 ≤ r1 < 1.
The proof for r − 1 ≤ r2 ≤ r1 and 0 < r1 ≤ 1 is exactly the same as the one for
−1 < r2 ≤ r1 and 0 < r1 < 1.
The proof for −1 ≤ r2 ≤ r1 < 0 is exactly the same as the one for −1 < r2 ≤ r1 < 0.
The proof for −1 ≤ r2 ≤ r1 = 0 is similar to the one for −1 < r2 ≤ r1 = 0. We explain
now how to modify the proof when r2 = −1 and r1 = 0. For ϵ > 0, we may split ⃗v2
⃗ .)∥∞ ≤ M (ϵ) < +∞. As
−1
in α
⃗ + β⃗ with sup0≤t≤T ∥⃗
α(t, .)∥Ḃ∞,∞
< ϵ and sup0≤t≤T ∥β(t,
on page 382, we write w
⃗ = ⃗u − ⃗v and compute
d
⃗ 2LP,σ .
dt ∥w∥
We have
Z
Aj,k,l = −
⃗ l ⃗u − ∆k⃗v · ∇∆
⃗ l⃗v ) dx
∆j w
⃗ · ∆j (∆k ⃗u · ∇∆
⃗ ∞ )∥∆l w∥
r2
≤C2j ∥∆j w∥
⃗ 2 ((2−kr2 ∥⃗
α∥Ḃ∞,∞
+ ∥β∥
⃗ 2 + ∥⃗u∥Ḃ 0
∞,∞
∥∆k w∥
⃗ 2)
and
Z
Bj = −
⃗ u − Sj+5⃗v · ∇⃗
⃗ v )) dx
∆j w
⃗ · ∆j (Sj+5 ⃗u · ∇⃗
≤C∥∆j w∥
⃗ 2 ∥Sj+5 w∥
⃗ 2 2j ∥⃗u∥Ḃ 0
∞,∞
⃗ ∞ )∥∇S
⃗ j+8 w∥
−1
+ C∥∆j w∥
⃗ 2 (2j ∥⃗
α∥Ḃ∞,∞
+ ∥β∥
⃗ 2.
We obtain finally
d
⃗ ⊗ w∥
∥w∥
⃗ 2LP,σ ≤ − ν∥∇
⃗ 2LP,σ
dt
1/2

+ C(∥⃗u∥Ḃ 0
∞,∞
⃗ ∞ ) ∥S0 w∥
+ ∥β∥
⃗ 22 +
X
22j(1−σ) ∥∆j w∥
⃗ 22 
∥w∥
⃗ LP,σ
j∈N


∥S0 w∥
−1
+C∥⃗
α∥Ḃ∞,∞
⃗ 22 +
X
22j(1−σ) ∥∆j w∥
⃗ 22  .
j∈N
We use again Bernstein’s inequality and Young’s inequality to get
1 ⃗
d
⃗ 2 )∥w∥
∥w∥
⃗ 2LP,σ ≤ − ν∥∇
⊗ w∥
⃗ 2LP,σ + C∥S0 w∥
⃗ 22 + C(∥⃗u∥2Ḃ 0
+ ∥β∥
⃗ 2LP,σ .
∞
∞,∞
dt
2


X
+ C∥⃗
α∥ −1 ∥S0 w∥
⃗ 22 +
22j(1−σ) ∥∆j w∥
⃗ 22 
Ḃ∞,∞
j∈N
1 ⃗
≤ − ν∥∇
⊗ w∥
⃗ 2LP,σ + C∥S0 w∥
⃗ 22 + C(∥⃗u∥2Ḃ 0
+ M (ϵ)2 )∥w∥
⃗ 2LP,σ .
∞,∞
2
⃗ ⊗ w∥
+ Cϵ(∥S0 w∥
⃗ 2 + ∥∇
⃗ 2 ).
2
LP,σ
388
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ = 0,
We choose ϵ such that Cϵ ≤ ν2 , and then use the Grönwall lemma and get that w
hence ⃗u = ⃗v .
Chen, Miao and Zhang [115] could further generalize Theorem 12.7 to the case ⃗u ∈
1
1
L∞ L2 ∩ L2 H 1 ∩ L1 Ḃ∞,∞
and ⃗v ∈ L∞ L2 ∩ L2 H 1 ∩ L1 Ḃ∞,∞
. The proof was based on the
losing regularity estimate for transportation through a Log-Lipschitz field (Chemin and
Lerner [113], Danchin [142, 15])
Theorem 12.9.
Let ⃗u0 ∈ L2 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, +∞), H 1 ). Assume that ⃗u and ⃗v are two
solutions of the Navier–Stokes equations on (0, T )
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u)),
⃗u(0, .) = ⃗u0
1
. Then ⃗u = ⃗v .
such that ⃗u and ⃗v belong to L∞ ((0, T ), L2 ) ∩ L2 ((0, T ), H 1 ) ∩ L1 Ḃ∞,∞
1
(if
Proof. Remark that ∥S0 ⃗u∥L∞ L∞ ≤ C∥⃗u∥L∞ L2 , so that ⃗u and ⃗v will belong to L1 B∞,∞
T < +∞). Let w
⃗ = ⃗u − ⃗v . One more time, we would like to compute the norm
X
∥w∥
⃗ LP,σ = (∥S0 w∥
⃗ 22 +
2−2jσ ∥∆j w∥
⃗ 22 )1/2
j∈N
but we add some flexibility by allowing the regularity to worsen as time increases: we
compute more precisely
∥w∥
⃗ LP,σ,η = e−η−1 (t) ∥S0 w∥
⃗ 2 + sup e−ηj (t) 2−jσ ∥∆j w∥
⃗ 2
j∈N
where ηj a time-dependent non-negative (increasing) function such that ηj (0) = 0. We write
d −2ηj
dηj −2ηj
(e
∥∆j w(t,
⃗ .)∥22 ) = − 2
e
∥∆j w(t,
⃗ .)∥22
dt
dt
Z
⃗ ⊗ ∆j w∥
−2e−2ηj (ν∥∇
⃗ 22 +
⃗ u − ⃗v · ∇⃗
⃗ v )) dx)
∆j w
⃗ · ∆j (⃗u · ∇⃗
We write again
Z
Z
⃗ u − ⃗v · ∇⃗
⃗ v ) dx = ∆j w
⃗ u − Sj+5⃗v · ∇⃗
⃗ v )) dx
∆j w
⃗ · ∆j (⃗u · ∇⃗
⃗ · ∆j (Sj+5 ⃗u · ∇⃗
Z
X X
⃗ l ⃗u − ∆k⃗v · ∇∆
⃗ l⃗v ) dx
+
∆j w
⃗ · ∆j (∆k ⃗u · ∇∆
k≥j+5 |k−l|≤3
We have, for k ≥ j + 5 and |l − k| ≤ 3,
Z
⃗ l ⃗u − ∆k⃗v · ∇∆
⃗ l⃗v ) dx
Aj,k,l = − ∆j w
⃗ · ∆j (∆k ⃗u · ∇∆
≤C2j ∥∆j w∥
⃗ 2 (2−l ∥⃗v ∥Ḃ 1
∞,∞
∥∆l w∥
⃗ 2 + 2−k ∥⃗u∥Ḃ 1
∞,∞
∥∆k w∥
⃗ 2 ).
Leray’s Weak Solutions
389
⃗ jw
Now, we write (as ∆j w.(S
⃗ j+5⃗v · ∇∆
⃗ dx = 0)
Z
⃗ u − Sj+5⃗v · ∇⃗
⃗ v )) dx
B j = − ∆j w
⃗ · ∆j (Sj+5 ⃗u · ∇⃗
Z
Z
⃗ j+8 ⃗u) dx − ∆j w
⃗ j+8 w)
= − ∆j w
⃗ · ∆j (Sj+5 w
⃗ · ∇S
⃗ · ([∆j , Sj+5⃗v ].∇S
⃗ dx
Z
Z
⃗ j w)
⃗ j+8 w)
= Sj+8 ⃗u · (∆j (Sj+5 w
⃗ · ∇∆
⃗ dx − ∆j w
⃗ · ([∆j , Sj+5⃗v ].∇S
⃗ dx
R
⃗ ⊗ Sj+8 ⃗u∥∞
≤C∥∆j w∥
⃗ 2 ∥Sj+5 w∥
⃗ 2 ∥∇
⃗ ⊗ Sj+5⃗v ∥∞ ∥∇
⃗ ⊗ Sj+8 w∥
+ C∥∆j w∥
⃗ 2 2−j ∥∇
⃗ 2
As
d −2ηj
d
(e
∥∆j w(t,
⃗ .)∥22 ) = 2e−ηj ∥∆j w(t,
⃗ .)∥2 ) (e−ηj ∥∆j w(t,
⃗ .)∥2 )
dt
dt
we get (by dividing with 2e−ηj ∥∆j w∥
⃗ 2)
d −ηj
dηj −ηj
(e ∥∆j w(t,
⃗ .)∥2 ) ≤ −
e ∥∆j w(t,
⃗ .)∥2
dt
dt
X
1
1
+ Ce−ηj (∥⃗u∥B∞,∞
+ ∥⃗v ∥B∞,∞
)
2j−k ∥∆k w∥
⃗ 2
k≥j+2
−ηj
+Ce
⃗
⃗ ⊗ Sj+8⃗v ∥∞ )(∥S0 w∥
(∥∇⊗S
u∥∞ + ∥∇
⃗ 2+
j+8 ⃗
X
∥∆k w∥
⃗ 2)
0≤k≤j+7
We take
Z t
X
X
⃗ ⊗ S0 ⃗u∥∞ +
⃗ ⊗ ∆k ⃗u∥∞ + ∥∇
⃗ ⊗ S0⃗v ∥∞ +
⃗ ⊗ ∆k⃗v ∥∞ ds
ηj (t) = λ
∥∇
∥∇
∥∇
0
0≤k≤j+7
0≤k≤j+7
for some λ ≥ 0 large enough (we shall fix the value of λ later). We have ηj ≤ λ(j +
1
1
9)(∥⃗u∥L1 B∞,∞
+ ∥⃗v ∥L1 B∞,∞
).
Let τ > 0 (we shall fix the value of τ later). If A(τ ) = sup0<t<τ ∥w∥
⃗ LP,σ,η , we find for
0<t<τ
Z t
dηj −jσ −ηj
−jσ −ηj (t)
2
e
∥∆j w(t,
⃗ .)∥2 +
2
e ∥∆j w∥
⃗ 2 ds
0 dt
Z t
X
1
1
≤ CA(τ )
(∥⃗u∥B∞,∞
+ ∥⃗v ∥B∞,∞
)
2(j−k)(1−σ) eηk −ηj ds
0
+
k≥j+2
C
λ
Z
t
0
dηj −jσ −ηj
2
e (∥S0 w∥
⃗ 2+
dt
X
∥∆k w∥
⃗ 2 ) ds
0≤k≤j+7
=I + II.
Rt
1
1
As, for k > j, ηk (t) − ηj (t) ≤ λ(k − j) 0 ∥⃗u∥B∞,∞
+ ∥⃗v ∥B∞,∞
ds, we find that there exists
tλ > 0 (which does not depend on j nor k) such that
for t ∈ [0, tλ ],
ηk (t) − ηj (t) ≤ ϵ(k − j)
with 2−(1−σ) eϵ < 1 and 2−σ eϵ < 1, and thus, for 0 < t < τ ≤ tλ ,
Z τ
1
1
I ≤ CA(τ )
∥⃗u∥B∞,∞
+ ∥⃗v ∥B∞,∞
ds.
0
390
The Navier–Stokes Problem in the 21st Century (2nd edition)
Similarly, we define
t
Z
B(τ ) = sup (
0<t<τ
0
dη−1 −η−1
e
∥S0 j w∥
⃗ 2 ds + sup
dt
j≥0
Z
t
0
dηj −jσ −ηj
2
e ∥∆j w∥
⃗ 2 ds).
dt
B is well defined and satisfies B ≤ C∥w∥
⃗ 2 . We have
II ≤ III + IV + V
where
t
C
III =
λ
Z
C
IV =
λ
Z
0
X C Z t dηk
dη−1 −jσ −ηj
2
e ∥S0 w∥
⃗ 2 ds +
2−jσ e−ηj ∥∆k w∥
⃗ 2 ) ds
dt
λ 0 dt
0≤k≤j
t
0
d(ηj − η−1 ) −jσ −ηj
2
e ∥S0 w∥
⃗ 2 ds
dt
X C Z t d(ηj − ηk )
+
2−jσ e−ηj ∥∆k w∥
⃗ 2 ) ds
λ 0
dt
0≤k≤j
and
C
V =
λ
t
Z
0
dηj −jσ −ηj
2
e (
dt
X
∥∆k w∥
⃗ 2 ) ds.
j+1≤k≤j+7
For 0 < t < τ ≤ tλ , we find
III ≤
X
C
C′
B(τ )
(2−σ eϵ )j−k ≤
B(τ )
λ
λ
−1≤k≤j
and, since
dηj
dt
≤
dηj
dt
when j ≤ k,
V ≤
C
B(τ )
λ
X
j+1≤k≤j+7
(2σ eϵ )k−j ≤
C′
B(τ ).
λ
Finally, we have
Z t
X
C
d(ηj − ηk ) −ηj +ηk
C′
−σ(j−k)
IV ≤ A(τ )
2
e
ds ≤
A(τ )
λ
dt
λ
0
−1≤k≤j
as
Z
0
t
d(ηj − ηk ) −ηj +ηk
e
ds = 1 − e−ηj (t)+ηk (t) ≤ 1
dt
when k ≤ j.
Similar estimates hold on S0 w.
⃗ Finally, we find that, for 0 < t < τ ≤ tλ , we have
Z τ
C1
1
1
A(τ ) + B(τ ) ≤ C0 A(τ )
∥⃗u∥B∞,∞
+ ∥⃗v ∥B∞,∞
ds +
(A(τ ) + B(τ ))
λ
0
where the constants C0 and C1 do not
fix λ such that Cλ1 < 12 ,
R τ depend on λ nor τ . We then
1
1
then we fix τ ∈ (0, tλ ] such that C0 0 ∥⃗u∥B∞,∞
+ ∥⃗v ∥B∞,∞
ds ≤ 14 , and we obtain A(τ ) = 0.
Thus, we have local uniqueness: ⃗u = ⃗v on [0, τ ]. This uniqueness propagates to the whole
interval [0, T ], as t 7→ w(t,
⃗ .) is weakly continuous in L2 : thus the maximal interval [0, T ∗ ] on
which w
⃗ = 0 is closed, and as it must be open by local uniqueness, we conclude T ∗ = T .
Leray’s Weak Solutions
12.6
391
Weak Perturbations of Mild Solutions
We have seen, up to now, essentially two classes of solutions: weak ones (obtained by
mollification and then by the use of Rellich’s theorem) and mild solutions (obtained through
Picard’s method). Sometimes, it is useful to combine the two approaches, i.e., to compute
the solution ⃗u of
∂t ⃗u = ν∆⃗u + P(f⃗ − div(⃗u ⊗ ⃗u)), ⃗u(0, .) = ⃗u0
as the sum of a mild solution w
⃗ of
∂t w
⃗ = ν∆w
⃗ + P(f⃗ − div(w
⃗ ⊗ w)),
⃗
w(0,
⃗ .) = w
⃗0
and a weak solution ⃗v of
∂t⃗v = ν∆⃗v − P div(⃗v ⊗ ⃗v + w
⃗ ⊗ ⃗v + ⃗v ⊗ w),
⃗
⃗v (0, .) = ⃗v0
For instance, Calderón [77] considered the case of an initial value ⃗u0 ∈ Lp with 2 <
p < 3 (and a forcing term f⃗ = 0). Then he could show existence of a solution by splitting
⃗u0 into ⃗v0 + w
⃗ 0 , with the norm of w
⃗ 0 small in L3 (so that global existence of the mild
2
solution w
⃗ is granted) and ⃗v0 ∈ L . This kind of “mixed initial-values” which pave the way
to a combination of weak and mild solutions was discussed by Lemarié-Rieusset (in the
paper [310] and in the concluding chapter of [313]) and recently extended by Cui [134] who
−1+r,
2
1−r
+ L2 (where Xr is the space of pointwise
considered an initial value in B−1(ln)
∞∞ + BXr
multipliers that map H r to L2 ).
In this section, we address the stability of mild solutions through some L2 perturbation of
the initial value. This issue has been recently considered by Karch, Pilarczyk, and Schonbek
[251].
Existence of permanent solutions
Theorem 12.10.
Let X be a Banach space of Lebesque measurable functions such that
• the pointwise product is bounded from L∞ × X to X.
• the Hardy–Littlewood maximal function is a bounded operator on X
• the Riesz transforms are bounded on X
• the bilinear operator (u, v) 7→
√ 1 (uv)
−∆
• the bilinear operator (u, v) 7→
1
1
(u (−∆)
1/4 v)
(−∆)1/4
is bounded on X
is bounded on X
Assume that the forcing term f⃗ corresponds to a permanent regime: f⃗ is steady (i.e.,
does not depend on time) or is time-periodic (f⃗(t + T, x) = f⃗(t, x) for some positive T )
and that f⃗ is small enough: for some ϵ0 > 0 (depending only on X), we have
• in the steady case: f⃗ = ∆F⃗ with F⃗ ∈ X and ∥F⃗ ∥X < ϵ0 ν 2
• in the time periodic case:
392
The Navier–Stokes Problem in the 21st Century (2nd edition)
RT
– f⃗ belongs to L1per X with 0 ∥f⃗∥X dt < ϵ0 ν
RT
– the mean value f⃗0 = T1 0 f⃗(s, .) ds can be written as f⃗0 = ∆F⃗ with F⃗ ∈ X
with ∥F⃗ ∥X < ϵ0 ν 2 .
⃗ on (0, +∞) × R3 of the problem
Then, there exists a unique permament solution U
(
⃗ − div(U
⃗ ⊗U
⃗ ) = ν∆U
⃗ + f⃗ − ∇p
⃗
∂t U
(12.35)
⃗
div U = 0
such that
⃗ is stationary (∂t U
⃗ = 0), U
⃗ ∈ X and
• in the steady case: U
⃗ ∥X ≤ C0 1 ∥F⃗ ∥X
∥U
ν
⃗ is time-periodic, U
⃗ ∈ L∞
• in the time-periodic case: U
per X and more precisely
⃗ (t, .)| ∥X ≤ C0 (
∥ sup |U
t∈R
Z
0
T
∥f⃗(s, .)∥X +
1 ⃗
∥F ∥X ).
ν
where the constant C0 > 0 depends only on X.
Stability of permanent solutions
Theorem 12.11.
Let X satisfy the assumptions of Theorem 12.10. Assume that, for some ϵ1 > 0 (depending only on X), we have the following assumptions on f⃗ and ⃗u0 :
• in the steady case: f⃗ = ∆F⃗ with F⃗ ∈ X and ∥F⃗ ∥X < ϵ1 ν 2
• in the time-periodic case:
RT
– f⃗ belongs to L1per X with 0 ∥f⃗∥X dt < ϵ1 ν
RT
– the mean value f⃗0 = T1 0 f⃗(s, .) ds can be written as f⃗0 = ∆F⃗ with F⃗ ∈ X
with ∥F⃗ ∥X < ϵ1 ν 2 .
• ⃗u0 can be written as the sum of two divergence-free vector fields ⃗u0 = ⃗v0 + w
⃗ 0,
with ⃗v0 ∈ L2 , w
⃗ 0 ∈ X and ∥w
⃗ 0 ∥X < ϵ1 ν.
⃗ is granted by
We assume that ϵ1 ≤ ϵ0 , so that existence of a permanent solution U
Theorem 12.10.
Then, there exists at least one solution ⃗u on (0, +∞) × R3 of the problem

⃗
∂t ⃗u − div(⃗u ⊗ ⃗u) = ν∆⃗u + f⃗ − ∇p
(12.36)
div ⃗u = 0

⃗u(0, .) = ⃗u0
such that
Leray’s Weak Solutions
393
• ⃗u = ⃗v + w
⃗ with w
⃗ ∈ L∞ X and ⃗v ∈ L∞ L2 ∩ L2 Ḣ 1
⃗ (t, .)| ∥X and
• ∥ supt∈R |w(t,
⃗ .)| ∥X ≤ C1 ∥ supt∈R |U
⃗ (t, .))∥X ≤ C1 ν −1/4 ∥ sup |U
⃗ (t, .)| ∥X
sup t1/4 ∥(−∆)1/4 (w(t,
⃗ .) − U
t>0
(12.37)
t∈R
(where the constant C1 > 0 depends only on X)
• limt→+∞ ∥⃗v (t, .)∥2 = 0.
⃗ (t, .) converges to 0 in S ′ (and in L2 ) as t goes to +∞.
In particular, ⃗u(t, .) − U
loc
Proof. Due to Theorems 10.11 and 10.15, we know that the existence result (Theorem 12.10)
holds, as well as the stability result (Theorem 12.11) in the case ⃗v0 = 0.
When ⃗v0 ̸= 0, we begin by solving the Navier–Stokes problem with initial value w
⃗ 0 and
find a solution w.
⃗ Then, we study the problem
∂t⃗v = ν∆⃗v − P div(⃗v ⊗ ⃗v + w
⃗ ⊗ ⃗v + ⃗v ⊗ w),
⃗
⃗v (0, .) = ⃗v0 .
The problem will be solved just as for the classical Navier–Stokes problem (i.e., when w
⃗ = 0).
Step 1: Leray’s mollification.
As for Theorem 12.2, we study the problem associated to a mollifier θϵ
⃗ v ) − P div(w
∂t⃗v = ν∆⃗v − P((θϵ ∗ ⃗v ) · ∇⃗
⃗ ⊗ ⃗v + ⃗v ⊗ w),
⃗
⃗v (0, .) = ⃗v0 .
We have X ⊂ M(Ḣ 1 7→ L2 ) = M(L2 7→ Ḣ −1 ) and ∥w∥
⃗ L∞ X ≤ 2C1 C0 ϵ1 ν. Thus, using
again the norm
√
∥⃗u∥ν,T0 = ∥⃗u∥L∞ ((0,T0 ),L2 ) + ν∥⃗u∥L2 (0,T0 ),Ḣ 1 )
and the inequalities
∥Wνt ∗ ⃗v0 ∥ν,T0 ≤ C2 ∥⃗v0 ∥2
and
Z
∥
0
t
1
Wν(t−s) ∗ ⃗g ds∥ν,T0 ≤ C2 √ ∥⃗g ∥L2 Ḣ −1
ν
we get local existence of the solution ⃗v : as we have
p
−3/2
⃗ v∥ 2
∥(⃗u ∗ θϵ ) · ∇⃗
∥⃗u∥L∞ L2 ∥⃗v ∥L∞ L2
L ((0,T0 ),Ḣ −1 ) ≤C3 T0 ϵ
p
−3/2
≤C3 T0 ϵ
∥⃗u∥ν,T0 ∥⃗v ∥ν,T0
and
∥ div(w
⃗ ⊗ ⃗v + ⃗v ⊗ w)∥
⃗ L2 ((0,T0 ),Ḣ −1 ) ≤C3 ∥w∥
⃗ L∞ X ∥⃗v ∥L2 Ḣ 1
√
≤2C0 C1 C3 ϵ1 ν∥⃗v ∥ν,T0 ,
we find that the Picard iterate shall converge to a solution
⃗vϵ if ϵ1 is small enough
√
(2C0 C1 C3 ϵ1 < 1/4) and T0 is small enough (2C22 √1ν C3 T0 ϵ−3/2 ∥⃗v0 ∥2 < 1/4).
394
The Navier–Stokes Problem in the 21st Century (2nd edition)
We easily check that ⃗vϵ is indeed a global solution: it is enough to show that the L2
norm of ⃗vϵ remains bounded (as the existence time T0 is controlled by the L2 norm
of the Cauchy data). We have
Z
d
2
∥⃗vϵ ∥2 =2 ∂t⃗vϵ · ⃗vϵ dx
dt
= − 2ν∥⃗vϵ ∥2Ḣ 1 − 2⟨div(⃗vϵ ⊗ w)|⃗
⃗ vϵ ⟩H −1 ,H 1
(12.38)
≤ − 2(ν − ∥w∥
⃗ M(Ḣ 1 7→L2 ) )∥⃗uϵ ∥2Ḣ 1
≤ − (2 − C4 ϵ1 )ν∥⃗uϵ ∥2Ḣ 1
Thus, if ϵ1 is small enough (C4 ϵ1 < 1), we find that
Z t
2
∥⃗vϵ (t, .)∥2 + ν
∥⃗vϵ ∥2Ḣ 1 ds ≤ ∥⃗v0 ∥22
(12.39)
0
Using this energy inequality, we find that we may then use the Rellich–Lions theorem
(Theorem 12.1) and find a sequence ϵn → 0 and a function ⃗v such that:
ˆ on every bounded subinterval of [0, +∞], ⃗v(ϵn ) is *-weakly convergent to ⃗v in
L∞ L2 and in L2 Ḣ 1
ˆ ⃗v(ϵn ) is strongly convergent to ⃗v in L2loc ((0, +∞) × R3 ).
Moreover, the weak limit ⃗v satisfies
∂t⃗v = ν∆⃗v − P div(⃗v ⊗ ⃗v + w
⃗ ⊗ ⃗v + ⃗v ⊗ w)
⃗
and the Leray energy inequality for every t ∈ (0, +∞), we have
Z t
Z t
2
2
2
⃗ ⊗ ⃗v ⟩L2 ,L2 ds
∥⃗v (t, .)∥2 + 2ν
∥⃗v ∥Ḣ 1 ds ≤ ∥⃗v0 ∥2 + 2
⟨⃗v ⊗ w|
⃗∇
0
(12.40)
0
It even fulfills the strong Leray energy inequality: for almost every t0 in (0, +∞) and
for every t ∈ (t0 , +∞), we have
Z t
Z t
⃗ ⊗ ⃗v ⟩L2 ,L2 ds
∥⃗v (t, .)∥22 + 2ν
∥⃗v ∥2Ḣ 1 ds ≤ ∥⃗v (t0 )∥22 + 2
⟨⃗v ⊗ w|
⃗∇
(12.41)
t0
t0
Step 2: Higher regularity estimates.
The proof of limt→+∞ ∥⃗v (t, .)∥2 = 0 then follows the proof of Corollary 12.1. However,
we have a little difficulty to overcome: w
⃗ is not regular enough to ensure that, when ⃗v0
is regular, then ⃗v is regular (we use the boundedness in H 1 in the proof of Corollary
12.1).
Thus, we shall study the behavior of ⃗v in a smaller space: the homogeneous Besov
1/2
space Ḃ2,∞ . Let us remark that, as well as we have X ⊂ M(Ḣ 1 7→ L2 ) = V 1 , we have
1/2
Ḃ2,∞ ⊂ V 1 : interpolating the Sobolev embeddings Ḣ 0 = L2 ⊂ L2 and Ḣ 1 ⊂ L6 , we
find
1/2
Ḃ2,∞ = [Ḣ 0 , Ḣ 1 ]1/2,∞ ⊂ [L2 , L6 ]1/2,∞ ⊂ L3,∞ ⊂ V 1 .
Another useful remark is that V 1 = M(Ḣ 1 7→ L2 ) coincides with M(L2 7→ Ḣ −1 ) (by
duality, as pointwise multiplication is a self-adjoint operator) and thus (by interpola1/2
−1/2
tion) V 1 ⊂ M(Ḃ2,∞ 7→ Ḃ2,∞ ).
Leray’s Weak Solutions
395
A final remark is an inequality we already used (on page 147) when proving the
uniqueness theorem for C([0, T ], L3 ) solutions:
Z
∥
0
t
1
Wν(t−s) ∗ ⃗g ds∥L∞ Ḃ −1/2 ≤ C5 ∥⃗g ∥L∞ Ḃ −3/2 .
2,∞
2,∞
ν
Thus, writing the inequalities
∥Wνt ∗ ⃗v0 ∥L∞ Ḃ 1/2 ≤ ∥⃗v0 ∥Ḃ 1/2
2,∞
2,∞
∥ div(⃗u ⊗ ⃗v )∥L∞ Ḃ −3/2 ≤C6 ∥⃗u∥L∞ X ∥⃗v ∥L∞ Ḃ 1/2
2,∞
2,∞
≤C7 ∥⃗u∥L∞ Ḃ 1/2 ∥⃗v ∥L∞ Ḃ 1/2
2,∞
2,∞
and
∥ div(w
⃗ ⊗ ⃗v + ⃗v ⊗ w)∥
⃗ L∞ Ḃ −3/2 ≤C6 ∥w∥
⃗ L∞ X ∥⃗v ∥L2 Ḃ 1/2
2,∞
2,∞
≤C7 ϵ1 ν∥⃗v ∥L∞ Ḃ 1/2
2,∞
Thus, if ϵ1 is small enough (C5 C7 ϵ1 < 1/4) and ⃗v0 is small enough (2C5 C7 ∥⃗v0 ∥Ḃ 1/2 <
2,∞
1/2
ν/4), we have a global solution in L∞ Ḃ2,∞ .
Of course, for ϵ1 small enough and ⃗v0 small enough, this solution will still be in
L∞ L2 ∩ L2 Ḣ 1 : just write
∥ div(⃗u ⊗ ⃗v )∥L2 Ḣ −1 ≤C8 min(∥⃗u∥L∞ X ∥⃗v ∥L2 Ḣ 1 , ∥⃗u∥L2 Ḣ 1 ∥⃗v ∥L∞ X )
≤C9 min(∥⃗u∥L∞ Ḃ 1/2 ∥⃗v ∥L2 H 1 , ∥⃗u∥L2 H 1 ∥⃗v ∥L∞ Ḃ 1/2 )
2,∞
2,∞
and
∥ div(w
⃗ ⊗ ⃗v + ⃗v ⊗ w)∥
⃗ L2 Ḣ −1 ≤C8 ∥w∥
⃗ L∞ X ∥⃗v ∥L2 H 1
≤C9 ϵ1 ν∥⃗v ∥L2 H 1
to check that the Picard iterates will converge in L∞ L2 ∩ L2 Ḣ 1 .
Step 3: Weak-strong uniqueness.
As for the classical Navier–Stokes problem (Theorem 12.3), we may prove weak-strong
uniqueness for ⃗v . More precisely, assume that we have two solutions ⃗v1 and ⃗v2 in
1/2
L∞ L2 ∩ L2 Ḣ 1 and that ⃗v1 is small enough in L∞ Ḃ2,∞ (and w
⃗ small enough in L∞ X)
while ⃗v2 satisfies the Leray energy inequality. Then we find that
Z t
∥⃗v1 − ⃗v2 (t, .)∥22 ≤ − 2ν
∥⃗v1 − ⃗v2 ∥2Ḣ −1 ds
0
Z t
⃗ ⊗ (⃗v1 − ⃗v2 )⟩L2 ,L2 ds
−2
⟨(⃗v1 − ⃗v2 ) ⊗ w|
⃗∇
0
Z t
⃗ ⊗ (⃗v1 − ⃗v2 )⟩L2 ,L2 ds
−2
⟨(⃗v1 − ⃗v2 ) ⊗ ⃗v1 |∇
0
Z t
∞
≤ − 2(ν − C10 ∥w∥
⃗ L X − C10 ∥⃗v1 ∥L∞ Ḃ 1/2 )
∥⃗v1 − ⃗v2 ∥2Ḣ 1 ds.
2,∞
0
396
The Navier–Stokes Problem in the 21st Century (2nd edition)
Step 4: End of the proof.
The proof then follows the proof of Corollary 12.1. We have just seen that if ϵ1 is
small enough (ϵ1 < C11 ) and ⃗v (t0 , .) is small enough (∥⃗v (t0 , .)∥Ḃ 1/2 < C11 ν), then the
2,∞
equations
∂t⃗v = ν∆⃗v − P div(⃗v ⊗ ⃗v + w
⃗ ⊗ ⃗v + ⃗v ⊗ w),
⃗
⃗v (t0 , .) = ⃗v0
⃗ that belongs to L∞ ((t0 , +∞), L2 ) ∩ L2 ((t0 , +∞), Ḣ 1 ) ∩
has a solution V
1/2
∞
⃗.
L ((t0 , +∞), Ḃ2,+∞ ) and that we have weak-strong uniqueness for V
1/2
2
2 1
4
As ⃗v belongs to L∞
t L ∩ Lt Ḣ , it belongs to Lt Ḃ2,+∞ . Thus, the set of times t such
that ∥⃗v (t, .)∥Ḃ 1/2 ≥ C11 ν is of finite measure. As the set of Lebesgue points of
2,+∞
t 7→ ∥⃗v (t, .)∥2 has a complement of null measure, we may find a time t0 such that
ˆ ∥⃗v (t0 , .)∥Ḣ 1/2 < C11 ν
ˆ ⃗v is a weak Leray solution on (t0 , +∞): for every t ∈ (t0 , +∞), we have
Z t
Z t
⃗ ⊗ ⃗v ⟩L2 ,L2 ds
⟨⃗v ⊗ w|
⃗∇
∥⃗v (t, .)∥22 + 2ν
∥⃗v ∥2Ḣ 1 ds ≤ ∥⃗v0 ∥22 + 2
0
0
Then, by weak-strong uniqueness, we find that ⃗v coincides on (t0 , +∞) with the mild
⃗.
solution V
We now write, for t0 < τ < t,
t
Z
∥⃗v (t, .)∥2 ≤ ∥Wν(t−τ ) ∗ ⃗v (τ, .)∥2 + C(
τ
∥P div(⃗v ⊗ ⃗v + w
⃗ ⊗ ⃗v + ⃗v ⊗ w)∥
⃗ 2Ḣ −1 ds)1/2
which gives
Z
⃗ .∥X )(
lim sup ∥⃗v (t, .)∥2 ≤ C(sup ∥⃗v (t, .)∥Ḃ 1/2 + sup ∥w(t,
t→+∞
2,∞
t>t0
t>t0
τ
+∞
∥⃗v ∥2Ḣ 1 ds)1/2 .
Letting τ go to +∞, we get
lim ∥⃗v (t, .)∥2 = 0.
t→+∞
12.7
Non-uniqueness of Weak Solutions
Very recently, some results have been published on the non-uniqueness of weak solutions.
We have seen that weak-strong uniqueness holds for Leray weak solutions in presence of a
mild solution, but other uniqueness issues could be considered:
• Q1) Does uniqueness hold for L∞ L2 ∩ L2 H 1 solutions in presence of a mild solution,
but without assuming Leray’s energy inequality?
• Q2) Does uniqueness hold for Leray weak solutions in absence of a mild solution?
• Q3) Does uniqueness hold for L∞ L2 solutions in presence of a mild solution?
While question Q1) is still open, answers to Q2) and Q3) have been proven to be negative
by Buckmaster and Vicol [71] in 2019 and by Albritton, Brué and Colombo [5] in 2021.
Leray’s Weak Solutions
397
Wild solutions on the torus
In this section, we will discuss solutions of the equations
⃗
∂t ⃗u = ν∆⃗u − div(⃗u ⊗ ⃗u) − ∇p
on (0, T ) × R3 with the conditions

⃗u(t, x + 2kπ) = ⃗u(t, x) for all k ∈ Z3






p(t, x + 2kπ) = p(t, x) for all k ∈ Z3





⃗u ∈ L∞ ((0, T ), L2 (R3 /Z3 ))

div ⃗u = 0







Z




div
⃗
u
=
0
and

0
⃗u(0, .) = ⃗u0
⃗u0 (x) dx = 0
[−π,π]3
R
The periodicity of p will ensure that [−π,π]3 ⃗u(t, x) dx = 0 for all t ∈ (0, T ) and that
X
1 ⃗
⃗ ∧ ⃗u), where, for a Fourier series f =
⃗u = − ∆
∇ ∧ (∇
eik·x ak ,
k∈Z3 ,k̸=0
X
1
f =−
∆
3
k∈Z ,̸=0
1 ik·x
e ak .
|k|2
Defining, for a periodic vector field f⃗ with zero mean,
1⃗
1⃗
⃗ ∧ f⃗),
Pf⃗ = f⃗ − ∇(div
f⃗) = − ∇
∧ (∇
∆
∆
we find that
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u).
The analysis for mild or weak periodic solutions is then very similar to the analysis on the
whole space. We start with the basic estimates:
R
• analysis of the heat kernel et∆ f = Wt ∗ f : if f ∈ L2 (R3 /Z3 ) with [−π,π]3 f (x) dx = 0,
then
∥eνt∆ f ∥L∞ ((0,+∞),L2 (R3 /Z3 )) = ∥f ∥L2 (R3 /Z3 )
and
⃗ νt∆ f ∥L2 ((0,+∞),L2 (R3 /Z3 )) = √1 ∥f ∥L2 (R3 /Z3 )
∥∇e
2ν
R
• Sobolev embedding: if f ∈ H 1 (R3 /Z3 ) with [−π,π]3 f (x) dx = 0, then
⃗ ∥L2 (R3 /Z3 ) ,
∥f ∥L6 (R3 /Z3 ) ≤ C∥∇f
R
and, if f ∈ H 2 (R3 /Z3 ) with [−π,π]3 f (x) dx = 0, then
1/2
1/2
⃗ ∥ 2 3 3 ∥∆f ∥ 2 3 3 .
∥f ∥L∞ (R3 /Z3 ) ≤ C∥∇f
L (R /Z )
L (R /Z )
R t ν(t−s)∆
⃗ ds belongs to
• if f ∈ L2 ((0, +∞), L2 (R3 /Z3 )), then F =
e
∇f
0
2
3
3
2
1
3
3
Cb ([0, +∞), L (R /Z )) ∩ L ((0, +∞), H (R /Z )) and
1
∥F ∥L2 ((0,+∞),L2 (R3 /Z3 )) ≤ C √ ∥f ∥L2 ((0,+∞),L2 (R3 /Z3 ))
ν
⃗ ⊗ F ∥L2 ((0,+∞),L2 (R3 /Z3 )) ≤ C 1 ∥f ∥L2 ((0,+∞),L2 (R3 /Z3 ))
∥∇
ν
398
The Navier–Stokes Problem in the 21st Century (2nd edition)
We thus easily control the bilinear operator
Z t
B(⃗u, ⃗v ) =
e(t−s)∆ P div(⃗u ⊗ w)
⃗ ds.
0
We get:
• control in Ḣ 1 norm:
∥B(⃗u, ⃗v )∥L∞ ((0,T ),Ḣ 1 ) +
√
ν∥B(⃗u, ⃗v )∥L2 ((0,T ),Ḣ 2 )
1
≤C √ ∥ div(⃗u ⊗ ⃗v )∥L2 ((0,T ),L2 )
ν
1
≤C √ ∥⃗u∥L4 ((0,T ),L∞ ) ∥⃗v ∥L4 ((0,T ),Ḣ 1 )
ν
√
T 1/4 q
≤C 3/4 ∥⃗u∥L∞ ((0,T ),Ḣ 1 ) ν∥⃗u∥L2 ((0,T ),Ḣ 2 ∥⃗v ∥L∞ ((0,T ),Ḣ 1 ) .
ν
This control gives us existence of a solution of the Navier–Stokes equations in
3
Cb ([0, T ), Ḣ 1 (R3 /Z3 )) ∩ L2 ((0, T ), Ḣ 2 (R3 /Z3 )) with T = O( ∥⃗u0ν∥4 ).
Ḣ 1
• control in L2 norm:
∥B(φϵ ∗ ⃗u, ⃗v )∥L∞ ((0,T ),L2 ) +
√
ν∥B(φϵ ∗ ⃗u, ⃗v )∥L2 ((0,T ),Ḣ 1 )
1
≤C √ ∥⃗u ⊗ ⃗v ∥L2 ((0,T ),L2 )
ν
1 1/2
≤C √ T ∥φϵ ∗ ⃗u∥L∞ ((0,T ),L∞ ) ∥⃗v ∥L∞ ((0,T ),L2 )
ν
≤C ′
T 1/2
∥⃗u∥L∞ ((0,T ),L2 ) ∥⃗v ∥L∞ ((0,T ),L2 ) .
ν 1/2 ϵ3/2
This control gives us existence of a solution ⃗uϵ of the mollified Navier–Stokes equations
3
in Cb ([0, Tϵ ), L2 (R3 /Z3 )) ∩ L2 ((0, Tϵ ), Ḣ 1 (R3 /Z3 )) with Tϵ = O( ∥⃗uνϵ0 ∥2 ).
2
• Then, we use the energy equality
Z
Z tZ
|⃗uϵ (t, x)|2 dx + 2ν
[−π,π]3
0
[−π,π]3
⃗ ⊗ ⃗uϵ (s, x)|2 dx ds =
|∇
Z
|⃗u0 (x)|2 dx
[−π,π]3
to extend ⃗uϵ as a global solution on (0, +∞) × R3 . Then, applying the Rellich–Lions
theorem, we get a weak solution ⃗u on (0, +∞) × R3 that satisfies Leray’s energy
inequality
Z
Z tZ
Z
⃗ ⊗ ⃗u(s, x)|2 dx ds ≤
|⃗u(t, x)|2 dx + 2ν
|∇
|⃗u0 (x)|2 dx
[−π,π]3
0
[−π,π]3
[−π,π]3
Finally, we check easily that we have weak-strong uniqueness: if ⃗u0 ∈ H 1 , and if a mild
solution ⃗u is defined on (0, T ), then every weak Leray solution ⃗v coincides with ⃗u on (0, T ).
Leray’s Weak Solutions
399
However, Buckmaster and Vicol [71] proved non-uniqueness in C([0, T ], L2 ):
Buckmaster and Vicol’s theorem
Theorem 12.12.
There exists β > 0, such that for any nonnegative smooth function e(t) : [0, T ] 7→
R+ , there exists ⃗v ∈ C([0, T ], Hxβ (R3 /Z3 )) a very weak solution ⃗v of the Navier-Stokes
equations, such that
Z
|⃗v (t, x)|2 dx = e(t) for all t ∈ [0, T ].
[−π,π]3
To quote Buckmaster and Vicol,
In particular, the above theorem shows that ⃗v = 0 is not the only weak solution which vanishes at a time slice, thereby implying the nonuniqueness of weak
solutions.
Buckmaster and Vicol’s proof relies on convex integration tools, a technique developed by
De Lellis and Székelyhidi Jr. [148, 149] for the study of Euler equations and the Onsager
conjecture which was eventually fully proved by P. Isett in 2018 [241] (see [70, 72] for a
survey).
Non-uniqueness results for Leray solutions.
Buckmaster and Vicol’s solutions are less regular than Leray solutions. (Their solutions
fulfill estimates in some Sobolev spaces H β (R3 ) with β << 1, but not in H 1 , whereas a
Leray solution should be controlled in L2 H 1 ). Albritton, Brué and Colombo [5] published
in 2022 the following result:
Albritton, Brué and Colombo’s theorem
Theorem 12.13.
There exist T > 0, f⃗ ∈ L1 ((0, T ), L2 (R3 )), and two distinct suitable Leray solutions ⃗u1 ,
⃗u2 to the Navier–Stokes equations on (0, T ) × R3 with body force f⃗ and initial condition
⃗u0 = 0.
Non-uniqueness is to be seen at time t = 0: we have an asymptotic estimats that
1
∥⃗u1 (t, .) − ⃗u2 (t, .)∥ = Ω(ta+ 4 ) for some positive a as t → 0. Of course, f⃗ is not regular near
t = 0: if we had f⃗ ∈ L1 ((0, T ), H 1/2 (R3 )), then we would have, for a small time T0 > 0, a
mild solution ⃗v in C([0, T0 ], H 1/2 ) and, by weak-strong uniqueness, ⃗u1 = ⃗v = ⃗u2 ; as a matter
of fact, the force involved in the proof of Theorem 12.13 is such that ∥f (t, .)∥H 1/2 ∼ Ct−1 .
Albritton, Brué and Colombo’s strategy of proof follows the idea developed by Guillod,
Jia and Šverák [246, 223] to derive non-uniqueness from linear unstability of a self-similar
⃗ (x) is a divergence-free
profile for the underlying linearized problem. More precisely, if U
⃗ ( √x ) and f⃗ = ∂t ⃗u − ∆⃗u + P(⃗u · ∇⃗
⃗ u) (so that
vector field on R3 , define ⃗u(t, x) = √1t U
t
1 ⃗
1 ⃗ √
f⃗(t, x) = 3/2
f (1, √x )) [or equivalently define f⃗(t, x) = 3/2
F ( x ), where
t
t
t
t
1 ⃗
⃗U
⃗ ) − ∆U
⃗ + P(U
⃗ ·∇
⃗U
⃗ )].
F⃗ = − (U
+x·∇
2
400
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ is enough smooth and decaying (for instance, U
⃗ ∈ H 2 and x2 U
⃗ ∈ L2 ), then f⃗ is in
If U
1
2
L (]0, T [, L ) for T < +∞, and ⃗u is a Leray solution of the Cauchy problem with body
force f⃗ and initial value ⃗u0 = 0. The idea for finding another solution ⃗v of the same
Cauchy problem is to write the problem in the variables ξ = √xt and τ = ln t; writing
⃗ (ξ) + V
⃗ (τ, ξ)), we find
⃗v (t, x) = √1 (U
t
⃗ = 1 (V
⃗ +x·∇
⃗V
⃗ ) + ∆V
⃗ − P(U
⃗ ·∇
⃗V
⃗ +V
⃗ ·∇
⃗U
⃗ +V
⃗ ·∇
⃗V
⃗)
∂τ V
2
(12.42)
whose linearization gives
⃗ +x·∇
⃗W
⃗ ) + ∆W
⃗ − P(U
⃗ ·∇
⃗W
⃗ +W
⃗ ·∇
⃗U
⃗ ).
⃗ = 1 (W
∂τ W
2
(12.43)
⃗ +x·∇
⃗W
⃗ ) + ∆W
⃗ − P(U
⃗ ·∇
⃗W
⃗ +W
⃗ ·∇
⃗U
⃗ ) and if L is linearly unstable,
If L(W ) = 12 (W
writing λ = a + ib an eigenvalue of L with maximal real part a > 0, the authors pick up an
⃗ = ℜ(eλτ η) is a solution of (12.43) and we
eigenvector η of L such that Lη = λη. Then W
⃗ of (12.42) such that ∥V
⃗ (τ, .) − W
⃗ (τ, .)∥2 ≤ Ce2aτ as τ → −∞ while
can find a solution V
aτ
⃗
⃗
∥W (τ, .)∥2 = Ω(e ). Thus, V =
̸ 0, and we shall have two solutions of the same Cauchy
problem.
⃗ such that L is unstable is not easy [5]. It is derived from an
The construction of U
unstable steady solution of the 2D Euler equations studied in two very recent papers of
Višik [490, 491].
12.8
The Inviscid Limit
The Navier–Stokes equations for a viscous incompressible homogeneous Newtonian fluids
(with no forcing term) read as
(
⃗ u = ν∆⃗u − ∇p,
⃗
∂t ⃗u + ⃗u · ∇⃗
div ⃗u = 0.
In the case of an inviscid fluid (ν = 0), we have the Euler equations:
(
⃗ u = −∇p,
⃗
∂t ⃗u + ⃗u · ∇⃗
div ⃗u = 0.
The inviscid limit problem is the study of the convergence of solutions ⃗uν to the Cauchy
problem for the Navier–Stokes equations with initial value ⃗u0 and viscosity ν when ν goes
to 0; in particular do we have convergence to a solution of the Cauchy problem for the Euler
equations with initial value ⃗u0 ?
This is a difficult problem when considering the problem in a domain Ω with a boundary
∂Ω, as the natural boundary conditions for the Navier–Stokes equations and for the Euler
equations are different: usually, the boundary condition for the Navier–Stokes equations is
the no-slip conditions ⃗uν = 0 on ∂Ω, whereas the boundary condition for the Euler equations
is the no-flux condition ⃗u ·⃗n = 0 on ∂Ω (where ⃗n is the normal vector to ∂Ω). The curvature
of the boundary may play a role in the convergence, as proved by Beirão da Veiga and
Crispo in 2012 [33].
Leray’s Weak Solutions
401
As we consider the problem on the whole space, we may ignore this discrepancy between
the boundary conditions, and the solution is easy in case of regular flows. Page 72, we have
already presented the results of Swann [458] on existence of a solution ⃗uν on a time interval
(0, T ) with T independent of ν when ⃗u0 is regular enough. In that case, the inviscid limit
is easy to prove [458, 254]:
Theorem 12.14.
Let ⃗u0 ∈ H s (R3 ) with s > 5/2 and div ⃗u0 = 0 and F ∈ L2 ((0, +∞), H s+1 ). Then there
exists T > 0 such that, for every ν > 0, the Cauchy problem

⃗
⃗

∂t ⃗uν + ⃗uν · ∇⃗uν = ν∆⃗uν − ∇pν + div F,
div ⃗uν = 0,


⃗uν (0, .) = ⃗u0
has a unique solution ⃗uν in C([0, T ], H s ) ∩ L2 ((0, T ), H s+1 ).
Moreover, the Euler equations

⃗
⃗

∂t ⃗u + ⃗u · ∇⃗u = ∇p + div F,
div ⃗u = 0,


⃗u(0, .) = ⃗u0
have a unique solution ⃗u ∈ C[0, T ], L2 )∩L∞ ((0, T ), H s ) and we have strong convergence
of ⃗uν to ⃗u in C([0, T ], H σ ) for every σ < s.
Proof. Recall that we proved in Theorem 7.3 existence of a unique ⃗uν on a time interval
(0, Tν ). More precisely, we have the inequalities
⃗ ⊗ (Wνt ∗ ⃗u0 )∥L2 H s ≤ √1 ∥⃗u0 ∥H s
∥Wνt ∗ ⃗u0 ∥L∞ H s ≤ ∥⃗u0 ∥H s , ∥∇
2ν
Z t
1
∥
Wν(t−s) ∗ P div F ds∥L∞ H s ≤ √ ∥F∥L2 H s ,
2ν
0
Z t
1
⃗ ⊗(
∥∇
Wν(t−s) ∗ P div F ds)∥L2 H s+1 ≤ ∥F∥L2 H s ,
ν
0
and (since H s is an algebra)
1/2
∥⃗u ⊗ ⃗v ∥L2 ((0,T0 ),H s ) ≤ C0 T0 ∥⃗u∥L∞ ((0,T0 ),H s ) ∥⃗v ∥L∞ ((0,T0 ),H s ) .
Thus, we find a solution ⃗uν in C([0, Tν ], H s ) ∩ L2 ((0, Tν ), H s+1 ) with
Tν = 2ν
4C0 (∥⃗u0 ∥H s +
√1 ∥F∥L2 H s )
2ν
2 .
From local-in-time existence and uniqueness of solutions, we find that we have a solution
⃗uν on a maximal time interval (0, Tν∗ ) (which belongs to C([0, T ], H s ) ∩ L2 ((0, T ), H s+1 ) for
every 0 < T < Tν∗ ). To prove that Tν∗ > T0 > 0, where T0 does not depend on ν, we must
prove that ⃗u remains bounded in H s on (0, min(Tν∗ , T0 )). More precisely, we shall prove that
∥⃗uν ∥H s ≤ C1 on (0, T0 ), where neither T0 nor C1 depend on ν (but depend on ⃗u0 and F).
402
The Navier–Stokes Problem in the 21st Century (2nd edition)
We write the energy balance in H s norm:
Z
d
2
s/2
2
(∥⃗uν ∥2 +∥(−∆) ⃗uν ∥2 ) = 2 ∂t ⃗uν · (⃗uν + (−∆)s ⃗uν ) dx
dt
Z
⃗ ⊗ ⃗uν |2 + |∇
⃗ ⊗ (−∆)s/2 ⃗uν |2 dx
= − 2ν |∇
Z
Z
+ 2 ⃗uν · div F dx + 2 (−∆)s/2 ⃗uν · (−∆)s/2 div F dx
Z
Z
s/2
⃗ uν ) dx − 2 (−∆)s/2 ⃗uν · (⃗uν · ∇(−∆)
⃗
− 2 ⃗uν · (⃗uν · ∇⃗
⃗uν ) dx
Z
+2
s/2
⃗
(−∆)s/2 ⃗uν · (⃗uν · ∇(−∆)
⃗uν ) dx − 2
Z
⃗ uν ) dx
(−∆)s/2 ⃗uν · (−∆)s/2 (⃗uν · ∇⃗
As div ⃗uν = 0, we have
Z
Z
s/2
⃗ uν ) dx− = (−∆)s/2 ⃗uν · (⃗uν · ∇(−∆)
⃗
⃗uν · (⃗uν · ∇⃗
⃗uν ) dx = 0.
Moreover, we have
Z
s
s
3
∥(−∆) 2 (u∂k v) − u∂k (−∆) 2 v∥2 =(2π)− 2 ∥ ηk û(ξ − η)v̂(η)(|ξ|s − |η|s ) dη∥2
Z
≤ C∥ |ηk ||û(ξ − η)||v̂(η)||ξ − η|(|ξ − η|s−1 + |η|s−1 ) dη∥2
≤C(∥|ξ|û∥1 ∥|ξ|s v̂∥2 + ∥|ξ|s û∥2 ∥|ξ|v̂∥1 )
≤C ′ ∥u∥H s ∥v∥H s .
Thus, we find that
d
(∥⃗uν ∥22 +∥(−∆)s/2 ⃗uν ∥22 )
dt
≤∥⃗uν ∥22 + ∥(−∆)s/2 ⃗uν ∥22 + ∥F∥2H s+1 + C0 (∥⃗uν ∥22 + ∥(−∆)s/2 ⃗uν ∥22 )3/2
and we may conclude that
∥⃗uν ∥22 + ∥(−∆)s/2 ⃗uν ∥22 ≤ 4(∥⃗u0 ∥22 + ∥(−∆)s/2 ⃗u0 ∥22 + 1)
on (0, T0 ), as long as T0 is small enough to grant that
RT
ˆ 0 0 ∥F∥2H s+1 dt ≤ 1
ˆ T0 ≤
1
4
ˆ T0 (∥⃗u0 ∥22 + ∥(−∆)s/2 ⃗u0 ∥22 + 1)1/2 ≤ 1.
Further, we may estimate ⃗uµ − ⃗uν for small µ and ν:
Z
d
∥⃗uν − ⃗uµ ∥22 =2 (⃗uν − ⃗uµ ) · (∂t ⃗uν − ∂t ⃗uµ ) dx
dt
Z
=2 (⃗uν − ⃗uµ ) · (ν∆⃗uν − µ∆⃗uµ ) dx
Z
⃗ uν − ⃗uµ )) dx
− 2 (⃗uν − ⃗uµ ) · (⃗uν · ∇(⃗
Z
⃗ uµ ) dx
− 2 (⃗uν − ⃗uµ ) · ((⃗uν − ⃗uµ ) · ∇⃗
≤2∥⃗uν − ⃗uµ ∥2 (µ∥∆⃗uµ ∥2 + ν∥∆⃗uν ∥2 )
⃗ ⊗ ⃗uµ ∥∞ ∥⃗uµ − ⃗uν ∥22
+ 2∥∇
Leray’s Weak Solutions
403
so that, on (0, T0 ), writing M0 = 2(∥⃗u0 ∥22 + ∥(−∆)s/2 ⃗u0 ∥22 + 1)1/2 , we get
d
∥⃗uν − ⃗uµ ∥22 ≤ 4M02 (µ + ν) + C0 M0 ∥⃗uν − ⃗uµ ∥22
dt
and
∥⃗uν − ⃗uµ ∥22 ≤ T0 4M02 (µ + ν)eC0 T0 .
Thus, ⃗uµ is strongly convergent in C([0, T0 ], L2 ) as µ goes to 0, to a limit ⃗u. As ⃗uµ is
bounded in L∞ ((0, T0 ), H s ), it is strongly convergent in C([0, T0 ]H σ ) for every σ < s. In
particular, µ∆⃗uµ − P⃗uµ converges in C([0, T0 ], L2 ) to −P⃗u, and ⃗u is a solution to the Euler
equations.
Chapter 13
Partial Regularity Results for Weak Solutions
13.1
Interior Regularity
In this chapter, we shall work on the local behavior of the solution ⃗u of the Navier–Stokes
equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0
(13.1)
We assume that f⃗ is given on an open set Q = I × Ω, where I = (a, b) is an interval of R
and Ω = B(x0 , r) an open ball of R3 .
The solution ⃗u is defined on Q and belongs to L∞ (I, L2 (Ω)) ∩ L2 (I, H 1 (Ω)) and the
equation (13.1) is fulfilled in a weak sense: for every φ
⃗ ∈ D(Q) with div φ
⃗ = 0, we have
⃗ u − f⃗|⃗
⟨∂t ⃗u − ν∆⃗u + ⃗u · ∇⃗
φ⟩D′ ,D = 0
(13.2)
If ⃗u is a solution of (13.2), then it is easy to prove that there exists a distribution p ∈ D′ (Q)
such that (13.1) is fulfilled in D′ . Serrin [434] studied the local regularity of such solutions.
His theory is based on the following theorem:
Local regularity theory
Theorem 13.1.
Let Q = I ×Ω, where I = (a, b) and Ω = B(x0 , r). Let ⃗u ∈ L∞ (I, L2 (Ω))∩L2 (I, H 1 (Ω)),
f⃗ ∈ L2 (I, L2 (Ω)) and p ∈ D′ (Q), and assume that ⃗u is a weak solution on Q of the
Navier–Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Then, if moreover ⃗u ∈ L∞ (Q), we have, for every a < c < b and 0 < ρ < r:
• ⃗u ∈ L∞ ((c, b), H 1 (B(x0 , ρ)) ∩ L2 ((c, b), H 2 (B(x0 , ρ))
• if k ∈ N and f⃗ ∈ L2 (I, H k (Ω)), then ⃗u ∈ L∞ ((c, b), H k+1 (B(x0 , ρ)) ∩
L2 ((c, b), H k+2 (B(x0 , ρ))
In particular, if f⃗ ∈ C ∞ (Q), then ⃗u is smooth on Q with respect to the space variable x.
Definition 13.1 (Serrin’s regularity).
A solution ⃗u of the Navier–Stokes equations on Q = I × Ω is regular in the sense of Serrin
if ⃗u belongs to L∞ (I, L2 (Ω)) ∩ L2 (I, H 1 (Ω)) and if moreover ⃗u ∈ L∞ (Q).
Proof. Step 1: Equation on the vorticity.
DOI: 10.1201/9781003042594-13
404
Partial Regularity Results for Weak Solutions
405
To get rid of the unknown pressure p, we take the curl of the Equation (13.1) and get
the following equation on ω
⃗ = curl ⃗u:
⃗ ∧ (⃗u · ∇⃗
⃗ u)
∂t ω
⃗ = ν∆⃗
ω + curl f⃗ − ∇
As ⃗u ∈ L2 H 1 and div ⃗u = 0, we may develop
⃗ ∧ (⃗u · ∇⃗
⃗ u) = ∇
⃗ ∧ ( 1 ∇|⃗
⃗ u|2 + ω
⃗ω−ω
⃗ u = div(⃗u ⊗ ω
∇
⃗ ∧ ⃗u) = ⃗u · ∇⃗
⃗ · ∇⃗
⃗ −ω
⃗ ⊗ ⃗u)
2
and we obtain
∂t ω
⃗ = ν∆⃗
ω + curl f⃗ − div(⃗u ⊗ ω
⃗ −ω
⃗ ⊗ ⃗u)
Thus, ω
⃗ is solution of a linear heat equation
∂t ω
⃗ = ν∆⃗
ω + ⃗g
(13.3)
with ⃗g ∈ L2 (I, H −1 (Ω)) and ω
⃗ ∈ L2 (Q).
Step 2: The heat equation.
We are going to prove that if ω ∈ L2 (Q) is solution of
∂t ω = ν∆ω + g
with g ∈ L2 (I, H k−1 (Ω)) for some k ∈ N, then, for a < c < b and 0 < ρ < r, we have
ω ∈ L∞ ((c, b), H k (Ω)) ∩ L2 ((c, b), H k+1 (Ω)).
As ∂t (∂j ω) = ν∆(∂j ω) + ∂j g, this is done by induction on k, and we have just to
consider the case k = 0.
We consider now a function ϕ ∈ D(R × R3 ) which is equal to 1 on [c, b] × B(x0 , ρ) and
r+ρ
is supported in [ a+c
2 , b + 1] × B(x0 , 2 ). We define ϖ = ϕω. We have:
ˆ ϖ ∈ L2 ((a, b) × R3 ) ∩ C([a, b], H −2 (R3 ))
ˆ ϖ(a.) = 0
⃗ · ∇ϕ
⃗ + ω∂t ϕ ∈ L2 ((a, b), H −1 (R3 ))
ˆ ∂t ϖ = ν∆ϖ + h with h = ϕg − νω∆ϕ − 2ν ∇ω
Writing
Z
t
Wν(t−s) ∗ h(s, .) ds
ϖ=
a
we see that ϖ ∈ L∞ ((0, L2 )) ∩ L2 H 1 . Thus, ω is locally regular.
Step 3: regularity of ⃗u.
We write
⃗ ∧ω
⃗
∇
⃗ = −∆⃗u + ∇(div
⃗u) = −∆⃗u.
If f⃗ ∈ L2 (I, H k ) and ⃗u ∈ L∞ L2 ∩ L2 H 1 ∩ L∞
t,x on Q, then we shall see that
for every a′ ∈ (a, b) and r′ ∈ (0, r), we have ω
⃗ ∈ L∞ ((a′ , b), H k (B(x0 , r′ )) ∩
2
′
k+1
′
∞
′
L ((a , b), H
(B(x0 , r )). Thus ∆⃗u ∈ L ((a , b), H k−1 (B(x0 , r′ )) ∩ L2 ((a′ , b), H k
′
(B(x0 , r )). We then pick up a function ϕ ∈ D(R3 ) such that φ = 1 for |x − x0 | < c
′
and = 0 for |x − x0 | > a 2+c . We have
∆(φ⃗u) = φ∆⃗u + (∆φ)⃗u + 2
3
X
i=1
∂i φ ∂i ⃗u.
406
The Navier–Stokes Problem in the 21st Century (2nd edition)
We know that ⃗u ∈ L∞ L2 ∩ L2 H 1 ; if ⃗u is L∞ H l ∩ L2 H l+1 for some 0 ≤ l ≤ k on
(a′ , b) × B(x0 , r′ ), then we find that ∆(φ⃗u) is L∞ H l−1 ∩ L2 H l and ⃗u is L∞ H l+1 ∩
L2 H l+2 . Thus, ⃗u will be L∞ H k+1 ∩ L2 H k+2 .
It remains to show that
ω
⃗ ∈ L∞ ((a′ , b), H k (B(x0 , r′ )) ∩ L2 ((a′ , b), H k+1 (B(x0 , r′ )).
We start from the equation
∂t ω
⃗ = ν∆⃗
ω + ⃗g
with
⃗g = curl f⃗ − div(⃗u ⊗ ω
⃗ −ω
⃗ ⊗ ⃗u).
We know that ω
⃗ ∈ L2 L2 and ⃗u ∈ L∞ , so that ⃗g ∈ L2 H −1 and that, locally, ω
⃗ ∈ L∞ L2 ∩
2 1
∞ 1
2 2
∞ l
L H and ⃗u ∈ L H ∩ L H . By induction, we assume that ⃗u ∈ L H ∩ L2 H l+1 ,
for some 0 ≤ l ≤ k. As
⃗ ∧ div(⃗u ⊗ ⃗u)
⃗g = curl f⃗ − ∇
and that, for ψ supported in (a′ , b] × B(x0 , r′ ),
∥ψ 2 ⃗u ⊗ ⃗u∥L2 H l+1 ≤ C∥ψ⃗u∥L∞ (Q) ∥ψ⃗u∥L2 H l+1 ,
we can see that ⃗g is locally L2 H l−1 , so that ω
⃗ is locally L∞ H l ∩ L2 H l+1 and ⃗u is
∞ l+1
2 l+2
locally L H
∩L H .
The theorem is thus proved.
It is important to notice that Theorem 13.1 does not convey any information on the
time regularity of ⃗u, because of the presence of the unknown pressure p: the control of p
is equivalent to the control of ∂t ⃗u. Serrin gave the following example: if ψ is a harmonic
⃗
function on R3 and α a bounded function on R, define ⃗u on (0, 1) × B(0, 1) as ⃗u = α(t)∇ψ.
Then we have div ⃗u = α(t)∆ψ = 0, curl ⃗u = 0, ∆⃗u = 0 and
2
2
⃗ u = ∇(
⃗ |⃗u| ) + curl ⃗u ∧ ⃗u = ∇(
⃗ |⃗u| ).
⃗u · ∇⃗
2
2
We get
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0
2
− |⃗u2|
with f⃗ = 0 and p =
− ∂t α ψ. Moreover, ⃗u ∈ L∞ L2 ∩ L2 H 1 ∩ L∞
t,x on (0, 1) × B(0, 1);
but, if α is not regular, ⃗u has no regularity with respect to time.
13.2
Serrin’s Theorem on Interior Regularity
In view of Theorem 13.1, it is important to show that ⃗u is locally bounded in time
and space variables. This may be done locally under the assumption f⃗ ∈ L2 (I, H 1 (Ω)) and
⃗u ∈ Lp (I, Lq (Ω)) with 2/p + 3/q = 1 and q > 3. The case 2/p + 3/q < 1 was first proved by
Serrin [434]; the case 2/p + 3/q = 1 was then proved by Struwe [455] and Takahashi [459].
In order to state quite a general theorem, we use the space of multipliers introduced in
Theorem 12.4:
Partial Regularity Results for Weak Solutions
407
2
2 1
2 2
ˆ X is the space of pointwise multipliers on R × R3 from L∞
t L ∩ Lt H to Lt Lx , normed
with
∥u∥X =
sup
∥uv∥L2 L2 ;
∥v∥L∞ L2 +∥v∥L2 H 1 ≤1
t
t
ˆ X(0) is the space of multipliers u in X such that, for every t0 ∈ R,
lim ∥1(t0 −ϵ,t0 +ϵ) (t)u(t, x)∥X = 0.
ϵ→0+
Lemma 13.1.
If u ∈ X and v ∈ L2 L2 , then uv ∈ L1t L2 + L2t H −1 .
2
2 1
1 2
2 −1
Proof. The dual of L1t L2 + L2t H −1 is L∞
,
t L ∩ Lt H . Thus, for w ∈ Lt L + Lt H
2 + ∥z∥L2 H 1 ≤ 1}
∥w∥L1t L2 +L2t H −1 ≈ sup{|⟨w|z⟩| / ∥z∥L∞
t L
t
If v ∈ D(R × R3 ), then uv belongs to L2 L2 and has compact support, hence belong to L1 L2 .
Moreover, we have
2 + ∥z∥L2 H 1 ≤ 1} ≤ ∥v∥L2 L2 ∥u∥X
∥uv∥L1t L2 +L2t H −1 ≈ sup{|⟨v|uz⟩| / ∥z∥L∞
t L
t
We then conclude by the density of D in L2 L2 and the completeness of L1t L2 + L2t H −1 .
Interior regularity
Theorem 13.2 (Serrin’s theorem).
Let Q = I ×Ω, where I = (a, b) and Ω = B(x0 , r). Let ⃗u ∈ L∞ (I, L2 (Ω))∩L2 (I, H 1 (Ω)),
f⃗ ∈ L2 (I, L2 (Ω)) and p ∈ D′ (Q), and assume that ⃗u is a weak solution on Q of the
Navier–Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Then, if moreover f⃗ ∈ L2 (I, H 1 (Ω)) and 1Q ⃗u ∈ X(0) , we have, for every a < c < b and
0 < ρ < r, ⃗u ∈ L∞ (((c, b), H 2 (B(x0 , ρ)), so that ⃗u is locally bounded in time and space
variables.
Proof. First step: a linear heat equation.
Let ω
⃗ = curl ⃗u. We consider a function ϕ ∈ D(R × R3 ) which is equal to 1 on [c, b] ×
r+ρ
B(x0 , ρ) and is supported in [ a+c
⃗ = ϕ⃗
ω . We have:
2 , b + 1] × B(x0 , 2 ). We define w
ˆ w
⃗ ∈ L2 ((a, b) × R3 )
ˆ w(t,
⃗ .) = 0 for a < t <
b+c
2
ˆ ∂t w
⃗ = ν∆w
⃗ − div(1Q ⃗u ⊗ w
⃗ −w
⃗ ⊗ 1Q ⃗u) + ⃗g with
⃗ u − (⃗u · ∇ϕ)⃗
⃗ ω − (∆ϕ)⃗
⃗g = (⃗
ω · ∇ϕ)⃗
ω−2
3
X
(∂i ϕ) ∂i ω
⃗ + ∂t ϕ⃗
ω + ϕ curl f⃗
i=1
From Lemma 13.1, we can see that ⃗g ∈ L2 ((a, b), H −1 (R3 )) + L1 ((a, b), L2 (R3 )).
We now consider the solutions ⃗z ∈ L2 ((a, b) × R3 ) of the linear heat equation
∂t ⃗z = ν∆⃗z − div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u) + ⃗g
(13.4)
408
The Navier–Stokes Problem in the 21st Century (2nd edition)
with ⃗g ∈ L2 ((a, b), H −1 (R3 )) + L1 ((a, b), L2 (R3 )). Using Lemma 13.1 again, we can
see that ∂t ⃗z ∈ L1 ((a, b), H −2 (R3 )), so that ⃗z ∈ C([a, b], H −2 (R3 )) and ⃗z(a, .) is well
defined. Moreover, we have the following results:
ˆ uniqueness: if ⃗z1 and ⃗z2 are two solutions in L2 ((a, b) × R3 ) of Equation (13.4)
and if ⃗z1 (a, .) = ⃗z2 (a, .), then ⃗z1 = ⃗z2 .
ˆ regularity: if ⃗z is a solution in L2 ((a, b) × R3 ) of Equation (13.4) and if ⃗z(a, .) ∈
L2 (R3 ), then ⃗z ∈ C([a, b], L2 (R3 )) ∩ L2 ((a, b), H 1 (R3 )).
This can be easily checked. First, we see that there exists a constant C0 such that, for
every t0 ∈ R and every δ ∈ (0, 1), the map
⃗h 7→
Z
t
Wν(t−s) ∗ ⃗h ds = Lt0 (⃗h)
t0
satisfies:
ˆ Lt0 is bounded from L2 ((t0 , t0 + δ), H −2 (R3 )) to L2 ((t0 , t0 + δ), L2 (R3 )) and
∥Lt0 (⃗h)∥L2 L2 ≤ C0 ∥⃗h∥L2 H −2
ˆ Lt0 is bounded from L1 ((t0 , t0 + δ), H −1 (R3 )) to L2 ((t0 , t0 + δ), L2 (R3 )) and
∥Lt0 (⃗h)∥L2 L2 ≤ C0 ∥⃗h∥L2 H −1
ˆ Lt0 is bounded from L2 ((t0 , t0 +δ), H −1 (R3 )) to C([t0 , t0 +δ], L2 (R3 ))∩L2 ((t0 , t0 +
δ), H 1 (R3 )) and
∥Lt0 (⃗h)∥L∞ L2 + ∥Lt0 (⃗h)∥L2 H 1 ≤ C0 ∥⃗h∥L2 H −1
ˆ Lt0 is bounded from L1 ((t0 , t0 + δ), L2 (R3 )) to C([t0 , t0 + δ], L2 (R3 )) ∩ L2 ((t0 , t0 +
δ), H 1 (R3 )) and
∥Lt0 (⃗h)∥L∞ L2 + ∥Lt0 (⃗h)∥L2 H 1 ≤ C0 ∥⃗h∥L1 L2
Second, 1Q ∈ X(0) ; hence, by compactness of [a, b], for every ϵ > 0, we may find a
η(ϵ) ∈ (0, 1) such that, for every t0 ∈ [a, b], ∥1[t0 ,t0 +η(ϵ)] (t)1Q (t, x)⃗u∥X < ϵ.
We now prove our claims on the solutions of (13.4):
ˆ uniqueness: if ⃗z1 = ⃗z2 on [a, t0 ] with t0 < b, we write on [t0 , t0 +β] with t0 +β ≤ b:
⃗z1 − ⃗z2 = ν∆(⃗z1 − ⃗z2 ) − div(1Q ⃗u ⊗ (⃗z1 − ⃗z2 ) − (⃗z1 − ⃗z2 ) ⊗ 1Q ⃗u).
Thus,
⃗z1 − ⃗z2 = −Lt0 (div(1Q ⃗u ⊗ (⃗z1 − ⃗z2 ) − (⃗z1 − ⃗z2 ) ⊗ 1Q ⃗u))
If β < 1, we get
∥⃗z1 − ⃗z2 ∥L2 ((t0 ,t0 +β),L2 ) ≤ C0 ∥1[t0 ,t0 +β] 1Q ⃗u∥X ∥⃗z1 − ⃗z2 ∥L2 ((t0 ,t0 +β),L2 )
Thus, if β = min(b − t0 , η( 2C1 0 )), we get ⃗z1 = ⃗z2 on [t0 , t0 + β]. Finally, we see
that ⃗z1 = ⃗z2 , by propagating the equality from [a, a + kη] to [a, a + (k + 1)η] for
k ≥ 0.
Partial Regularity Results for Weak Solutions
409
ˆ regularity: For t0 ∈ [a, b], ⃗z0 ∈ L2 and η = η( 4C1 0 ) we consider the equation on
(t0 , t0 + η) × R3
⃗ − Lt (div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u))
⃗z = Z
0
with
⃗ = Wν(t−t ) ∗ ⃗z0 +
Z
0
Z
t
Wν(t−s) ∗ 1[t0 ,t0 +η] (s)⃗g (s, .) ds.
t0
⃗ belongs to C([t0 , t0 + η], L2 (R3 )) ∩ L2 ((t0 , t0 + η), H 1 (R3 )). Moreover, ⃗z 7→
Z
Lt0 (div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u)) is bounded on C([t0 , t0 + η], L2 (R3 )) ∩ L2 ((t0 , t0 +
η), H 1 (R3 )) and
1
(∥⃗z∥L∞ L2 ∩L2 H 1 )
2
By Banach’s contraction principle, we can see that there exists one and only one
solution ⃗z on [t0 , t0 + η].
Now, starting from t0 = 0 and ⃗z0 = ⃗z(0, .), we construct our solution on [0, η],
then we reiterate the construction for t0 = η and ⃗z0 = ⃗z(η, .) and get a solution
on [η, 2η], and so on. Finally, we get a solution of (13.4) on the whole interval [a, b]
with ⃗z ∈ C([a, b], L2 (R3 )) ∩ L2 ((a, b), H 1 (R3 )). By uniqueness of the solutions in
L2 ((a, b), L2 ), we see that a solution ⃗z of (13.4) that belongs to L2 ((a, b), L2 ) and
satisfies ⃗z(a, .) ∈ L2 must belong to C([a, b], L2 (R3 )) ∩ L2 ((a, b), H 1 (R3 )).
∥Lt0 (div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u))∥L∞ L2 +∩L2 H 1 ≤
Second step: regularity estimates on the vorticity ω
⃗.
From our study of Equation (13.4), we have found that ϕ⃗
ω belongs to C([a, b], L2 (R3 ))∩
2
1
3
L ((a, b), H (R )). As ∆⃗u = − curl ω
⃗ , we find that for every a < c < b and 0 < ρ < r,
⃗u belongs to C([c, b], H 1 (B(x0 , ρ))) ∩ L2 ((c, b), H 2 (B(x0 , ρ))).
We write again
∂t ω
⃗ = ν∆⃗
ω + ⃗g
with
⃗g = curl f⃗ − div(⃗u ⊗ ω
⃗ −ω
⃗ ⊗ ⃗u).
We find that ⃗g is locally L2 H −1/2 : for all ϕ ∈ D((a, +∞) × B(x0 , r))),
1(a,b) ϕ⃗g ∈ L2 H −1/2 . This gives more local regularity on ω
⃗ and ⃗u: for all ϕ ∈
D((a, +∞) × B(x0 , r))), 1(a,b) ϕ⃗
ω ∈ C([a, b], H 1/2 (R3 ) ∩ L2 ((a, b), H 3/2 ) and 1(a,b) ϕ⃗u ∈
C([a, b], H 3/2 (R3 ) ∩ L2 ((a, b), H 5/2 ).
Those estimates on ω
⃗ and ⃗u give in turn more regularity on ⃗g : ⃗g is locally L2 L2 , so
∞ 2
that ⃗u is locally L H : the theorem is proved.
Of course, we find a proposition similar to Proposition 12.3:
Proposition 13.1.
Serrin’s theorem on interior regularity holds in the following cases:
• 1Q ⃗u ∈ Lpt Lqx with
[455, 459]
2
p
+
3
q
= 1 and 2 ≤ p < +∞ (this is the Struwe-Takahashi criterion
• more generally, 1Q ⃗u ∈ Lpt Ṁx2,q with
2
p
+
3
q
= 1 and 2 ≤ p < +∞
• 1Q ⃗u ∈ C([a, b], L3 )
• 1Q ⃗u ∈ C([a, b], , V01 ), where V01 is the closure of L3 in M(Ḣ 1 7→ L2 )
410
The Navier–Stokes Problem in the 21st Century (2nd edition)
13.3
O’Leary’s Theorem on Interior Regularity
For α > 0, let Xα be the space of homogenous type (R × R3 , δα , µ), where δα is the
parabolic (quasi)-distance
δα ((t, x), (s, y)) = |t − s|1/α + |x − y|
(13.5)
and µ is the Lebesgue measure dµ = dt dx. Then the homogeneous dimension Q of Xα
is equal to α + 3. Recall that we defined Morrey spaces Mp,q
= Ṁ p,q (Xα ) on Xα for
α
p
p,q
< +∞, where
1 < p < q < +∞ by u ∈ Mα if and only if u is locally Lt,x and ∥u∥Mp,q
α
∥u∥Mp,q
=
α
1
sup
x0 ∈R3 ,t0 ∈R,R>0
R(3+α)(1−p/q)
!1/p
ZZ
p
|u(t, x)| dt dx
.
|t−t0 |<Rα ,|x−x0 |<R
In particular, the space Mp,5
2 , 2 < p ≤ 5, was discussed on page 98 in Chapter 5.
O’Leary [379] gave the following variant of Serrin’s theorem:
Theorem 13.3.
Let Q = I × Ω, where I = (a, b) and Ω = B(x0 , r). Let ⃗u ∈ L∞ (I, L2 (Ω)) ∩ L2 (I, H 1 (Ω)),
f⃗ ∈ L2 (I, H 1 (Ω)) and p ∈ D′ (Q), and assume that ⃗u is a weak solution on Q of the Navier–
Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
3
Then, if moreover 1Q ⃗u ∈ Mp,q
2 (R × R ) for some q > 5 and some 2 < p ≤ q, we have, for
∞
every a < c < b and 0 < ρ < r, ⃗u ∈ L ((c, b) × B(x0 , ρ)).
Proof. With no loss of generality, we may assume, as Q is bounded, that 5 < q ≤ 6. We
write λ = 1 − q−5
5q ∈ (0, 1). We are going to show that, for a < c < b and 0 < ρ < r and
Q0 = (c, b) × B(x0 , ρ), we have 1Q0 ⃗u ∈ Mp20 ,q with p0 = min( λp , q).
r+ρ
One more time, we pick up functions ϕ, ψ ∈ D(R×R3 ) supported in ( a+c
2 , b)×B(x0 , 2 ))
and such that ψϕ = ϕ and such that ϕ(t, x) = 1 on (c, b) × B(x0 , ρ)), and we write, for
⃗ = ϕ⃗u,
U
!!
3
X
1
1
⃗ = ψ( ∆(ϕ⃗u)) = ψ
∂i ((∂i ϕ)⃗u)
.
U
ϕ∆⃗u − (∆ϕ)⃗u + 2
∆
∆
i=1
⃗ = ψ( 1 (ϕ∆⃗u)), we find that
If V
∆
⃗ −V
⃗ ) ∈ L∞ L6
1a<t<b (U
t
and thus
⃗ −V
⃗ ) ∈ Lq = Mq,q ⊂ Mp0 ,q .
1Q (U
2
2
⃗ = −ψ( 1 (ϕ curl(ψ⃗
⃗ = ψ⃗
Now, we write V
ω )), where ω
⃗ = curl ⃗u. Let W
ω . We have
∆
3
X
⃗ = (∂t ψ)⃗
∂t W
ω + νψ∆(⃗
ω ) − ψ curl(
∂i (ui ⃗u)) + ψ curl f⃗,
i=1
which we rewrite as
3
3
X
X
⃗ = ν∆W
⃗ + (∂t ψ)⃗
∂t W
ω − ψ curl(
∂i (ui ⃗u)) + ψ curl f⃗ + ν(∆ψ)⃗
ω − 2ν
∂i ((∂i ψ)⃗
ω ).
i=1
i=1
Partial Regularity Results for Weak Solutions
411
This gives
⃗ ) = ν∆(curl W
⃗ )+
∂t (curl W
7
X
⃗k
R
k=1
with
⃗ 1 = curl(ψ curl f⃗), R
⃗ 2 = curl((∂t ψ + ν∆ψ)⃗
R
ω)
3
3
X
X
⃗ 3 = −2ν curl(
⃗ 4 = curl(
⃗ ∧ (ui ⃗u)))
R
∂i ((∂i ψ)⃗
ω )), R
∂i (∇ψ
i=1
i=1
3
6
X
X
⃗ 5 = − curl(curl( (∂i ψ)ui ⃗u)). R
⃗ 6 = curl( (∇∂
⃗ i ψ) ∧ (ui ⃗u))
R
i=1
i=1
3
X
⃗ 7 = − curl(curl(
R
∂i (ψui ⃗u))).
i=1
Thus, we have
⃗ =
W
7
X
⃗i =
S
i=1
7 Z
X
i=1
t
⃗ i (s, .) ds.
Wν(t−s) ∗ R
0
⃗ 1, R
⃗ 2 and R
⃗ 3 belong to L2 H −2 , we have that ϕS
⃗ 1 , ϕS
⃗2 , and ϕS
⃗3 belong to L∞ H −1 ,
As R
∞ −1
∞ 1
and even to L Ḣ (as pointwise multiplication by ψ maps L H to L∞ Ḣ −1 ) and we get
1
⃗1 ), ψ 1 (ϕS
⃗2 ), and ψ 1 (ϕS
⃗3 ) belong to L∞ L6 , and, being supported in a
finally that ψ ∆
(ϕS
∆
∆
q
q,q
3
compact subset of R × R , to Lt,x = M2 ⊂ Mp20 ,q .
1
⃗i ), for i = 4, . . . , 7. For i = 4, . . . , 6,
We must now estimate the non-linear terms ψ ∆
(ϕS
⃗i = curl T⃗i and ϕS
⃗i = curl(ϕT⃗i ) − ∇ϕ
⃗ ∧ T⃗i . The estimations will be done with the
we write S
Riesz potentials Iβ,α on Xα (0 < β < 3 + α) defined by
ZZ
1
f (s, y) ds dy
Iβ,α f (t, x) =
1/α
+ |x − y|)3+α−β
R×R3 (|t − s|
We have |T⃗4 | ≤ CQ I1,2 (1Q |⃗u|2 ), |T⃗5 | ≤ CQ I1,2 (1Q |⃗u|2 ) and |T⃗6 | ≤ CQ I2,2 (1Q |⃗u|2 ).
Using Adams’s inequality (see the Corollary 5.1 of Adams–Hedberg’s inequality), we
get:
p/2,q/2
ˆ I1,2 maps M2
to Mp21 ,q1 with
1
q1
=
2
q
−
1
5
p/2,q/2
ˆ Hence, as Q is bounded, 1Q I1,2 maps M2
ˆ Let r < 5/2 close enough to 5/2 to get that
Mp22 ,r2
with
1
r2
=
1
r
−
2
5
=
1
rµ
with µ = 1 −
= 1q λ and
1
p1
= p1 λ
to Mp20 ,q
min(p/2,r)
> q.
1− 2r
5
2r
1
5 < 1 and p2
min(p/2,r),r
I2,2 maps M2
=
1
min(p/2,r) µ
p/2,q/2
ˆ Hence, as Q is bounded and q > 5, f 7→ I2,2 (1Q f ) maps M2
p/2,q/2
p0 ,q
f 7→ 1Q I2,2 (1Q f ) maps M2
to Mq,q
.
2 ⊂ M2
<
to
1
q.
to Mp22 ,r2 and
1
Moreover L1 : f 7→ ψ ∂∆k (ϕf ) and L2 : f 7→ ψ ∆
((∂k ϕ)f ) are bounded on Mp20 ,q , as
Z
|Li f (t, x)| ≤ A(x − y)|f (t, y)| dy
with A ∈ L1 (R3 ); as the norm of Mp20 ,q is invariant by translation, we have ∥A ∗ f ∥Mp20 ,q ≤
⃗4 ), ψ 1 (ϕS
⃗5 ), and ψ 1 (ϕS
⃗6 ) belong to Mp0 ,q .
∥A∥1 ∥f ∥ p0 ,q . Thus, we find that ψ 1 (ϕS
M2
∆
∆
∆
2
412
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗7 = ∆T⃗7 with
Finally, we write S
T⃗7 = −
3 Z
X
i=1
t
0
1
curl(curl(∂i (Wν(t−s) ∗ (ψui ⃗u)))) ds)
∆
p/λ,q/λ
We have |T⃗7 | ≤ CQ I1,2 (1Q |⃗u|2 ), so that T⃗7 ∈ M2
. Moreover, we have
ψ
3
X
1
∂i
1
(ϕ∆T⃗7 ) = ϕT⃗7 + ψ ((∆ϕ)T⃗7 ) − 2
ψ ((∂i ϕ)T⃗7 )
∆
∆
∆
i=1
p/λ,q/λ
1
⃗7 ) ∈ M
(ϕS
, and, as it is compactly supported, it belongs to Mp20 ,q .
and finally ψ ∆
2
min(p/λ,q),q
Thus, we have seen that 1Q0 ⃗u belongs to M2
. As limn→+∞ p/λn = +∞, we
see that we may reiterate the proof in finitely many steps on some smaller cylinders and
q q
get 1Q0 ⃗u ∈ Mq,q
2 = Lt Lx . Thus, we may apply Serrin’s theorem on interior regularity.
13.4
Further Results on Parabolic Morrey Spaces
While O’Leary’s results deal with the condition ⃗u ∈ Mp,q
2 , 2 < p ≤ q and q > 5, which
is subcritical with respect to the natural scaling of the Navier–Stokes equations, the case of
critical scaling has been dealt with by Chen and Price [118], when 1Q ⃗u is small enough in
7
3
Mp,5
2 (R × R ) for some 2 < p ≤ 5; as we shall see, the result is true when 2 < p ≤ 5, and
3
even when the parabolic Morrey space Mp,5
2 (R × R ) is replaced with the multiplier space
1/2
V 2,1 = M(L2t Ḣx1 ∩ L2x Ḣt 7→ L2t L2x ) described in Chapter 5. (Recall that for 2 < p ≤ 5, we
p,5
5
1,2
have Lt,x ⊂ M2 ⊂ V (R × R3 ) ⊂ M2,5
2 ).
Parabolic multipliers and interior regularity
Theorem 13.4.
Let Q = I ×Ω, where I = (a, b) and Ω = B(x0 , r). Let ⃗u ∈ L∞ (I, L2 (Ω))∩L2 (I, H 1 (Ω)),
f⃗ ∈ L2 (I, H 1 (Ω)) and p ∈ D′ (Q), and assume that ⃗u is a weak solution on Q of the
Navier–Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
1/2
Then, if moreover 1Q ⃗u ∈ V 2,1 (R × R3 ) = M(L2t Ḣx1 ∩ L2x Ḣt →
7 L2t L2x ) and ∥1Q ⃗u∥V 2,1
is small enough, we have, for every a < c < b and 0 < ρ < r, ⃗u ∈ L∞ ((c, b) × B(x0 , ρ)).
Proof. This theorem is a direct generalization of Theorem 13.2, and its proof is quite similar:
First step: uniqueness for a linear heat equation.
Let ω
⃗ = curl ⃗u. We consider a function ϕ ∈ D(R × R3 ) which is equal to 1 on [c, b] ×
r+ρ
⃗ = ϕ⃗
ω . We have:
B(x0 , ρ) and is supported in [ a+c
2 , b + 1] × B(x0 , 2 ). We define w
ˆ w
⃗ ∈ L2 ((a, b) × R3 )
ˆ w(t,
⃗ .) = 0 for a < t <
b+c
2
Partial Regularity Results for Weak Solutions
413
ˆ ∂t w
⃗ = ν∆w
⃗ − div(1Q ⃗u ⊗ w
⃗ −w
⃗ ⊗ 1Q ⃗u) + ⃗g with
⃗ u − (⃗u · ∇ϕ)⃗
⃗ ω − (∆ϕ)⃗
⃗g = (⃗
ω · ∇ϕ)⃗
ω−2
3
X
(∂i ϕ) ∂i ω
⃗ + ∂t ϕ⃗
ω + ϕ curl f⃗
i=1
We now prove uniqueness of the solutions ⃗z ∈ L2 ((a, b) × R3 ) of this linear heat
equation
∂t ⃗z = ν∆⃗z − div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u) + ⃗g
(13.6)
Let ⃗z1 and ⃗z2 be two solutions in L2 ((a, b) × R3 ) of equation (13.6) with ⃗z1 (a, .) =
⃗z2 (a, .). We write:
⃗z1 − ⃗z2 = ν∆(⃗z1 − ⃗z2 ) − div(1Q ⃗u ⊗ (⃗z1 − ⃗z2 ) − (⃗z1 − ⃗z2 ) ⊗ 1Q ⃗u).
Thus,
⃗z1 − ⃗z2 = −L(div(1Q ⃗u ⊗ (⃗z1 − ⃗z2 ) − (⃗z1 − ⃗z2 ) ⊗ 1Q ⃗u))
where L is defined by
L(⃗h) =
Z
t
Wν(t−s) ∗ ⃗h ds.
a
We have seen in Chapter 5 that
|L(div H)| ≤ C0 I1 (|H|)
where I1 is the parabolic Riesz potential
ZZ
1
dy ds.
I1 f =
f (s, y)
(|t − s|1/2 + |x − y|)4
Let
p
p−1
< r ≤ 2; we have L2t,x = M2,2
⊂ Mr,2
2
2 ; pointwise multiplication with a
rp
rp
, 10
, 10
7rp
,2
p+r 7
, while I1 maps M2p+r 7 to M25(p+r) ; if
function in Mp,5
maps Mr,2
2
2 into M2
p
2p > 5r (such a choice of r is possible when 2p > 5 p−1
, i.e. p > 7/2), we find
∥⃗z1 − ⃗z2 ∥Mr,2 ≤ C1 ∥⃗z1 − ⃗z2 ∥Mr,2 ∥1Q ⃗u∥Mp,5 .
2
2
2
1
z1 −⃗z2 = 0, hence uniqueness
C1 ), we have ⃗
2 1/2
2 1
= Lt Ḣx ∩ Lx Ḣt . We know that I1 maps
1,2
If ∥1Q ⃗u∥Mp,5 is small enough (∥1Q ⃗u∥Mp,5 <
2
2
for the solutions of equation (13.6). Let W
L2t,x to W , hence maps W ′ to L2 by transposition; similarly, we know that for V ∈ V ,
pointwise multiplication with V maps W to L2 , hence by transposition maps L2 to
W ′ . This gives:
∥⃗z1 − ⃗z2 ∥L2 ((a,b),L2 (R3 )) ≤2C0 ∥I1 (1Q |⃗u||⃗z1 − ⃗z2 |)∥L2 L2
≤C1 ∥1Q |⃗u||⃗z1 − ⃗z2 |∥W ′
≤C2 ∥1Q ⃗u∥V 1,2 ∥⃗z1 − ⃗z2 ∥L2 ((a,b),L2 (R3 ))
Thus, when ∥1Q ⃗u∥V 1,2 is small enough (∥1Q ⃗u∥V 1,2 <
uniqueness for the solutions of Equation (13.6).
1
C2 ),
we have ⃗z1 − ⃗z2 = 0, hence
414
The Navier–Stokes Problem in the 21st Century (2nd edition)
Second step: regular solutions for the linear heat equation.
We consider the equation on (a, b) × R3
⃗ − L(div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u))
⃗z = Z
with
⃗=
Z
Z
t
Wν(t−s) ∗ ⃗g (s, .) ds.
a
We are going to search solutions extended to R × R3 by extending ⃗g to 0 outside from
(a, b) and defining L as
Z t
L(⃗h) =
Wν(t−s) ∗ ⃗h ds.
−∞
If ⃗h is equal to 0 on (−∞, a), we find that on (a, b) the new definition of L(⃗h) coincides
with the old one.
We have ⃗g = ⃗g1 + ⃗g2 , with
!
3
X
⃗
⃗g1 = 1(a,b) (t) −(∆ϕ)⃗
ω−2
(∂i ϕ) ∂i ω
⃗ + ∂t ϕ⃗
ω + ϕ curl f ∈ L2 (R, H −1 (R3 ))
i=1
⃗ u − 1Q (⃗u · ∇ϕ)⃗
⃗ ω ∈ W ′.
and ⃗g2 = 1Q (⃗
ω · ∇ϕ)⃗
Moreover, we take a function ψ ∈ D(R) such that ψ = 1 on (a, b), and we study the
solutions of
⃗ − L(div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u))
⃗z = Z
(13.7)
with
⃗ = ψ(t)
Z
Z
t
Z
t
⃗1 + Z
⃗2.
Wν(t−s) ∗ ⃗g2 (s, .) ds = Z
Wν(t−s) ∗ ⃗g1 (s, .) ds +
−∞
−∞
Then, we have:
2
⃗ 1 ∈ L∞
ˆ since ⃗g1 ∈ L2 (H −1 (R3 )) and is equal to 0 for t < a, we know that Z
t L ∩
L2 H 1 moreover, we have
Z t
⃗
⃗
∂t Z1 = ν∆Z1 + ⃗g1 + ∂t ψ
Wν(t−s) ∗ ⃗gi (s, .) ds ∈ L2t H −1 .
−∞
⃗1 ∈ W .
This gives Z
ˆ Let us remark that, for a non-negative function f , we have
I2 f ≤ CI1 (I1 f ).
It is equivalent to prove that
ZZ
1
1
dy ds
≤
C
= A(t, x).
1/2
3
1/2
4
1/2
(|t| + |x|)
(|t − s| + |x − y|) (|s| + |y|)4
If |t|1/2 ≤ |x|, we write (|t|1/21+|x|)3 ≤
ZZ
A(t, x) ≥
|x|
|y|<|s|1/2 < 2
If |t|1/2 ≥ |x|, we write
ZZ
A(t, x) ≥
1
(|t|1/2 +|x|)3
|t|
|y|<|s|1/2 <
≤
1/2
2
1
|x|3
and
1
1
c
dy ds =
.
4
1/2
4
(3|x|) (2|s| )
|x|3
1
|t|3/2
and
1
1
c
dy ds = 3/2 .
(3|t|1/2 )4 (2|s|1/2 )4
|t|
Partial Regularity Results for Weak Solutions
ˆ We have
⃗2| ≤ C
|Z
Z
(|t −
s|1/2
415
1
|⃗
ω |1Q |⃗u| ds dy
+ |x − y|)3
so that
⃗ 2 | ≤ CI2 (|⃗
|Z
ω |1Q |⃗u|) ≤ C ′ I1 (I1 (|⃗
ω |1Q |⃗u|)).
We have 1Q ω
⃗ ∈ L2t,x and 1Q ⃗u ∈ V 1,2 , so that 1Q |⃗u||⃗
ω | ∈ W ′ ; as I1 maps W ′ to
⃗2 ∈ W .
L2t,x and L2t,x to W , we find that Z
Now, if ⃗z ∈ W , we have
∥L(div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u))∥W ≤ C∥1Q ⃗u ⊗ ⃗z∥L2 L2 ≤ C0 ∥1Q ⃗u∥V 1,2 ∥⃗z∥W .
If ∥1Q ⃗u∥V 1,2 is small enough (∥1Q ⃗u∥V 1,2 < C10 ), the Banach contraction principle gives
us the existence and uniqueness of solutions ⃗z ∈ W of Equation (13.7).
Third step: regularity estimates on the vorticity ω
⃗.
Recall that w
⃗ = ϕ⃗
ω is a solution on (a, b) × R3 of Equation (13.6). We know that there
⃗ . Then W
⃗ is another
exists a solution ⃗z in W of Equation (13.7). Let 1(a,b) (t) ⃗z = W
⃗ belongs to L2 ((a, b), L2 ):
solution on (a, b) × R3 of Equation (13.6). Moreover, W
ˆ We have the Sobolev embedding W ⊂ L2t Ḣx1 ⊂ L2t L6x , so that 1(a,b) (t)1B(x0 ,3r) ⃗z ∈
L2t L2x .
ˆ As ⃗z is computed through a Picard iteration, we find that ⃗z = 0 for t < a. We
have
1|x−x0 |≥3r (x)|L(div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u)| ≤
Z tZ
1
C
1 (y)|⃗u(s, y)||⃗z(s, y)| dy ds
4 Q
a
|−y|>2r |x − y|
As 1Q (y)|⃗u||⃗z| ∈ L2 L2 , we see that
1(a,b) (t)1|x−x0 |≥3r (x)L(div(1Q ⃗u ⊗ ⃗z − ⃗z ⊗ 1Q ⃗u) ∈ L2t L2x .
⃗ 1 ∈ L∞ L2 , so that 1(a,b) (t)1|x−x |≥3r Z
⃗ 1 ∈ L2 L2 .
ˆ We already know that Z
0
ˆ We have ⃗u ∈ L∞ L2 and ω
⃗ ∈ L2 L2 , thus ⃗g2 ∈ L2 L1 . As we have
Z tZ
1
⃗
1|x−x0 |≥3r (x)|Z2 | ≤C
|⃗g2 (s, y)| dy ds
|x
−
y|3
a
|−y|>2r
⃗ 2 ∈ L2 L2 .
we find that 1(a,b) (t)1|x−x0 |≥3r Z
⃗ ∈ L2 L2 , and by uniqueness of solutions to Equation (13.6), we have w
⃗.
Thus, W
⃗ =W
2
2
2 ∞
In particular, ϕ⃗u ∈ L ((a, b), H ) ⊂ L L , and we may then finish the proof by
applying Serrin’s theorem on interior regularity.
416
The Navier–Stokes Problem in the 21st Century (2nd edition)
13.5
Hausdorff Measures
In the following sections, we shall recall the proofs that the set of singular points of a
Leray solution is small, this smallness will be expressed in terms of Hausdorff dimensions.
Let (X, δ, µ) be a space of homogeneous type and Q its homogeneous dimension (see
Definition 5.1). In particular:
ˆ there is a positive constant κ such that:
for all x, y, z ∈ X, δ(x, y) ≤ κ(δ(x, z) + δ(z, y))
ˆ there exists postive numbers 0 < A0 ≤ A1 which satisfy:
Z
Q
for all x ∈ X, for all r > 0, A0 r ≤
dµ(y) ≤ A1 rQ
δ(x,y)<r
A basic useful property of spaces of homogeneous type is the Vitali covering lemma
[215, 313].
Proposition 13.2 (The Vitali covering lemma).
Let E ⊂ X be decomposed as a union of balls E = ∪α∈A B(xα , rα ), where (B(xα , rα ))α∈A
is a family of balls so that supα rα < ∞. Then there exists a (countable) subfamily of
balls (B(xα , rα ))α∈B (B ⊂ A) so that α =
̸ β ⇒ B(xα , rα ) ∩ B(xβ , rβ ) = ∅ and so that
E ⊂ ∪α∈B B(xα , 5κ2 rα ).
We may now introduce the Hausdorff measures on X.
Definition 13.2 (Hausdorff measure).
Let (X, δ, µ) be a separable space of homogeneous type (see Definition 5.1).
(i) For a sequence of open balls P
B = (B(xi , ri ))i∈N of X and for α > 0, we define
r(B) = supi∈N ri and σα (B) = i∈N riα .
(ii) The Hausdorff measure Hα on X is defined for a Borel subset B ⊂ X by
Hα (B) = lim min{σα (B) / B = (B(xi , ri ))i∈N , B ⊂ ∪i∈N B(xi , ri ), r(B) < ϵ}
ϵ→0
We have obviously, if α < β, σβ (B) ≤ r(B)β−α σα (B); thus,
ˆ Hα (B) < +∞ ⇒ Hβ (B) = 0
ˆ Hβ (B) > 0 ⇒ Hα (B) = +∞
Moreover, we have:
α > Q ⇒ Hα (B) = 0.
Indeed, if B = B(x0 , r), we use the Vitali lemma on the collection (B(x, ϵ))x∈B to exhibit a
family of disjoint balls (B(xi , ϵ)) such that B ⊂ ∪i B(xi , 5κ2 ϵ). Let Bϵ = (B(xi , 5κ2 ϵ)). We
have:
X
A1
A1
σQ (Bϵ ) ≤
(5κ2 )Q
µ(B(xi , ϵ) ≤
(5κ2 )Q A1 (κ(r + ϵ))Q
A0
A
0
i
and thus
HQ (B(x0 , r)) ≤
A21
(5rκ3 )Q < +∞.
A0
Partial Regularity Results for Weak Solutions
417
If α > Q and B ⊂ X, we write, for a x0 ∈ X,
Hα (B) ≤ Hα (X)) ≤
+∞
X
Hα (B(x0 , N )) = 0.
N =1
Definition 13.3 (Hausdorff dimension).
The Hausdorff dimension dH (B) of a Borel subset of X is defined as
dH (B) = inf{α > 0 / Hα (B) = 0}.
If dH (B) > 0, it may be defined as well as
dH (B) = sup{α > 0 / Hα (B) = +∞}.
13.6
Singular Times
A classical result (which goes back to the description of the structure of turbulent
solutions by Leray [328]) states that the set of singular times for a Leray solution is very
small.
We consider the Navier–Stokes problem
∂t ⃗u = ν∆⃗u + P div(F div(⃗u ⊗ ⃗u)),
⃗u(0, .) = ⃗u0
(13.8)
where ⃗u0 ∈ L2 with div ⃗u = 0 and the tensor F is smooth on (0, +∞) × R3 :
F ∈ ∩k∈N H k ((0, +∞) × R3 ) = H ∞ ((0, +∞) × R3 )
(13.9)
We have seen in Proposition 12.1 that the solution ⃗u constructed in Theorem 12.2
satisfies the strong Leray energy inequality: for almost every t0 in (0, T ) and for every
t ∈ (t0 , T ), we have
Z t
Z t
2
2
2
∥⃗u(t, .)∥2 + 2ν
∥⃗u∥Ḣ 1 ds ≤ ∥⃗u(t0 )∥2 + 2
(13.10)
⟨div F|⃗u⟩H −1 ,H 1 ds
t0
t0
Singular times
Theorem 13.5.
Let ⃗u0 ∈ L2 with div ⃗u = 0 and F ∈ H ∞ ((0, +∞) × R3 ). Let ⃗u be a weak Leray solution
of the Navier–Stokes Equations (13.8) on (0, ∞) × R3 which satisfies the strong energy
inequality. Then there is compact set Σt ⊂ [0, ∞) so that:
(i) ⃗u is smooth outside from Σt × R3
(ii) H1/2 (Σt ) = 0 (where H1/2 is the Hausdorff measure on R).
Proof. Let t0 be a Lebesgue point of the map t 7→ ∥⃗u(t, .)∥22 such that ⃗u(t0 , .) ∈ H 1 . From
Theorem 7.1, we know that there exists a t1 > t0 a local solution ⃗v on (t0 , t1 ) of the
Navier–Stokes problem with initial value ⃗u(t0 , .) at t = t0 such that ⃗v ∈ C([t0 , t1 ], (H 1 )3 ) ∩
418
The Navier–Stokes Problem in the 21st Century (2nd edition)
L2 ((t0 , t1 ), (H 2 )3 ). If t2 is the maximal existence time of this solution (for every T < t2 ,
RT
⃗v ∈ C([t0 , T ], (H 1 )3 ) ∩ L2 ((t0 , T ), (H 2 )3 )) and T < +∞, then t0 ∥⃗v (s, .)∥2Ḣ 3/2 ds = +∞.
Moreover, by induction on k, we see that for every k ∈ N and t1 ∈ (t0 , t2 ), ⃗v (t1 , .) ∈ H k and
thus (from Theorem 7.3) ⃗v ∈ C([t0 , T ], (H k )3 ) ∩ L2 ((t0 , T ), (H k+1 )3 ) for every T ∈ (t1 , t2 ).
⃗ = P(div(F − ⃗u ⊗
Moreover, by weak-strong uniqueness, we have ⃗u = ⃗v on [t0 , t2 ]. As ∇p
⃗u)), we find by induction on k that for every k ∈ N and every m ∈ N, and for every t0 <
t1 < T < t2 , ∂tk ⃗u ∈ L2 ((t1 , T ), H m ); thus, for every t0 < t1 < T < t2 , ⃗u ∈ H ∞ ((t1 , T ) × R3 ),
and ⃗u is smooth on (t0 , t2 ) × R3 .
Let I be the collection of open intervals I ⊂ (0, +∞) such that ⃗u ∈ H ∞ (J × R3 ), O =
∪I∈I I and Σt = [0, +∞) \ O. By construction Σt is a closed subset of [0, +∞) and ⃗u is
smooth outside Σt × R3 .
In order to check that Σt is compact, it is enough to show that it is bounded. Let us
recall what we proved in RTheorem 7.2 : there exists a positive constant ϵ0 , such that, if
+∞
∥⃗u(t0 , .)∥Ḣ 1/2 < ϵ0 ν and t0 ∥F(s, .)∥2 1 ds < ϵ20 ν 3 , then there is a global solution ⃗v on
Ḣ 2
(t0 , +∞) of the Navier–Stokes problem with initial value ⃗u(t0 , .) at t = t0 such that ⃗v ∈
C([t0 , +∞], (H 1 )3 ) ∩ L2 ((t0 , +∞), (Ḣ 2 )3 ). If moreover t0 is a Lebesgue point of the map t 7→
∥⃗u(t, .)∥22 , then ⃗u = ⃗v on [ts0 , +∞) (by weak–strong uniqueness). As F ∈ L2 ((0, +∞), H 1/2 ),
R +∞
there exists a time T such that T ∥F(s, .)∥2 1 ds < ϵ20 ν 3 . As ⃗u ∈ L∞ L2 ∩L2 Ḣ 1 ⊂ L4 Ḣ 1/2 ,
Ḣ 2
we find that the measure of the set of points t such that ∥⃗u(t, .)∥Ḣ 1/2 ≥ ϵ0 ν is finite; as
almost every time is a Lebesgue point of the map t 7→ ∥⃗u(t, .)∥22 , we find that there exists a
time t0 > T from which ⃗u will belong to C([t0 , +∞], (H 1 )3 ) ∩ L2 ((t0 , +∞), (Ḣ 2 )3 ). We may
conclude that Σt ⊂ [0, t0 ], and thus Σt is compact.
Now, we are going to estimate the Hausdorff dimension of Σt . Let τ ∈ Σt and let s < τ
be a Lebesgue point of the map t 7→ ∥⃗u(t, .)∥22 such that ⃗u(s, .) ∈ H 1 . By Theorem 7.1, we
may find a local solution ⃗v on (s, s + T ) of the Navier–Stokes problem with initial value
⃗u(s, .) at t = s such that ⃗v ∈ C([s, s + T ], (H 1 )3 ) ∩ L2 ((s, s + T ), (H 2 )3 ), where the existence
time is given by inequality 7.17:
T = min(1, Cν
1
).
(∥⃗u(s, .)∥H 1 + ∥F|L2 ((s,s+1),H 1 )4
Since ⃗u is a Leray solution on (s, s + T ), we find that ⃗u = ⃗v on (s, s + T ) (due to weakstrong uniqueness). Thus, ⃗u is smooth on (s, s + T ) × R3 and s + T < τ . As T ≥ Cν,F (1 +
1/4
∥⃗u(s, .)∥H 1 )−4 , we find that 1 + ∥⃗u(s, .)∥H 1 ≥ Cν,F (τ − s)−1/4 .
Let ϵ > 0 with ϵ < 16 Cν,F . We write
1
1
Σt ⊂ [0, ϵ) ∪ ∪τ ∈Σt ,τ ≥ϵ (τ − ϵ, τ + ϵ).
5
5
By the Vitali covering lemma, we may find N ∈ N and τ1 , . . . , τN ∈ Σt ∩ [ϵ, ∞) so that
2
Σt ⊂ [0, ϵ) ∪N
i=1 (τi − ϵ, τi + ϵ) while min1≤i<j≤N |τi − τj | ≥ 2ϵ/5. On (τi − 5 ϵ, τi ), we have
∥⃗u(s, .)∥H 1 ≥
and thus
Z
(τi − 25 ϵ,τi )
1 1/4
C (τ − s)−1/4
2 ν,F
∥⃗u(s, .)∥2H 1 ds ≥
1 1/2 2 1/2
C ( ϵ)
2 ν,F 5
Partial Regularity Results for Weak Solutions
419
Let B = ((τi − ϵ, τi + ϵ))0≤i≤N with τ0 = 0. We have
σ1/2 (B) = (N + 1)ϵ1/2
≤ϵ1/2 +
N
X
√
−1/2
10Cν,F
i=1
1/2
≤ϵ
+
√
−1/2
10Cν,F
Z
τi
τi −2ϵ/5
∥⃗u∥2H 1 ds
Z
∥⃗u∥2H 1 ds.
d(s,Σt )≤2ϵ/5
√
−1/2 R
∥⃗u∥2H 1 ds. In
Since we know that ⃗u ∈ (L2 H 1 )3 , we find that H1/2 (Σt ) ≤ 10Cν,F
Σt
particular,
H1/2 (Σt ) < ∞; hence, the Lebesgue measure of Σt is equal to 0; this gives
R
2
∥⃗u∥H 1 ds = 0 and finally H1/2 (Σt ) = 0.
Σt
13.7
The Local Energy Inequality
Scheffer studied the partial regularity of the Leray weak solutions. More precisely, he
has been interested in the set Σ which is the complement in (0, +∞) × R3 of the set of
regular points of the solution ⃗u, i.e., of points (t, x) in the neighborhood of which ⃗u is a
continuous function of time and space variables. We have, of course, Σ ⊂ Σt × R3 .
In Scheffer [425], he considered a maximal interval of regularity I = (t0 , t1 ) such that
t0 , t1 ∈ Σt and I ∩ Σt = ∅ and he showed that H1 (Σ ∩ ({t1 } × R3 )) < ∞. In [426],
Scheffer then introduced the so-called local energy inequality and he was able to prove that
H2 (Σ) < ∞. This local energy inequality turned out to be a fundamental tool in the partial
regularity theory of Caffarelli, Kohn and Nirenberg [74], in Lemarié-Rieusset’s theory of
uniformly locally square integrable solutions [313] and in Jia and Šverak’s theory of selfsimilar solutions [245].
Local energy inequality
Theorem 13.6.
Let ⃗u0 ∈ L2 with div ⃗u0 = 0 and f⃗ ∈ L2 ((0, T ), H −1 ). The solution ⃗u of the Navier–
Stokes problem with initial value ⃗u0 and forcing term f⃗ constructed in Theorem 12.2
satisfies the local Leray energy inequality: there exists a non-negative locally finite measure µ on (0, T ) × R3 such that
⃗ ⊗ ⃗u|2 − div((|⃗u|2 + 2p)⃗u) + 2⃗u · f⃗ − µ
∂t |⃗u|2 = ν∆|⃗u|2 − 2ν|∇
(13.11)
Proof. Recall that ⃗u is constructed as the limit of ⃗u(ϵn ) , a sequence of solutions of the
mollified equation, such that:
ˆ on every bounded subinterval of [0, T ], ⃗u(ϵn ) is *-weakly convergent to ⃗u in L∞ L2 and
in L2 Ḣ 1
ˆ ⃗u(ϵn ) is strongly convergent to ⃗u in L2loc ((0, T ) × R3 ).
We write
⃗ ⃗u(ϵ) − ∇p
⃗ (ϵ)
∂t ⃗u(ϵ) = ν∆⃗u(ϵ) + f⃗ − (⃗u(ϵ) ∗ θϵ ) · ∇
420
The Navier–Stokes Problem in the 21st Century (2nd edition)
with
p(ϵ) =
1
⃗ ⃗u(ϵ )
div f⃗ − (⃗u(ϵ) ∗ θϵ ) · ∇
∆
and we write
∂t (
|⃗u(ϵ) |2
|⃗u(ϵ) |2
⃗ ⊗ ⃗u(ϵ) |2 + f⃗ · ⃗u(ϵ)
) = ν∆(
) − ν|∇
2
2
|⃗u(ϵ) |2
− div(
(⃗u(ϵ) ∗ θϵ )) − div(p(ϵ) ⃗u(ϵ) )
2
We know that ⃗u(ϵn ) converge strongly to ⃗u in L2loc ((0, T ) × R3 ); as the family is bounded
10/3
3/5
10/3
10/3
in Lt Hx ⊂ Lt Lx , we find that we have strong convergence in L3loc ((0, T ) × R3 ) as
well. Thus, we have the following convergence results in D′ ((0, T ) × R3 ): ∂t |⃗u(ϵn ) |2 → ∂t |⃗u|2 ,
∆|⃗u(ϵn ) |2 → ∆|⃗u|2 , div(|⃗u(ϵn ) |2 (⃗u(ϵn ) ∗ θϵ )) → div(|⃗u|2 ⃗u) and ⃗u(ϵn ) · f⃗ → ⃗u · f⃗. Similarly, we
have that


3 X
3
X
1
∂
∂
j
l
√
√
div 
div f⃗ +
(u(ϵn ),j (u(ϵn ,l) ∗ θϵn )) ⃗u(ϵn ) 
∆
−∆
−∆
j=1
l=1
1
converges in D′ to div ∆
div f +
Thus far, we have got that
P3
j=1
∂
à j öl
u
l=1 −∆ −∆ (uj ul ) ⃗
P3
.
∂t |⃗u|2 = ν∆|⃗u|2 − div((|⃗u|2 + 2p)⃗u) + 2⃗u · f⃗ − νT
with
⃗ ⊗ ⃗u(ϵ ) |2 .
T = lim 2|∇
n
ϵn →0
√
⃗ ⊗ ⃗u(ϵ ) is weakly convergent
Let ϕ ∈ D′ ((0, T ) × R3 ) be a non-negative function. As ϕ ∇
√
√
√n
2
2
2
⃗ ⊗ ⃗u in Lt Lx , we find that ∥ ϕ ∇
⃗ ⊗ ⃗u∥ ≤ lim inf ϵ →0 ∥ ϕ ∇
⃗ ⊗ ⃗u(ϵ ) ∥2 . Thus, we
to ϕ ∇
2
2
n
n
have
ZZ
⃗ ⊗ ⃗u(ϵ ) |2 ϕ(t, x) dt d
⟨T |ϕ⟩D′ ,D =2 lim
|∇
n
ϵn →0
ZZ
⃗ ⊗ ⃗u|2 ϕ(t, x) dt dx.
≥2
|∇
⃗ ⊗ ⃗u|2 + µ, where µ is a non-negative locally finite measure.
Thus, νT = 2ν|∇
A natural question is to find a criterion when we have indeed local energy equality. An
easy criterion is the following one:
Proposition 13.3.
Let Q = I × Ω, where I = (a, b) and Ω = B(x0 , r). Let ⃗u ∈ L∞ (I, L2 (Ω)) ∩ L2 (I, H 1 (Ω)),
f⃗ ∈ L2 (I, H −1 (Ω)) and p ∈ D′ (Q), and assume that ⃗u is a weak solution on Q of the
Navier–Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
If p ∈ L1t,x (Q), then
• for every 0 < ρ < r, p ∈ L1 ((a, b), L2 (B(x0 , ρ)))
div ⃗u = 0.
Partial Regularity Results for Weak Solutions
421
• the quantity
⃗ ⊗ ⃗u|2 − div((|⃗u|2 + 2p)⃗u) + 2⃗u · f⃗
µ = −∂t |⃗u|2 + ν∆|⃗u|2 − 2ν|∇
is well defined in D′ (Q)
• if moreover ⃗u ∈ L4t,x (Q), then µ = 0
Proof. We first check the regularity of p. We take the divergence of the Navier–Stokes
equations and get that, on Q, we have
∆p = −
3 X
3
X
∂i ∂j (ui uj ) + div f⃗
i=1 j=1
We then pick up a function ϕ ∈ D(R3 ) supported in B(x0 r+,ρ
2 ) such that ϕ(t, x) = 1 on
r+3ρ
B(x0 , 4 ).We write
∆(ϕp) = −ϕ
3 X
3
X
∂i ∂j (ui uj ) + ϕ div f⃗ − p∆ϕ + 2
i=1 j=1
3
X
∂i (p∂i ϕ).
i=1
We have:
P3 P3
ˆ R1 = −ϕ i=1 j=1 ∂i ∂j (ui uj ) ∈ L1 ((a, b), H −3/2 (R3 )) and is supported in the
1
fixed compact set B̄(x0 , r), hence R1 ∈ L1 ((a, b), Ḣ −2 + Ḣ −1 (R3 )) and ∆
R1 ∈
1
1
2
1
1 2
L ((a, b), L + H ) so that 1Q ∆ R1 ∈ L L
ˆ similarly, R2 = ϕ div f⃗ ∈ L1 ((a, b), H −2 (R3 )) and is supported in the fixed compact
1
set B̄(x0 , r), so that R2 ∈ L1 ((a, b), Ḣ −2 + Ḣ −1 (R3 )) and thus 1Q ∆
R2 ∈ L1 L2
1
1
ˆ R3 = −p∆ϕ ∈ L1 L1 , so that ∆
R3 ∈ L1 ((a, b), L3,∞ ) so that 1Q ∆
R3 ∈ L1 L2
P3
1
ˆ for estimating R4 = 2 i=1 ∂i (p∂i ϕ), we write p∂i ϕ ∈ L1 L1 so that ∆
R4 ∈
1
1
3/2,∞
1 6/5
L ((a, b), L
) so that 1Q ∆ R4 ∈ L L
Thus far, we have just obtained that ϕp ∈ L1 L6/5 . But then reiterating the argument on a
smaller ball, we find that, in estimating R4 , we may replace p∂i ϕ ∈ L1 L1 by p∂i ϕ ∈ L1 L6/5
1
and find that 1Q ∆
R4 ∈ L1 L2 .
Thus, we find that ϕp ∈ L1 L2 and thus that µ is well defined. Moreover, if we consider
a relatively compact open subset O = (c, d) × B(x, ρ) of Q and a mollifier θϵ with ϵ < r − ρ,
2
we may define on O ⃗uϵ = ⃗u ∗ θϵ ; we have on O ⃗uϵ ∈ L∞
uϵ ∈ L1t L2x , so that
t Lx and ∂t ⃗
2
⃗uϵ ∈ C([c, d], L ); using the density of smooth functions in {⃗v / ⃗uϵ ∈ C([c, d], L2 ) and ∂t⃗v ∈
L1 L2 }, we find that
∂t |⃗uϵ |2 = 2⃗uϵ ∂t ⃗uϵ
and thus
⃗ ⊗ ⃗uϵ |2 + 2⃗uϵ ∗ (θϵ ∗ f⃗)
∂t |⃗uϵ |2 =ν∆|⃗uϵ |2 − 2ν|∇
− 2 div((p ∗ θϵ )⃗uϵ ) − 2⃗uϵ .θϵ ∗ div(⃗u ⊗ ⃗u)
⃗ ⊗ ⃗uϵ to ∇
⃗ ⊗ ⃗u in L2 L2 (O), of
We have the strong convergence of ⃗uϵ to ⃗u in L2 L2 (O), of ∇
2 1
2 −1
1 2
⃗
⃗
⃗uϵ to ⃗u in L H (O), of θϵ ∗ f to f in L H (O), of p ∗ θϵ to p in L L (O) and the *-weak
convergence of ⃗uϵ to ⃗u in L∞ L2 so that we find
µ = − div(|⃗u|2 ⃗u) + 2 lim ⃗uϵ .θϵ ∗ div(⃗u ⊗ ⃗u)
ϵ→0+
(13.12)
Of course, when ⃗u ∈ L4 L4 , we find that θϵ ∗ div(⃗u ⊗ ⃗u) converges strongly to div(⃗u ⊗ ⃗u) in
L2 H −1 (O) so that µ = 0.
422
The Navier–Stokes Problem in the 21st Century (2nd edition)
Of course, if ⃗u satisfies the hypotheses of Theorem 13.2 on interior regularity and if
p ∈ L1 L1 , then we find µ = 0: we have ⃗u ∈ L∞ L2 ∩ L2 H 1 and ⃗u ∈ X, hence |⃗u|2 ∈ L2 L2 , so
that ⃗u ∈ L4 L4 . . .
However, one may find a weaker assumption on ⃗u that grants that µ = 0. This assumption has been described by Duchon and Robert [159], in a paper that generalizes the results
of Constantin, E and Titi [127] on Onsager’s conjecture [381]. This result of Duchon and
Robert underlines the link between a minimal regularity of ⃗u and the fact that µ = 0:
Energy equality
Theorem 13.7.
Let Q = I ×Ω, where I = (a, b) and Ω = B(x0 , r). Let ⃗u ∈ L∞ (I, L2 (Ω))∩L2 (I, H 1 (Ω)),
f⃗ ∈ L2 (I, H −1 (Ω)) and p ∈ L1t,x (Q), and assume that ⃗u is a weak solution on Q of the
Navier–Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Let µ be the distribution
⃗ ⊗ ⃗u|2 − div((|⃗u|2 + 2p)⃗u) + 2⃗u · f⃗.
µ = −∂t |⃗u|2 + ν∆|⃗u|2 − 2ν|∇
Let us define, for 0 < ρ < r,
A(ρ) = lim inf
+
ϵ→0
1
ϵ4
Z
a
b
Z
x∈B(x0 ,ρ)
Z
|⃗u(t, x) − ⃗u(t, x + y)|3 dt dx dy.
y∈B(0,ϵ)
If A(ρ) = 0 for every ρ ∈ (0, r), then µ = 0.
Proof. We start from Equality (13.12) expresssing µ as a limit:
µ = − div(|⃗u|2 ⃗u) + 2 lim+ ⃗uϵ .θϵ ∗ div(⃗u ⊗ ⃗u)
ϵ→0
We introduce the distribution
3 Z
X
Tϵ =
∂k θϵ (y)(uk (x − y) − uk (x))|⃗u(x − y) − ⃗u(x)|2 dy
k=1
which is well defined on (a, b) × B(x0 , ρ) for ϵ < r−ρ
u ∈ L∞ L2 ∩ L2 H 1 on Q, so that
2 , as ⃗
4 3
⃗u ∈ L L on (a, b) × B(x0 , (r + ρ)/2). R
P3
On (a, b) × B(x0 , ρ), we have k=1 ∂k θϵ (y)(uk (x − y) − uk (x)) dy = 0 (as div ⃗u = 0),
so that
3 Z
X
Tϵ =
∂k θϵ (y)(uk (x − y) − uk (x))(|⃗u(x − y)|2 − 2⃗u(x − y) · ⃗u(x)) dy
k=1
Moreover,
lim θϵ ∗ (|⃗u|2 ⃗u) − (θϵ ∗ |⃗u|2 )⃗u = 0 in L1t,x ((a, b) × B(x0 , ρ))
ϵ→0
so that
3 Z
X
∂k θϵ (y)(uk (x − y) − uk (x))|⃗u(x − y)|2 dy = div(θϵ ∗ (|⃗u|2 ⃗u) − (θϵ ∗ |⃗u|2 )⃗u) → 0
k=1
where the limit is taken in D′ ((a, b) × B(x0 , ρ)).
Partial Regularity Results for Weak Solutions
423
Similarly, we introduce the distribution
Sϵ =
3 Z
X
∂k θϵ (y)(uk (x − y) − uk (x))((⃗u(x − y) − ⃗u(x)).(⃗uϵ (x) − ⃗u(x)) dy
k=1
which is equal as well to
Sϵ =
3 Z
X
∂k θϵ (y)(uk (x − y) − uk (x))(⃗u(x − y).(⃗uϵ (x) − ⃗u(x)) dy
k=1
We thus have 2Sϵ − Tϵ = Aϵ + Bϵ + Cϵ with
Aϵ = 2
3 Z
X
∂k θϵ (y)uk (x − y)⃗u(x − y) · ⃗uϵ (x) dy = 2⃗uϵ .θϵ ∗ div(⃗u ⊗ ⃗u)
k=1
Bϵ = −2
3 Z
X
⃗ uϵ )
∂k θϵ (y)uk (x)⃗u(x − y) · ⃗uϵ (x) dy = − 2⃗uϵ .(⃗u · ∇⃗
k=1
= − div(|⃗uϵ |2 ⃗u) → − div(|⃗u|2 ⃗u)
and
lim Cϵ = 0 in D′ .
ϵ→0+
Thus, we find that
µ = lim 2Sϵ − Tϵ .
ϵ→0
This is the formula given by Duchon and Robert. We have
Z
1
|Tϵ (t, x)| ≤ C 4
|⃗u(t, x + y) − ⃗u(t, x)|3 dy.
ϵ |y|<ϵ
R
Similarly, writing ⃗uϵ (t, x) − ⃗u(t, x) = θϵ (y)(⃗u(t, x − y) − ⃗u(t, x)) dy, we get
Z
Z
1
|Sϵ (t, x)| ≤C 7 (
|⃗u(t, x + y) − ⃗u(t, x)|2 dy)(
|⃗u(t, x + y) − ⃗u(t, x)| dy)
ϵ
|y|<ϵ
|y|<ϵ
Z
′ 1
≤C 4
|⃗u(t, x + y) − ⃗u(t, x)|3 dy.
ϵ |y|<ϵ
RR
Thus, if A(ρ) = 0, we find that limϵ→0+ (a,b)×B(x0 ,ρ) |2Sϵ − Tϵ | dt dx = 0 and µ = 0.
1/3
Thus, we can see that the equality µ = 0 is granted when locally ⃗u belongs to L3t b3,∞ ,
1/3
1/3
1/3
where b3,∞ is the closure of D in the Besov space Ḃ3,∞ : if ϕ⃗u ∈ L3t b3,∞ , then
R
ˆ |ϕ(t, x)⃗u(t, x) − ϕ(t, x + y)⃗u(t, x + y)|3 dx ≤ C∥ϕ⃗u∥3 1/3 |y| and
Ḃ3,∞
lim
y→0
1
|y|
Z
|ϕ(t, x)⃗u(t, x) − ϕ(t, x + y)⃗u(t, x + y)|3 dx = 0
ˆ by dominated convergence, we get
ZZZ
1
lim
|ϕ(t, x)⃗u(t, x) − ϕ(t, x + y)⃗u(t, x + y)|3 dt dx dy = 0.
ϵ→0+ ϵ4
|y|<ϵ
424
The Navier–Stokes Problem in the 21st Century (2nd edition)
In particular, we may check that Duchon and Robert’s criterion is based on a weaker
1/3
assumption than ⃗u ∈ L4t,x (Q): if v ∈ Ḣ 1 ∩ L4 then v ∈ b3,∞ and
1/3
2/3
∥v∥Ḃ 1/3 ≤ C∥v∥Ḣ 1 ∥v∥4 .
(13.13)
3,∞
Indeed let Ip =
that
RR
|v(t, x) − v(t, x + y)|p dt dx. We have I2 ≤ ∥v∥2Ḣ 1 |y|2 and I4 ≤ 16∥v∥44 so
I3 ≤ (I2 )1/2 (I4 )1/2 ≤ 4∥v∥Ḣ 1 ∥v∥24 |y|
Thus, (13.13) is proved.
13.8
The Caffarelli-Kohn-Nirenberg Theorem on Partial
Regularity
The celebrated regularity criterion of Caffarelli, Kohn and Nirenberg [74] states that if
⃗u is a solution of the Navier–Stokes equations in a neighborhood of a point (t0 , x0 ) ∈ R × R3
which satisfies “some conditions” on the velocity ⃗u, the pressure p and the force f⃗ and if
the number
ZZ
1
⃗ ⊗ ⃗u|2 ds dx
lim sup
|∇
r→0+ r
(t0 −r 2 ,t0 +r 2 )×B(x0 ,r)
is “small enough,” then (t0 , z0 ) is a “regular point.” The definitions of a regular point and
the choice of the admissible conditions on ⃗u, p and f⃗ have been discussed by many authors.
In the original paper of Caffarelli, Kohn and Nirenberg [74], assumptions on ⃗u, p and f⃗
were:
1. ⃗u, p and f⃗ are defined on a cylinder Q = (T, T + R2 ) × B(X, R)
2 1
2
2. on Q, ⃗u belongs to L∞
t Lx ∩ Lt Hx
RR
3. on Q, Q |p(t, x)|5/4 dt dx < +∞
4. on Q,
RR
Q
|f⃗(t, x)|q dt dx < +∞ for some q > 5/2
5. ⃗u is a solution of the Navier–Stokes equations on Q: div ⃗u = 0 and
⃗ u + f⃗ − ∇p
⃗ in D′ (Q)
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
(13.14)
6. ⃗u is a suitable solution, i.e., there exists a non-negative distribution µ such that
⃗ ⊗ ⃗u|2 − div((|⃗u|2 + 2p)⃗u) + 2⃗u · f⃗ − µ
∂t |⃗u|2 = ν∆|⃗u|2 − 2ν|∇
(13.15)
7. regularity of ⃗u at (t0 , z0 ) is meant in the sense of Definition 13.1: ⃗u is bounded in the
neighborhood of (t0 , z0 )
5/4
The reason for the exponent 5/4 in the assumption p ∈ Lt,x (Q) was that 5/4 was at
that time the best exponent one could prove when exhibiting suitable solutions for the
Navier–Stokes equations on a bounded domain associated with a square–integrable initial
3/2
value ⃗u0 . But Lin [335] proved the existence of suitable solutions with p ∈ Lloc ((0, T ) × Ω)
for bounded domains Ω, by using regularity estimates for the pressure obtained by Sohr
Partial Regularity Results for Weak Solutions
425
and von Wahl [444]. The computations were much easier with the hypothesis p ∈ L3/2 (Q),
so Lin could give a simplified proof of the Caffarelli–Kohn–Nirenberg criterion.
While the pressure for P
LerayPsolutions on the whole space is entirely determined by
3
3
the equation ∆p = div f⃗ − i=1 j=1 ∂i ∂j (ui uj ), this is no longer the case when studying
a local solution of the Navier–Stokes equations. Thus, pressure has to be dealt with very
carefully. Some variants of the Caffarelli, Kohn and Nirenberg theorem involve different
assumptions on the pressure: for instance, Vasseur [487] gave a proof (with f⃗ = 0) under
the assumption p ∈ Lqt L1x (Q) with a different method (instead of estimating quadratic means
of ⃗u on small cylinders, as in the other references, he used an à la Di Giorgi method and
estimated the measure of level sets {(t, x) / |⃗u| > λ}). Wolf [504] considered an extended
version of suitable solutions in order to include in the pressure the harmonic term that is
deleted when applying the divergence operator to the equation.
In this section, we are going to make the following assumptions on ⃗u, f⃗ and p:
Hypotheses for the Caffarelli-Kohn-Nirenberg regularity criterion
Definition 13.4.
We call (HCKN ) the following set of hypotheses:
1. ⃗u, p and f⃗ are defined on a domain Ω ⊂ R × R3
2 1
2
2. on Ω, ⃗u belongs to L∞
t Lx ∩ Lt Ḣx :
Z
ZZ
⃗ ⊗ ⃗u|2 dt dx < +∞
sup
|⃗u(t, x)|2 dx < +∞ and
|∇
t∈R
(t,x)∈Ω
Ω
3. for some q0 > 1, p belongs to Lqt 0 L1x (Ω):
Z Z
(
|p(t, x)| dx)q0 dt < +∞
R
(t,x)∈Ω
10/7
4. on Ω, f⃗ is a divergence free vector field in Lt,x (Ω):
ZZ
⃗
div f = 0 and
|f⃗(t, x)|10/7 dt dx < +∞
Ω
5. ⃗u is a solution of the Navier–Stokes equations on Ω: div ⃗u = 0 and
⃗ u + f⃗ − ∇p
⃗ in D′ (Ω)
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
(13.16)
We have, of course, some further estimates on ⃗u and p that we can deduce from (HCKN ).
If I is a bounded interval of R and B = B(xB , rB ) a ball of R3 such that I ×B(xB , 2rB ) ⊂ Ω,
then we have the following properties:
RR
ˆ I×B |⃗u||f⃗| dt dx < +∞: by Sobolev inequality, we have
Z Z
Z Z
( |⃗u(t, x)|6 dx)1/3 dt ≤C
I
B
I
B
|u|2
⃗ ⊗ ⃗u|2 dx dt
+ |∇
|B|2/3
|I|
⃗ ⊗ ⃗u∥2 2 2 )
≤ C( 2/3 ∥⃗u∥2L∞
+ ∥∇
2
Lt Lx (Ω)
t Lx (Ω)
|B|
(13.17)
426
The Navier–Stokes Problem in the 21st Century (2nd edition)
By interpolation between L∞ L2 and L2 L6 , we find that
10/3
⃗u ∈ Lt,x (I × B)
ˆ
(13.18)
RR
|⃗u||p| dt dx < +∞: taking the divergence of the Navier–Stokes equations and
using div ⃗u = div f⃗ = 0, we get:
I×B
⃗ u) = −
∆p = − div(⃗u · ∇⃗
3 X
3
X
∂i ∂j (ui uj ).
(13.19)
i=1 j=1
Now, we introduce a function ω ∈ D(R3 ) with ω = 1 on B(0, 5/4) and with Supp ω ⊂
B
B(0, 7/4) and we define ζB (x) = ω( x−x
rB ). Let G be the fundamental solution of −∆
(so that −∆G = δ):
1
G=
.
4π|x|
We have ζB p = G ∗ (−∆(ζB p)), with
−∆(ζB p) =p(−∆ζB ) − 2
3
X
∂i ζB ∂i p − ζB ∆p
i=1
=p(∆ζB ) − 2
3
X
∂i (p∂i ζB ) + ζB
i=1
=p(∆ζB ) − 2
3
X
+
i=1 j=1
∂i ∂j (ui uj )
i=1 j=1
∂i (p∂i ζB ) +
i=1
3 X
3
X
3 X
3
X
ui uj ∂i ∂j ζB − 2
3 X
3
X
∂i ∂j (ζB ui uj )
i=1 j=1
3 X
3
X
∂i (ui uj ∂j ζB )
i=1 j=1
We find:
ζB p = ϖB + pB + qB
with

3 X
3
X


ϖB =
∂j ∂l G ∗ (ζB uj ul )




j=1 l=1




3 X
3
3 X
3

X
X
qB = − 2
∂j G ∗ ((∂l ζB )uj ul ) +
G ∗ ((∂j ∂l ζB )uj ul )


j=1 l=1
j=1 l=1




3

X



p
=
−
2
∂j G ∗ ((∂j ζB )p) + G ∗ ((∆ζB )p

B

j=1
When (t, x) ∈ I × B, we find that

Z
3
3 X
X

1


|uj (t, y)ul (t, y)| dy

 |qB (t, x)| ≤C
r3
j=1 l=1 B B(xB ,2rB )
Z


1


|p(t, y)| dy
|pB (t, x)| ≤C 3
rB B(xB ,2rB )
(13.20)
Partial Regularity Results for Weak Solutions
427
5/3
Thus, on I ×B, we have p = ϖB +pB +qB with ϖB ∈ Lt,x (I ×B), pB ∈ Lqt 0 L∞
x (I ×B)
10/3
2
and qB ∈ L∞
u ∈ L∞
t,x (I × B) and we have, as ⃗
t Lx ∩ Lt,x (I × B),
ZZ
|p||⃗u| dx dt < +∞
(13.21)
I×B
Thus, the distribution
⃗ ⊗ ⃗u|2 + 2⃗u · f⃗ − div((|⃗u|2 + 2p)⃗u)
µ = −∂t |⃗u|2 + ν∆|⃗u|2 − 2ν|∇
(13.22)
is well defined on Ω.
Suitable solutions
Definition 13.5.
The solution ⃗u is suitable if the distribution µ is a non-negative locally finite measure
on Ω.
We are going to prove Caffarelli, Kohn and Nirenberg’s result in the setting of parabolic
Morrey spaces, following the papers by Ladyzhenskaya and Seregin [297] and by Kukavica
[286] (a clear survey is given in the lecture notes of Robinson [415]).
Let ρ2 be the parabolic “norm” given by ρ2 (t, x) = |t|1/2 + |x|. A function h on R × R3
is Hölderian of exponent α ∈ (0, 1) with respect to the parabolic distance if we have
|h(t, x) − h(s, y)| ≤ Ch (|t − s|1/2 + |x − y|)α .
A function h belongs to the parabolic Morrey space Mq,τ
(1 < q ≤ τ < +∞) if and
2
only if
∥h∥Mq,τ
< +∞
2
with
∥h∥qMq,τ
2
=
ZZ
1
sup
q
(t,x)∈R×R3 ,r>0
r5(1− τ )
|h(s, y)|q ds dy.
ρ2 (t−s,x−y)<r
Of course, one may replace in this definition the balls B((t, x), r) by the cylinders Qr (t, x) =
(t − r2 , t + r2 ) × B(x, r), as we have B((t, x), r) ⊂ Qr (t, x) ⊂ B((t, x), 2r).
Moreover, when a function h is defined on a cylinder Q0 = Qr0 (t,0 , x0 ), for estimating
the parabolic Morrey of 1q0 h (the function that is equal to h on Q0 and to 0 elsewhere),
i.e., to estimate
1
sup
I (t, x)1/q
5( 1 − 1 ) r
(t,x)∈R×R3 ,r>0 r q τ
with
ZZ
|h(s, y)|q ds dy,
Ir (t, x) =
Qr (t,x)∩Q0
there is no need to consider r > r0 : for r ≥ r0 , we may write Ir (t, x) ≤ Ir0 (t0 , x0 ). . .
Moreover, if r ≤ r0 and Qr (t, x) ∩ Q0 ̸= ∅, then there exists (t1 , x1 ) ∈ Q0 so that Ir (t, x) ≤
I2r (t1 , x1 ); thus, there is no need as well to consider (t, x) ∈
/ Q0 .
428
The Navier–Stokes Problem in the 21st Century (2nd edition)
We may now state the theorem:
Caffarelli-Kohn-Nirenberg regularity criterion
Theorem 13.8.
Let Ω be a domain of R × R3 . Let (⃗u, p) a weak solution on Ω of the Navier–Stokes
equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Assume that
• (⃗u, p, f⃗) satisfies the conditions (HCKN ): ⃗u ∈ L∞ L2 ∩ L2 Ḣ 1 (Ω), p ∈ Lq0 L1 (Ω)
(q0 > 1), div f⃗ = 0 and f⃗ ∈ L10/7 L10/7 (Ω)
• ⃗u is suitable
10/7,τ0
• 1Ω (t, x)f⃗ ∈ M2
for some τ0 > 5/2.
There exists a positive constant ϵ∗ which depends only on ν and τ0 such that, if for
some (t0 , x0 ) ∈ Ω, we have
ZZ
1
⃗ ⊗ ⃗u|2 ds dx < ϵ∗
|∇
lim sup
r→0 r
(t0 −r 2 ,t0 +r 2 )×B(x0 ,r)
then ⃗u is Hölderian (with respect to the quasi-norm δ(t, x) = |t|1/2 + |x|) in a neighborhood of (t0 , x0 ).
The proof relies on Campanato’s lemma on Hölderian functions [79] applied to the
regularity of solutions of parabolic equations:
Lemma 13.2 (Campanato’s lemma).
1/2
3
Let ρ2 (t, x)
x−y) < r} and MQr f (t, x) =
RR = |t| +|x|, Qr (t, x) = {(s, y) ∈ R×R } / ρ2 (t−s,
1
f (s, y) ds dy. Let p ∈ [1, +∞) and f ∈ Lploc (dt dx). Let 0 < α < 1. Then
|Qr (t,x)| |
QR (t,x)
f is Hölderian of exponent α with respect to the parabolic distance if and only if
ZZ
1
1
sup sup
(
|f (s, y) − MQr (t,x) f |p ds dy)1/p < +∞.
α |Q (t, x)|
r>0 t,x)∈R×R3 r
r
QR (t,x)
Proof. If f is Hölderian, just write
|f (s, y) − MQr (t,x) f | =
1
|
|Qr (t, x)|
ZZ
f (s, y) − f (t, x) ds dy| ≤ Cf rα .
Qr (t,x)
Conversely, let
Hf = sup
sup
r>0 t,x)∈R×R3
1
1
(
rα |Qr (t, x)|
ZZ
|f (s, y) − MQr (t,x) f |p ds dy)1/p .
QR (t,x)|
Let Φ ∈ D(R × R3 ) supported in Q1 (0, 0) with
RR
Φ dx dt = 1 and Φϵ =
Fϵ (t, x, s, y) = Φϵ ∗ f (t, x) − Φϵ ∗ f (s, y).
1
t x
ϵ5 Φ( ϵ2 , ϵ ).
Let
Partial Regularity Results for Weak Solutions
429
Then we have, for every t, x, y and ϵ,
Φϵ ∗ f (t, x) − Φϵ ∗ f (t, y)
ZZ
=
(Φϵ (t − σ, x − z) − Φϵ (s − σ, y − z))f (σ, z) dz dσ
ZZ
=
(Φϵ (t − σ, x − z) − Φϵ (s − σ, y − z))×
Qϵ+ρ2 (t−s,x−y) (t,x)
× (f (σ, z) − MQϵ+ρ2 (t−s,x−y) (t,x) f ) dz dσ
and
Φϵ ∗ f (t, x) − Φϵ/2 ∗ f (t, x)
ZZ
=
(Φϵ (t − σ, x − z) − Φϵ/2 (t − σ, x − z))f (σ, z) dz dσ
ZZ
=
(Φϵ (t − σ, x − z) − Φϵ/2 (t − σ, x − z))(f (σ, z) − MQϵ (t,x) f ) dz dσ
Qϵ (t,x)
We then write
|Φϵ (t − σ, x − z)| ≤
1
1
∥Φ∥∞ ≤ C
5
ϵ
|Qϵ (t, x)|
so that
|Φϵ ∗ f (t, x) − Φϵ/2 ∗ f (t, x)| ≤ CHf ϵα
and
|Φϵ (t − σ, x − z) − Φϵ (s − σ, y − z)|
≤ |Φϵ (t − σ, x − z)−Φϵ (t − σ, y − z)| + |Φϵ (t − σ, y − z) − Φϵ (s − σ, y − z)|
|t − s|1/2 p
|x − y| ⃗
≤
2∥Φ∥∞ ∥∂t Φ∥∞ +
∥∇Φ∥∞
6
ϵ
ϵ6
1
ρ2 (t − s, x − y)
≤C
.
|Qϵ (t, x)|
ϵ
so that, for ϵ > ρ2 (t − s, x − y), we have
|Φϵ ∗ f (t, x) − Φϵ ∗ f (t, y)| ≤ CHf ϵα
ρ2 (t − s, x − y)
.
ϵ
Now, let (t0 , x0 ) be a Lebesgue point of f . We have convergence in D′ (R × R3 ) of
X
(Φ2j ∗ f (t, x) − Φ2j ∗ f (t0 , x0 )) − (Φ2j+1 ∗ f (t, x) − Φ2j+1 ∗ f (t0 , x0 ))
j∈Z
to f − f (t0 , x0 ). Thus, as the series is uniformly convergent on every bounded subset of
R × R3 , the sum is a continuous function. Identifying f to the sum, we find finally that the
function f is Hölderian of exponent α, as
X
2αj min(1, 2−j ρ2 (t − s, x − y)) ≤ Cρ2 (t − s, x − y)α .
j∈Z
The lemma is proved.
We may now study the regularity of the heat equation. Regularity results on solutions
of parabolic equations may be found in many references, such as the classical book by
Ladyzhenskaya, Solonnikov and Uraltseva [298]. Here, we shall consider parabolic Hölderian
regularity.
430
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proposition 13.4.
0
1
Let f ∈ Mp,q
and g ∈ Mp,q
with 1 ≤ p ≤ q0 < q1 < +∞, q11 = 15 − α5 , q10 = 25 − α5 , 0 <
2
2
α < 1. Let σ be a smooth function on R3 \ {0}, homogeneous of exponent 1: σ(λξ) = λσ(ξ)
for λ > 0, and let σ(D) be the Fourier multiplier operator with symbol σ. Then the function
h equal to 0 for t ≤ 0 and to
Z t
Wν(t−s) ∗ (f (s, .) + σ(D)g(s, .)) ds
h(t, x) =
0
for t > 0 is Hölderian of exponent α with respect to the parabolic distance.
Proof. We may write h as a convolution in time and space variables
h = W+ ∗ (f+ + σ(D)g+ )
with W+ (t, x) = 1t>0 Wνt (x), f+ = 1t>0 f and g+ = 1t>0 g. The size estimates on W+ are
easily established (see Ladyzhenskaya et al. [298] for instance, or our estimates in Chapter
5). In particular,
|W+ (t, x)| ≤ Cρ2 (t, x)−3 and |σ(D)W+ (t, x)| ≤ Cρ2 (t, x)−4
|∂t W+ (t, x)| ≤ Cρ2 (t, x)−5 and |∂t σ(D)W+ (t, x)| ≤ Cρ2 (t, x)−6
and
−5
⃗ + (t, x)| ≤ Cρ2 (t, x)−4 and |∇σ(D)W
⃗
|∇W
.
+ (t, x)| ≤ Cρ2 (t, x)
We now estimate
1
|Qr (t, x)|
ZZ
|h(s, y) − MQr (t,x) h|p ds dy
QR (t,x)
ZZZZ
1
≤
|h(s, y) − h(σ, z)|p ds dy dσ dz
|Qr (t, x)|2
QR (t,x)×Qr (t,x)
P
Define Γj = Q2j+1 r (t, x) \ Q2j r (t, x), fj = 1t>0 1Γj f and gj = 1t>0 1Γj g, so that h = j∈Z hj
with hj = W+ ∗ (fj + σ(D)gj ). We have
RR
1
dη
ˆ for j ≤ 5, |hj (s, y)| ≤ C( 1ρ2 (τ,η)<33r ρ2 (τ,η)
3 |fj (s − τ, y − η)| dτ
RR
1
+ 1ρ2 (τ,η)<33r ρ2 (τ,η)4 |gj (s − τ, y − η)| dτ dη), so that
∥hj ∥p ≤ C(∥1ρ2 (τ,η)<33r
1
1
∥1 ∥fj ∥p + ∥1ρ2 (τ,η)<33r
∥1 ∥gj ∥p )
3
ρ2 (τ, η)
ρ2 (τ, η)4
and thus
1
|Qr (t, x)|2
!1/p
ZZZZ
p
|hj (s, y) − hj (σ, z)| ds dy dσ dz
QR (t,x)×Qr (t,x)
1
1
α 5j( p − q1 )
≤ C(∥f ∥Mp,q
g ∥Mp,q
0 + ∥⃗
1 )r 2
2
2
ˆ for j ≥ 6, (s, y) ∈ Qr (t, x) and (σ, z) ∈ Qr (t, x), we have
ZZ
ρ2 (s − σ, y − z)
|hj (s, y) − hj (σ, z)| ≤ C(
|fj (τ, η)| dτ dη
(2j r)4
ZZ
ρ2 (s − σ, y − z)
+
|⃗gj (τ, η)| dτ dη)
(2j r)5
Partial Regularity Results for Weak Solutions
431
and thus
1
|Qr (t, x)|2
!1/p
ZZZZ
p
|hj (s, y) − hj (σ, z)| ds dy dσ dz
Qr (t,x)×Qr (t,x)
α j(α−1)
≤ C(∥f ∥Mp,q
g ∥Mp,q
.
0 + ∥⃗
1 )r 2
2
2
Thus, we get
X
j∈Z
!1/p
ZZZZ
1
|Qr (t, x)|2
p
|hj (s, y) − hj (σ, z)| ds dy dσ dz
Qr (t,x)×Qr (t,x)
α
≤ C(∥f ∥Mp,q
g ∥Mp,q
0 + ∥⃗
1 )r
2
2
and the proposition is proved.
The strategy for the proof of the Caffarelli–Kohn–Nirenberg criterion is then clear. Let
r1 > 0 be fixed and Q1 = Qr1 (t0 , x0 ). We choose a non-negative function ω ∈ D(R × R3 )
such that ω is supported in (−1, 1) × B(0, 1) and is equal to 1 on (−1/4, 1/4) × B(0, 1/2),
and we define
t − t 0 x − x0
ψ(t, x) = ω( 2 ,
) and ⃗v (t, x) = ψ(t, x)⃗u(t, x)
r1
r1
⃗v is defined on R × R3 with support in Q1 and satisfies
∂t⃗v = ν∆⃗v + ⃗g +
3
X
∂i⃗hi
(13.23)
i=1
with
(
⃗ u + ψ ∇p
⃗ + ψ f⃗
⃗g = (∂t ψ)⃗u + ν(∆ψ)⃗u + (⃗u · ∇ψ)⃗
⃗hi =
−2ν(∂i ψ)⃗u − ψ ui ⃗u
q ,τ1 /2
As ⃗v coincides with ⃗u on Qr1 /2 (t0 , x0 ), we are going to estimate the Morrey norms M21
of ⃗g and Mq21 ,τ1 of ⃗hi with q1 > 1 and τ1 > 5, and conclude by using Proposition 13.4.
13.9
Proof of the Caffarelli–Kohn–Nirenberg Criterion
We list the quantities that we want to estimate for (t, x) ∈ Qr0 (t0 , x0 ) and r ≤ r0 (we
assume that r0 is small enough to grant that Q4r0 (t0 , x0 ) ⊂ Ω):
R
ˆ Ur (t, x) = sups∈(t−r2 ,t+r2 ) Br (t,x) |⃗u(s, y)|2 dx dy
ˆ Vr (t, x) =
RR
Qr (t,x)
ˆ Wr (t, x) =
RR
ˆ Ωr (t, x) =
RR
ˆ Pr (t, x) =
RR
⃗ ⊗ ⃗u(s, y)|2 ds dy
|∇
Qr (t,x)
Qr (t,x)
Qr (t,x)
|⃗u(s, y)|3 ds dy
|⃗u(s, y)|10/3 ds dy
|p(s, y)|q0 ds dy
432
The Navier–Stokes Problem in the 21st Century (2nd edition)
RR
⃗
ˆ Πr (t, x) = Qr (t,x) |∇p(s,
y)|q1 ds dy
ˆ Fr (t, x) =
RR
Qr (t,x)
|f⃗(s, y)|10/7 ds dy
By assumptions (HCKN ), we have ⃗u ∈ L∞ L2 ∩ L2 Ḣ 1 (Ω), so that Ur (t, x) and Vr (t, x)
are well defined on Qr0 (t0 , x0 ); moreover, we have seen that, in that case, ⃗u ∈
L10/3 L10/3 (Q4r0 (t0 , x0 )), so that Wr (t, x) and Ωr (t, x) are well defined on Qr0 (t0 , x0 ). We
10/7
have f⃗ ∈ Lt,x (Ω), so that Fr (t, x) is well defined on Qr0 (t0 , x0 ).
We have p ∈ Lqt 0 L1 with q0 > 1. With no loss of generality, we may assume q0 < 3/2. In
0
that case, using (13.20) for B = B(x0 , 2r0 ), we can see that we have p ∈ Lqt,x
(Q2r0 (t0 , x0 ))
so that Pr (t, x) is well defined on Qr0 (t0 , x0 ).
1
⃗ ∈ Lqt,x
Finally, differentiating (13.20), we can see that ∇p
(Q2r0 (t0 , x0 )) for every 1 <
q1 < min(q0 , 6/5) (since ⃗u is locally L3 L3 and L2 H 1 ). Thus, Πr (t, x) is well defined on
Qr0 (t0 , x0 ) for q1 small enough.
We are going to estimate those quantities Ur , Vr ,... in terms of Uρ , Vρ ,..., where 0 < r <
ρ/2 < r0 /2.
Step 1: The local energy inequality.
A consequence of the local energy inequality is that, for any smooth ψ ∈
D(Q4r0 (t0 , x0 )) with ψ ≥ 0, we have, for τ ∈ (t0 − 16r02 , t0 + 16r02 )
Z
Z
Z
⃗ ⊗ ⃗u(s, y)|2 dy ds
ψ(τ, y)|⃗u(τ, y)|2 dy+2ν
ψ(s, y)|∇
s<τ
Z
Z
≤
(∂t ψ(s, y) + ν∆ψ(s, y))|⃗u(s, y)|2 dy ds
s<τ
Z
Z
(13.24)
2
⃗
+
(|⃗u(s, y)| + 2p(s, y))⃗u(s, y) · ∇ψ(s, y) dy ds
s<τ
Z
Z
+2
ψ(s, y)⃗u(s, y) · f⃗(s, y) dy ds
s<τ
Of course, the problem is to choose a good test function ψ. The choice of ψ has been
given by Scheffer [426]: we choose a non-negative function ω ∈ D(R × R3 ) such that
ω is supported in (−1, 1) × B(0, 3/4) and is equal to 1 on (−1/4, 1/4) × B(0, 1/2),
a non-negative smooth function θ on R that is equal to 1 on (−∞, 1) and to 0 on
(2, +∞) and we define
ψ(s, y) = r3 ω(
s−t
s−t y−x
,
)θ( 2 )H(4r2 + t − s, x − y)
2
ρ
ρ
r
where 0 < r ≤ ρ/2 ≤ r0 /2 and H(t, x) = Wνt (x).
ψ enjoys many good properties (in the following estimates, C means some positive
constant which depends on ν):
ˆ ψ is smooth, non-negative and is supported in Qρ (t, x)
ˆ ψ(s, y) ≤ C on Qρ (t, x), and ψ(s, y) ≥
1
C
on Qr (t, x)
⃗
ˆ |∇ψ(s,
y)| ≤ C 1r on Qρ (t, x)
ˆ for s < t + r2 and (s, y) ∈
/ Qρ/2 (t, x), we have
H(4r2 + t − s, x − y) ≤ C
1
|4r2
+t−
s|3/2
+ |x −
y|3
≤ C′
1
ρ3
Partial Regularity Results for Weak Solutions
433
ˆ for s < t + r2 , we have
(∂s + ν∆y )(H(4r2 + t − s, x − y)) = 0
while, for (s, y) ∈ Qρ/2 (t, x)
(∂t + ν∆)(ω(
s−t y−x
,
)) = 0
ρ2
ρ
so that for (s, y) ∈ Qρ (t, x) with s < t + r2 , we have
| (∂s + ν∆y )ψ(s, y)| ≤ C
r3
ρ5
Moreover, as div ⃗u = 0, if Γρ,⃗u (s, t, x) is any function which does not depend on y, we
have
ZZ
ZZ
2
⃗
⃗ dy ds.
|⃗u(s, y)| ⃗u(s, y) · ∇ψ(s, y) dy ds =
(|⃗u|2 − Γρ,⃗u )⃗u · ∇ψ
s<τ
s<τ
We take
1
|B(x, ρ)|
Γρ,⃗u (s, t, x) =
Z
|⃗u(s, y)|2 dy.
B(x,ρ)
(as a matter of fact, it does not depend on t).
We obtain
ZZ
max(Ur (t, x), 2νVr (t, x)) ≤ C
Q (t,x)
Z Zρ
r3
|⃗u(s, y)|2 dy ds
ρ5
+C
Qρ (t,x)
1
|⃗u(s, y)|2 − Γρ,⃗u (s, t, s) |⃗u(s, y)| dy ds
r
ZZ
+C
(t−ρ2 ,t+ρ2 )×B(x, 34 ρ)
ZZ
1
|p(s, y)| |⃗u(s, y)| dy ds
r
|⃗u(s, y)| |f⃗(s, y)| dy ds
+C
Qρ (t,x)
The first term is easy to estimate:
ZZ
r3
r3
|⃗u(s, y)|2 dy ds ≤ 2 3 Uρ (t, x)
5
ρ
Qρ (t,x) ρ
(13.25)
For the second term, we write
ZZ
1
|⃗u(s, y)|2 − Γρ,⃗u (s, t, s) |⃗u(s, y)| dy ds
Qρ (t,x) r
ZZ
ZZ
1
3/2
2
2/3
≤C (
|⃗u(s, y)| − Γρ,⃗u (s, t, s)
dy ds) (
|⃗u(s, y)|3 dy ds)1/3
r
Qρ (t,x)
Qρ (t,x)
We write
ZZ
(
Qρ (t,x)
2/5
3/5
|⃗u(s, y)|10/3 dy ds)3/10 ≤∥⃗u∥L∞ L2 (Qρ (t,x)) ∥⃗u∥L2 L6 (Qρ (t,x))
2/5
≤ C∥⃗u∥L∞ L2 (Qρ (t,x)) ((
∥⃗u∥L2 L2 (Qρ (t,x)) 3/5
⃗ ⊗ ⃗u∥3/5
) + ∥∇
L2 L2 (Qρ (t,x)) )
ρ
≤C ′ (Uρ (t, x) + Vρ (t, x))1/2
434
The Navier–Stokes Problem in the 21st Century (2nd edition)
In particular, we have
ZZ
ZZ
3
1/3
1/6
(
|⃗u(s, y)| dy ds)
≤Cρ (
Qρ (t,x)
|⃗u(s, y)|10/3 dy ds)3/10
Qρ (t,x)
≤C ′ ρ1/6 (Uρ (t, x) + Vρ (t, x))1/2
Moreover, by the Gagliardo–Nirenberg inequality, we have
Z
Z
3/2
(
|⃗u(s, y)|2 − Γρ,⃗u (s, t, s)
dy)2/3 ≤ C
B(x,ρ)
⃗ u(s, y)|2 )| dy
|∇(|⃗
B(x,ρ)
so that
ZZ
(
|⃗u(s, y)|2 − Γρ,⃗u (s, t, s)
3/2
dy ds)2/3
Qρ (t,x)
⃗ ⊗ ⃗u∥L2 L2 (Q (t,x))
≤ C∥⃗u∥L6t L2x (Qρ (t,x)) ∥∇
ρ
t x
≤ Cρ1/3 Uρ (t, x)1/2 Vρ (t, x)1/2
and finally
ZZ
Qρ (t,x)
1
|⃗u(s, y)|2 − Γρ,⃗u (s, t, s) |⃗u(s, y)| dy ds ≤
r
ρ1/2
C
(Uρ (t, x) + Vρ (t, x))Vρ (t, x)1/2 .
r
(13.26)
The fourth term is then easy to control:
ZZ
|⃗u(s, y)| |f⃗(s, y)| dy ds ≤ ∥⃗u∥L10/3 L10/3 (Qρ (t,x)) ∥f⃗∥L10/7 L10/7 (Qρ (t,x))
Qρ (t,x)
and thus
ZZ
|⃗u(s, y)| |f⃗(s, y)| dy ds ≤ C(Uρ (t, x)+Vρ (t, x))1/2 Fρ (t, x)7/10
(13.27)
Qρ (t,x)
The third term is more delicate to deal with. We introduce a function θ ∈ D(R3 ) with
θ = 1 on B(0, 13/16) and with Supp θ ⊂ B(0, 15/16) and we define ζ(y) = θ( y−x
ρ ).
P3 P3
2
2
On (t − ρ , t + ρ ) × B(x, 3ρ/4), we have ζp = p. From ∆p = − i=1 j=1 ∂i ∂j (ui uj ),
we get
3 X
3
3
X
X
∆(ζp) = −ζ
∂i ∂j (ui uj ) + 2
∂j (p∂j ζ) − p∆ζ
i=1 j=1
j=1
and we may write ζ(y)p(s, y) = pρ,x (s, y) + qρ,x (s, y) with

P3 P3
qρ,x =
j=1
l=1 G ∗ (ζ∂j ∂l (uj ul ))

pρ,x = −2
P3
j=1
∂j G ∗ ((∂j ζ)p) + G ∗ ((∆ζ)p)
We may, of course, replace, in the definition of qρ,x , the term uj (s, y)ul (s, y) with
uj ul − Γρ,⃗u,j,l (s, x) with
Z
1
Γρ,⃗u,j,l (s, x) =
uj (s, y)ul (s, y) dy.
|B(x, ρ)| B(x,ρ)
Partial Regularity Results for Weak Solutions
435
For (s, y) ∈ (t − ρ2 , t + ρ2 ) × B(x, 3ρ/4), we have
Z
1
|pρ,x (s, y)| ≤ C 3
|p(s, z)| dz
ρ B(x,ρ)
so that
3
∥pρ,x ∥Lq0 L∞ ((t−ρ2 ,t+ρ2 )×B(x,3ρ/4)) ≤ Cρ− q0 Pρ (t, x)1/q0 .
As we have
∥⃗u∥
1
q0
L q0 −1
L1 ((t−ρ2 ,t+ρ2 )×B(x,3ρ/4))
≤ Cρ2(1− q0 ) ρ3/2 Uρ (t, x)1/2
we get
ZZ
(t−ρ2 ,t+ρ2 )×B(x, 34 ρ)
1
|pρ,x (s, y)| |⃗u(s, y)| dy ds ≤
r
3
5
1
C ρ2+ 2 − q0 Pr (t, x)1/q0 Uρ (t, x)1/2
r
(13.28)
For the term involving qρ,x , we write
qρ,x =
3 X
3
X
∂j ∂l G ∗ (ζ(uj ul − Γρ,⃗u,j,l ))
j=1 l=1
−2
3 X
3
X
∂j G ∗ ((∂l ζ)(uj ul − Γρ,⃗u,j,l ))
j=1 l=1
+
3 X
3
X
G ∗ ((∂j ∂l ζ)(uj ul − Γρ,⃗u,j,l ))
j=1 l=1
Thus, again for (s, y) ∈ (t − ρ2 , t + ρ2 ) × B(x, 3ρ/4), we have
|qρ,x (s, y)| ≤
3 X
3
X
|∂j ∂l G ∗ (ζ(uj ul − Γρ,⃗u,j,l ))(s, y)|
j=1 l=1
+C
3 X
3
X
MR3 (1Qρ (t,x) (uj ul − Γρ,⃗u,j,l ))(s, y)
j=1 l=1
where MR3 is the Hardy–Littlewood maximal function with respect to the space
variable. Thus, we get
⃗ ⊗ ⃗u∥L2 L2 (Q (t,x))
∥qρ,x ∥L3/2 L3/2 ((t−ρ2 ,t+ρ2 )×B(x,3ρ/4)) ≤C∥⃗u∥L6t L2x (Qρ (t,x)) ∥∇
ρ
t x
≤ Cρ1/3 Uρ (t, x)1/2 Vρ (t, x)1/2
and finally
ZZ
(t−ρ2 ,t+ρ2 )×B(x, 34 ρ)
1
|qρ,x (s, y)| |⃗u(s, y)| dy ds ≤
r
ρ1/2
C
(Uρ (t, x) + Vρ (t, x))Vρ (t, x)1/2 .
r
(13.29)
436
The Navier–Stokes Problem in the 21st Century (2nd edition)
Besides, let us notice that
Pr (t, x) ≤(∥pρ,x ∥Lq0 Lq0 (Qr (t,x) + ∥qρ,x ∥Lq0 Lq0 (Qr (t,x) )q0
q0
5(1−
≤C(r3 ∥pρ,x ∥L
q0 L∞ (Q (t,x) + r
r
≤C(
2q0
3
)
∥qρ,x ∥qL03/2 L3/2 (Qr (t,x) )
2q0
r3
Pρ (t, x) + r5(1− 3 ) ρq0 /3 Uρ (t, x)q0 /2 Vρ (t, x)q0 /2 )
3
ρ
Summarizing all those estimates, we have shown:
Lemma 13.3.
Assume that
• ⃗u ∈ L2t,x (Ω) ∩ L2t Ḣx1 (Ω)
0
• p ∈ Lqt,x
(Ω) with 1 < q0 ≤ 3/2)
10/7
• f⃗ ∈ Lt,x (Ω)
• ⃗u is suitable
then, for 0 < r ≤ ρ/2 ≤ r0 /2 and (t, x) ∈ Qr0 (t0 , x0 ),
r3
Uρ (t, x)
ρ3
ρ1/2
(Uρ (t, x) + Vρ (t, x))Vρ (t, x)1/2
+C
r
3
5
1
+C ρ2+ 2 − q0 Pρ (t, x)1/q0 Uρ (t, x)1/2
r
+C(Uρ (t, x)+Vρ (t, x))1/2 Fρ (t, x)7/10
(13.30)
2q0
r3
Pρ (t, x) + r5(1− 3 ) ρq0 /3 Uρ (t, x)q0 /2 Vρ (t, x)q0 /2 ).
ρ3
(13.31)
Ur (t, x) + Vr (t, x) ≤ C
and
Pr (t, x) ≤ C(
Step 2: Morrey estimates for the velocity and the pressure.
We are going to use the estimates of Lemma 13.3 to show the following result (inspired
from Kukavica [286]):
Lemma 13.4.
10/7
Let ⃗u be a suitable solution of the Navier–Stokes equations (with f⃗ ∈ Lt,x (Ω) and
10/7,τ0
p ∈ Lq0 (Ω) with 1 < q0 ≤ 3/2). Assume moreover that 1Ω f⃗ ∈ M
for some
t,x
τ0 > 5/3. Let τ2 be such that 1 <
2
τ2
5
∗
< min(q0 , 2) and 2 −
5
τ0
+
5
τ2
> 0.
There exists a positive constant ϵ which depends only on ν, q0 , τ0 and τ2 such that,
if (t0 , x0 ) ∈ Ω and
ZZ
1
⃗ ⊗ ⃗u(s, y)|2 ds dy < ϵ∗
lim sup
|∇
r→0 r
(t0 −r 2 ,t0 +r 2 )×B(x0 ,r)
2
then there exists a neighborhood Q2 = Qr2 (t0 , x0 ) of (t0 , x0 ) such that 1Q2 ⃗u ∈ M3,τ
2
q0 ,τ2 /2
and 1Q2 p ∈ M2
.
Partial Regularity Results for Weak Solutions
437
10/7,5/3+ϵ
Remark: Assumption 1Q0 f⃗ ∈ M2
(R×R3 )) is borrowed from Kukavica [286].
3/2,5/2
is underlined by Robinson [415] as
The results that 1Q0 ⃗u ∈ M3,5
2 and 1Q0 p ∈ M2
providing a much easier proof for the following step than the sole control on ⃗u ∈ L3 (Q0 )
and p ∈ L3/2 (Q0 ) provided by the original proof of Caffarelli, Kohn and Nirenberg
[74].
The conclusion that (1Q2 ⃗u ∈ M3,5+ϵ
(R × R3 )) is, of course, reminiscent of O’Leary’s
2
assumption in Theorem 13.3.
Proof. We want to prove, for r < r2 and (t, x) ∈ Q2 ,
2q0
3
Wr (t, x) ≤ Cr5(1− τ2 ) and Pr (t, x) ≤ Cr5(1− τ2
)
If 0 < κ < 1, it is enough to prove that, for every n ∈ N, we have
ZZ
1
sup
sup
|⃗u(s, y)|3 dy ds < +∞
5(1− τ3 )
n∈N (t,x)∈Qr2 (t0 ,x0 ) (κn r2 )
2
Qκn r2 (t,x)∩Q2
(13.32)
(13.33)
and
sup
sup
n∈N (t,x)∈Qr2 (t0 ,x0 )
ZZ
1
(κn r2 )5(1−
2q0
τ2
)
|p(s, y)|q0 dy ds < +∞.
(13.34)
Qκn r2 (t,x)∩Q2
We have seen that
Wr (t, x) ≤ Cr1/2 (Ur (t, x) + Vr (t, x))3/2
hence it will be enough to prove that
10
Ur (t, x) + Vr (t, x) ≤ Cr3− τ2
(13.35)
to get the control of Wr (t, x).
We start from the cylinder Q0 = Qr0 (x0 , t0 ) discussed in the previous step. We have
as assumptions that
10
10/7
Fr (t, x) ≤ C∥1Q0 f⃗∥
10/7,τ0
M2
and
r5(1− 7τ0 )
1
lim sup Vr (t0 , x0 ) < ϵ∗
r→0 r
We introduce the reduced quantities
αr (t, x, τ2 ) =
1
r
(Ur (t, x) + Vr (t, x))
3− τ10
2
1
pr (t, x, τ2 ) =
r
5(1−
and
βr (t, x) =
2q0
τ2
)
Pr (t, x)
1
Vr (t, x).
r
(13.36)
(13.37)
438
The Navier–Stokes Problem in the 21st Century (2nd edition)
We may rewrite the conclusions of Lemma 13.3 as
r 10
αr (t, x, τ2 ) ≤ C0 ( ) τ2 αρ (t, x, τ2 )
ρ
10
ρ
+C0 ( )4− τ2 αρ (t, x, τ2 )βρ (t, x)1/2
r
10
5
ρ
+C0 ( )4− τ2 ρ1− τ2 pρ (t, x, τ2 )1/q0 αρ (t, x, τ2 )1/2
r
ρ 3− τ10 2− τ5 + τ5
+C0 ( ) 2 ρ 0 2 αρ (t, x, τ2 )1/2 ∥1Q0 f⃗∥M10/7,τ0
2
r
(13.38)
r 10q0
pr (t, x, τ2 ) ≤C0 ( ) τ2 −2 pρ (t, x, τ2 )
ρ
5q0
2
2
ρ
+ C0 ( )5q0 ( 3 − τ2 ) ρ τ2 −q0 αρ (t, x, τ2 )q0 /2 βρ (t, x)q0 /2
r
(13.39)
and
where the constant C0 does not depend on r, ρ, τ0 nor τ2 . Inequality (13.39) cannot
1
0
be used directly, as the exponent 5q
τ2 − q0 is negative . We therefore introduce the
auxiliary quantity
5
qr (t, x, τ2 ) = rq0 (1− τ2 )−η pr (t, x, τ2 )
10
and rewrite our inequalities (since βρ ≤ ρ2− τ2 αρ ) as
r 10
αr (t, x, τ2 ) ≤ C0 ( ) τ2 αρ (t, x, τ2 )
ρ
10
ρ
+C0 ( )4− τ2 αρ (t, x, τ2 )βρ (t, x)1/2
r
10
ρ
+C0 ( )4− τ2 qρ (t, x, τ2 )1/q0 αρ (t, x, τ2 )1/2
r
ρ 3− τ10 2− τ5 + τ5
+C0 ( ) 2 ρ 0 2 αρ (t, x, τ2 )1/2 ∥1Q0 f⃗∥M10/7,τ0 ,
2
r
10q
5
0
r
qr (t, x, τ2 ) ≤C0 ( ) τ2 −2+q0 (1− τ2 ) qρ (t, x, τ2 )
ρ
5q0
2
2
ρ
+ C0 ( )5q0 ( 3 − τ2 )+ τ2 −q0 αρ (t, x, τ2 )q0 /2 βρ (t, x)q0 /2
r
(13.40)
(13.41)
10
and (since βρ ≤ ρ2− τ2 αρ ) as
r 10
αr (t, x, τ2 ) ≤ C0 ( ) τ2 αρ (t, x, τ2 )
ρ
10
5
ρ
+C0 ( )4− τ2 ρ1− τ 2 αρ (t, x, τ2 )3/2
r
10
ρ
+C0 ( )4− τ2 qρ (t, x, τ2 )1/q0 αρ (t, x, τ2 )1/2
r
10
5
5
ρ
+C0 ( )3− τ2 ρ2− τ0 + τ2 αρ (t, x, τ2 )1/2 ∥1Q0 f⃗∥M10/7,τ0 ,
2
r
0 −2+q (1− 5 )
r 10q
0
τ2
qr (t, x, τ2 ) ≤C0 ( ) τ2
qρ (t, x, τ2 )
ρ
5q0
q0
2
2
5
ρ
+ C0 ( )5q0 ( 3 − τ2 )+ τ2 −q0 ρ 2 (1− τ2 ) αρ (t, x, τ2 )3q0 /2
r
(13.42)
(13.43)
1 Thanks to D. Chamorro and J. He for letting me know that the exponent in equation (13.39) was wrong
0
0
in the first edition: it was estimated as the positive quantity 5q
− q20 instead of the negative quantity 5q
−q0 .
τ
τ
2
2
Partial Regularity Results for Weak Solutions
r 10q0
pr (t, x, τ2 ) ≤C0 ( ) τ2 −2 pρ (t, x, τ2 )
ρ
2
2
ρ
+ C0 ( )5q0 ( 3 − τ2 ) αρ (t, x, τ2 )q0 .
r
439
(13.44)
We first begin with inequalities (13.42), (13.43) and (13.44). Let λ be the positive
10q0
1
λ
exponent λ = min( 10
τ2 , τ2 − 2); we fix κ ∈ (0, 1/2) such that C0 κ ≤ 4 . For r = κρ
we obtain
1
αr (t, x, τ2 ) ≤ αρ (t, x, τ2 )
4
5
+ Cκ ρ1− τ 2 αρ (t, x, τ2 )3/2
(13.45)
+ Cκ qρ (t, x, τ2 )1/q0 αρ (t, x, τ2 )1/2
5
5
+ Cκ ρ2− τ0 + τ2 αρ (t, x, τ2 )1/2 ∥1Q0 f⃗∥M10/7,τ0 ,
2
1
qr (t, x, τ2 ) ≤ qρ (t, x, τ2 ) + Cκ ρ
4
q0
2
(1− τ5
2
)
αρ (t, x, τ2 )3q0 /2 ,
(13.46)
and
1
pr (t, x, τ2 ) ≤ pρ (t, x, τ2 ) + Cκ αρ (t, x, τ2 )q0 .
4
(13.47)
> 0, there exists a ρ0 = ρ0 (κ, τ0 , τ2 , f⃗) such that
q 0
q0
5
C κ ρ0
≤ 14 , Cκ ρ0
∥1Q0 f⃗∥M10/7,τ ≤ 14 and Cκ ρ 2 (1− τ2 ) ≤ 34 4c1κ
.
2
q0
Assume that for some ρ = r1 ≤ ρ0 , we have αρ (t, x, τ2 ) ≤ 1 and qρ (t, x, τ2 ) ≤ 4c1κ
.
n
Then the same will be true for κρ, and by induction for all κ ρ, n ∈ N. Moreover, we
will have pκn r1 (t, x, τ2 ) ≤ pr2 (t, x, τ2 ) + 43 Cκ .
q0
Thus, if αr1 (t, x, τ2 ) ≤ 1 and qr1 (t, x, τ2 ) ≤ 4c1κ
for every (t, x) ∈ Qr2 (t0 , x0 ) for
Now, since 5 < τ2 and 2 −
1− τ5
2
2− τ5
0
+ τ5
2
5
τ0
+
5
τ2
3/2,τ2 /2
2
some r3 > 0, we finally get that 1Qr2 (t0 ,x0 ) ⃗u ∈ M3,τ
and 1Qr2 (t0 ,x0 ) p ∈ M2
2
.
Thus, to finish the proof of Lemma 13.4, it is enough
that for r small enough,
to
qprove
0
we have αr (t0 , x0 , τ2 ) < 1 and qr (t0 , x0 , τ2 ) < 4c1κ
. This will be one by using
inequalities (13.40) and (13.41) and the assumption that limr→0 βr (t0 , x0 ) = 0. We
start from ρ0 and define
αn = ακn ρ0 (t0 , x0 , τ2 ) and qn = qκn ρ0 (t0 , x0 , τ2 ).
We rewrite inequalities (13.40) and (13.41) as
αn+1 ≤
1
αn + un αn + Cκ qn1/q0 αn1/2 + vn αn1/2
4
(13.48)
and
q0
1
qn + wn αn2
4
where un , vn and wn go to 0 when n goes to +∞.
qn+1 ≤
(13.49)
2
Let D be a large positive constant, and θn = αn + Dqnq0 . We have
θn+1
1
1
1
≤ αn + un αn + Cκ D− 2 θn + vn αn + vn + D
4
4
q0
1
qn + wn αn2
4
q2
0
.
440
The Navier–Stokes Problem in the 21st Century (2nd edition)
For a, b, ϵ > 0, we have
2
(a + b)
2
q0
2
2
≤ (a + b + 2ab)
1
q0
≤a
2
q0
+b
2
q0
+2
1
q0
a
1
q0
b
1
q0
≤ (1 + ϵ)a
2
q0
2 q0 q2
+ (1 +
)b 0
4ϵ
and we get
1
1
1
θn+1 ≤ αn + un θn + Cκ D− 2 θn + vn θn + vn
4
4
2
2
2
q
1 2
2 0
+ (1 + ϵ)( ) q 0 D qnq0 + (1 +
)Dwnq0 θn ,
4
4ϵ
so that
θn+1 ≤ Γθn + Xn θn + Yn
with
1
1
1 2
Γ = max( + Cκ D− 2 (1 + ϵ)( ) q0 ),
4
4
and
lim Xn = lim Yn = 0.
n→+∞
n→+∞
If D is large enough and ϵ small enough, we have
Γ<1
so that
lim θn = 0.
n→+∞
Thus, limn→+∞ αn = limn→+∞ qn = 0. Lemma 13.4 is proved.
Step 3: Further estimates on the pressure and the velocity.
We shall now use a more precise representation for the pressure and the velocity. Let
us first notice that the proof of Lemma 13.4 actually conveys more information on ⃗u:
we have indeed proved that
10
Vr (t, x) ≤ Cr3− τ2
⃗ ⊗ ⃗u belongs to M2,τ3 with
and thus that 1Q2 ∇
2
1
τ3
=
1
τ2
+ 15 , so that τ3 > 5/2.
Let Q3 = Qr3 (t0 , x0 ) of (t0 , x0 ) with r3 < r2 . We consider a function ϕ ∈ D(R × R3 )
which is equal to 1 on Q3 and is compactly supported in Q2 . We write
⃗v = ϕ⃗u
and
∂t⃗v = ν∆⃗v + ⃗g +
3
X
⃗
∂i⃗hi − ϕ∇p
i=1
with
⃗ u + ϕf⃗
⃗g = (∂t ϕ)⃗u + (∆ϕ)⃗u − ϕ⃗u · ∇⃗
and
⃗hi = −2(∂i ϕ)⃗u
(13.50)
Partial Regularity Results for Weak Solutions
441
⃗ We start from
Now, we estimate ϕ∇p.
∆p = −
3 X
3
X
∂j ∂l (uj ul )
j=1 l=1
and we consider a function ζ ∈ D(R × R3 ) which is equal to 1 on a neighborhood of
the support of ϕ and is compactly supported in Q2 . We have
⃗ = ⃗γ + ⃗η +
ϕ∇p
3 X
3
X
⃗ j ∂l G ∗ (ϕuj ul )
∇∂
(13.51)
j=1 l=1
with
⃗ ∗ ((∆ζ)p) + ϕ
⃗γ =ϕ∇G
3 X
3
X
⃗ ∗ ((∂j ∂l ζ)uj ul )
∇G
j=1 l=1
− 2ϕ
3
X
⃗ j ∗ ((∂j ζ)p) − ϕ
∇∂
j=1
3 X
3
X
⃗ j G ∗ ((∂l ζ)uj ul )
∇∂
j=1 l=1
and
⃗η = −
3 X
3
X
⃗ j ∂l
∇∂
[ϕ,
](ζuj ul ).
∆
j=1
l=1
We finally find
|⃗v | ≤ C1Q2 (I2 (|⃗g |) +
3
X
I1 (|⃗hi |) + I2 (|⃗γ |) + I2 (|⃗η |) +
i=1
3 X
3
X
I1 (ϕ|uj ul |))
(13.52)
j=1 l=1
(whre Iα is the Riesz potential on the parabolic space R × R3 introduced in Theorem
5.3). This will allow us to prove:
Lemma 13.5.
10/7
Let ⃗u be a suitable solution of the Navier–Stokes equations (with f⃗ ∈ Lt,x (Ω) and
q0
p ∈ Lt,x (Ω) with 1 < q0 ≤ 3/2). Assume moreover that on some neighborhood Q2 =
Qr2 (t0 , x0 ) of (t0 , x0 ), we have
10/7,τ0
• 1Q2 f⃗ ∈ M2
for some τ0 > 5/2
2
• 1Q2 ⃗u belongs to M3,τ
for some τ2 > 5
2
⃗ ⊗ ⃗u belongs to M2,τ3 with
• 1Q2 ∇
2
1
τ3
=
1
τ2
+
1
5
Then, for every r3 < r2 , we have 1Q3 ⃗u ∈ M3,σ
with
2
1
σ
+
1
τ2
< 15 .
Proof. We shall start from assumption 1Q2 ⃗u ∈ M3,τ with τ > 5 and prove that
1Q3 ⃗u ∈ M3,σ
with σ > τ . Some terms are easily controlled:
2
ˆ 1Q2 I2 (|⃗g |) ≤ 1Q2 I2 (A1 ) + 1Q2 I2 (A2 ) with A1 = |∂t ϕ ⃗u| + |∆ϕ ⃗u| + |ϕf⃗| and
⃗ u|. As A1 belongs to M10/7,ρ for every ρ < 5/2, we find that
A2 = |ϕ⃗u · ∇⃗
2
1Q2 I2 A1 belongs to M3,σ
for every σ ≥ 3.
2
442
The Navier–Stokes Problem in the 21st Century (2nd edition)
3,σ
⃗
ˆ as ⃗hi belongs to M3,ρ
2 for every ρ < 5, we find that 1Q2 I1 (|hi |) belongs to M2
for every σ ≥ 3.
5q0
0
q0 , 2
ˆ as 1Q2 p and 1Q2 |⃗u|2 belong to Lqt,x
, we find that ⃗γ belongs to Lqt 0 L∞
.
x ⊂M
q0 ,ρ
As ⃗γ is compactly supported, we find that it belongs to M2 for every ρ < 5/2,
so that 1Q2 I2 (|⃗γ |) belongs to M3,σ
for every σ ≥ 3.
2
Thus, we are left with the estimation of 1Q2 I2 (A2 ), 1Q2 (|⃗η |) and 1Q2 I1 (ϕ|uj ul |).
3/2,γ
3/2,δ
First, let us notice that, when 1Q2 |⃗u|2 belongs to M2
, then ⃗η belongs to M2
1
1
1
with δ = γ + 5 : indeed, using the Calderón commutator theorem, we see that
∂ ∂ ∂
w →
7 [ϕ, i ∆j k ]w is bounded on Lpt Lqx for 1 ≤ p ≤ ∞, 1 < q < +∞. If we want
RR
to estimate Qr (t,x) |⃗η |3/2 ds dy, it is enough to do it for r < r0 (as ⃗η belongs to
L3/2 L3/2 , hence the behavior on large cylinders is well controlled); then one writes
⃗η = ⃗ηr + ⃗η[r] with
⃗ηr = −
3 X
3
X
⃗ j ∂l
∇∂
](1Q2r (t,x) ζuj ul ).
∆
[ϕ,
j=1 l=1
and
⃗η[r] = −
3 X
3
X
⃗ j ∂l
∇∂
]((1 − 1Q2r (t,x) ζuj ul ).
∆
[ϕ,
j=1 l=1
Then, we have
ZZ
ZZ
|⃗ηr |3/2 ds dy ≤ C
Qr (t,x)
3
|⃗u|3 ds dy ≤ Cr5(1− 2γ ) ∥1Q2 |⃗u|2 ∥
Q2r (t,x)∩Q2
3/2
3/2,γ
M2
while, on Qr (t, x), we have
Z
1
1Q (y − z)|⃗u2 (s, y − z)|2 dz;
|z|4 2
|⃗η[r] (s, y)| ≤ C
|z|>r
thus,
Z
∥1Qr (t,x) ⃗η[r] ∥M3/2,γ ≤ C
2
|z|>r
dz
1
∥1Q2 |⃗u|2 ∥M3/2,γ = C ′ ∥1Q2 |⃗u|2 ∥M3/2,γ
4
2
2
|z|
r
and
ZZ
3
|⃗η[r] |3/2 ds dy ≤Cr−3/2 r5(1− 2γ ) ∥1Q2 |⃗u|2 ∥
Qr (t,x)
3
3/2
3/2,γ
M2
3/2
=Cr5(1− 2δ ) ∥1Q2 |⃗u|2 ∥
3/2,γ
M2
Let
1
τ2
=
1
5
− α. Assume that
6
⃗ u ∈ M5
ˆ ϕ⃗u · ∇⃗
2
6 σ
5 ρ ,σ
M2
with
,ρ
1
σ
with
=
1
ρ
−
1
ρ
2
5
1
τ
>
1
α.
Then, we have:
=
1
τ3
+
1
τ
=
1
τ
⃗ u|) belongs to
and I2 (|ϕ⃗u · ∇⃗
3/2,γ
ˆ 1Q2 |⃗u|2 belongs to M2
1
ρ
=
1
τ
+
2
5
=
1
τ
+
2
5
⃗ u|) belongs to
− α. Hence, I2 (|ϕ⃗u · ∇⃗
− α. Moreover,
1
ρ
>
3
τ
− α > 3 σ1 , so that 3 <
6σ
5 ρ,
M3,σ
2
with
1
γ
=
− α, and I2 (|⃗η |) belongs to
1
τ2
3
+ τ1 , hence ⃗η belongs to M22
M3,σ
2
with
1
σ
=
1
τ
− α.
,ρ
with
Partial Regularity Results for Weak Solutions
3σ
ˆ finally, we have that I1 (1Q2 |⃗u|2 ) belongs to M22γ
1
τ
−α<
1
2γ ,
2
and thus I1 (1Q2 |⃗u| ) belongs to
1
τ
We then iterate the estimate, changing
1
σ
< α: if τ =
1
α,
we write 1Q2 ⃗u ∈
M3,τ
2
′
1
τ
into
σ
443
with
1
γ
=
1
τ2
+
1
τ
1
σ
≤ α (and even
and
1
σ
=
M3,σ
2
− α, until we obtain
1
τ′
with α <
< 2α. . . ) .
Step 4: End of the proof.
We then end the proof with the lemma:
Lemma 13.6.
10/7
Let ⃗u be a suitable solution of the Navier–Stokes equations (with f⃗ ∈ Lt,x (Ω) and
q0
p ∈ Lt,x (Ω) with 1 < q0 ≤ 3/2). Assume moreover that on some neighborhood Q2 =
Qr2 (t0 , x0 ) of (t0 , x0 ), we have
10/7,τ0
• 1Q2 f⃗ ∈ M2
for some τ0 > 5/2
2
• 1Q2 ⃗u belongs to M3,τ
for some τ2 > 5
2
• 1Q2 ⃗u belongs to M3,σ
for some σ with
2
⃗ ⊗ ⃗u belongs to M2,τ3 with
• 1Q2 ∇
2
1
τ3
1
σ
1
τ2
=
+
1
τ2
+
1
5
<
1
5
Then, for every r3 < r2 , ⃗u is Hölderian on Qr3 (t0 , x0 ).
Proof. We write again
∂t⃗v = ν∆⃗v + ⃗g +
3
X
∂i⃗hi − ⃗γ − ⃗η −
i=1
3
XX
⃗ j ∂l ∗ (ϕuj ul )
∇∂
j=1 l=1
with
⃗ u + ϕf⃗, where (∂t ϕ)⃗u + (∆ϕ)⃗u + ϕf⃗ belongs to
ˆ ⃗g = (∂t ϕ)⃗u + (∆ϕ)⃗u − ϕ⃗u · ∇⃗
6
10/7,min(τ0 ,τ2 )
⃗ u belongs to M 5 ,ρ with
M2
(with min(τ0 , τ2 ) > 5/2) and ϕ⃗u · ∇⃗
2
1
1
1
1
1
1
2
ρ = τ3 + σ = τ2 + 5 + σ < 5 (so that ρ > 5/2)
2
ˆ ⃗hi = −2(∂i ϕ)⃗u belongs to M3,τ
with τ2 > 5
2
P
P
P3 ⃗
3
3
⃗
⃗
ˆ ⃗γ = ϕ∇G∗((∆ζ)p)+ϕ j=1 l=1 ∇G∗((∂
j ∂l ζ)uj ul )−2ϕ
j=1 ∇∂j ∗((∂j ζ)p)−
P3 P3 ⃗
5q
5q0
q0 , 20
ϕ j=1 l=1 ∇∂j G ∗ ((∂l ζ)uj ul ) belongs to M
(with 2 > 5/2)
P3 P3
⃗
3/2,ρ
∇∂ ∂
ˆ ⃗η = − j=1 l=1 [ϕ, ∆j l ](ζuj ul ): as ζuj ul belongs to M2
with ρ1 = τ12 + τ1 ,
3/2,σ
with
1
ρ
1
τ2
we find that ⃗η belongs to M2
ˆ ϕuj ul belongs to
3/2,ρ
M2
with
=
1
σ
+
1
τ
=
<
1
τ2
1
5
+
1
σ
+
1
5
<
2
5
(so that σ > 5/2)
(so that ρ > 5).
and we apply Proposition 13.4.
13.10
Parabolic Hausdorff Dimension of the Set of Singular Points
To more accurately describe the singularities in R × R3 , Caffarelli, Kohn, and Nirenberg
used a notion of parabolic Hausdorff dimension, adapted to the scaling properties of the
Navier–Stokes equations:
444
The Navier–Stokes Problem in the 21st Century (2nd edition)
Definition 13.6 (Parabolic Hausdorff measure).
(i) For a sequence of open parabolic cylinders Q = (Q((ti , xi ), ri ))i∈N of R × Rd (where
2
Q((t
for α > 0, we define σ̃α (Q) =
P i , xαi ), ri ) = {(t, x) / |t − ti | ≤ ri and |x − xαi | ≤ ri }) and
d
r
.
The
parabolic
Hausdorff
measure
P
on
R
×
R
is defined for a Borel subset
i∈N i
B ⊂ R × Rd by
P α (B) = lim min{σ̃α (Q) /B ⊂ ∪i∈N Q((ti , xi ), ri ), sup ri < δ}
δ→0
i∈N
(ii) The parabolic Hausdorff dimension dH (B) of a Borel subset of R × Rd is defined as
dP (B) = inf{α / P α (B) = 0} = sup{α / P α (B) = ∞}.
We may now state the result of Caffarelli, Kohn, and Nirenberg:
Caffarelli–Kohn–Nirenberg regularity theorem
Theorem 13.9 (Dimension of the singular set).
Let ⃗u be a weak solution for the Navier–Stokes equations on (0, T ) × R3 , which is a
suitable solution on the cylinder Q0 = (a, b) × B(x0 , r0 ) (with pressure p ∈ Lqt 0 L1x (Q)
10/7,τ0
with τ0 > 5/2). Let Σ be the smallest closed set in Q
with q0 > 1 and 1Q0 f⃗ ∈ M2
so that ⃗u is locally bounded on Q0 − Σ. Then P 1 (Σ) = 0.
Proof. Let δ > 0. Let (t, x) ∈ Σ. According to Theorem 13.8, we know that
ZZ
1
⃗ ⊗ ⃗u|2 dy ds ≥ ϵ∗ > 0.
lim sup
|∇
r→0 r
Q(x,r)
We fix ϵ0 such that 0 < ϵ0 < ϵ∗ . Then, we have Σ ⊂ ∪Q∈Qδ Q, where Qδ is the collection of
RR
⃗ ⊗
open cylinders Q((t, x), r) = (t − r2 , t + r2 ) × B(x, r) so that Q ⊂ Q0 , r < δ and Q |∇
p
⃗u|2 dy ds ≥ ϵ0 r. We use the parabolic distance dP ((t, x), (s, y)) = max(|y − x|, 2|t − s|);
for this distance, an open cylinder Q((t, x), r) is a ball B(t, x), r), hence we may apply the
Vitali covering lemma and find a countable subcollection Q[δ] = (Q((ti , xi ), ri ))i∈N of Qδ so
that Σ ⊂ ∪i∈N Q((ti , xi ), 5ri ) and i =
̸ j ⇒ Qi ∩ Qj = ∅. We then have:
Z Z
X
5
⃗ ⊗ ⃗u|2 dx dt.
σ̃1 (Q[δ] ) = 5
ri ≤
|∇
ϵ0
(t,x)∈Q0 ,dP ((t,x),Σ)≤δ
i∈N
⃗ ⊗ ⃗u|2 dx dt. In particular P 1 (Σ) < ∞; hence, the Lebesgue
This gives P 1 (Σ) ≤ ϵ5∗ Σ |∇
measure of Σ is equal to 0 and finally P 1 (Σ) = 0.
RR
13.11
On the Role of the Pressure in the Caffarelli, Kohn, and
Nirenberg Regularity Theorem
In June 2013, the 8th Japanese-German International Workshop on Mathematical Fluid
Dynamics was held at Waseda University (Japan). Choe [122] announced that no assumption
had to be made on the pressure to get the Caffarelli, Kohn and Nirenberg regularity theorem.
Partial Regularity Results for Weak Solutions
445
Of course, if no assumption is made on the pressure, one cannot speak of the local energy
inequality and of suitable solutions, as the term div(p⃗u) might be meaningless. Choe’s talk
explained that a new inequality, based on an idea of Jin about Cacciopolli inequalities for
the Stokes problem, was satisfied by any weak solution ⃗u ∈ L∞ L2 ∩ L2 H 1 and that this
inequality allowed to prove
the force was regular enough
RR that, under the assumption that
⃗ ⊗ ⃗u(s, y)|2 ds dy was small enough, ⃗u was
and that lim supr→0 1r (t0 −r2 ,t0 +r2 )×B(x0 ,r) |∇
Hölderian on a neighborhood of (t0 , x0 ).
Chamorro, Lemarié-Rieusset and Mayoufi [102] studied Choe’s ideas. As explained in
Mayoufi’s Ph.D. [356], the inequality may be not fulfilled by every weak solution (as Choe’s
RR
⃗ u) dx dt =
proof implicitly used the equality, for a non-negative test function φ,
φ⃗u.(⃗u · ∇⃗
RR
1
2 ⃗
|⃗u| ⃗u·∇⃗
φ dx dt, while the term on the left-hand side of the equality is not well defined).
−2
Moreover, there is no hope of proving that ⃗u is locally Hölderian without any assumption
on the pressure, as Serrin’s counterexample (presented on page 406) has no regularity with
respect to the time variable. Chamorro, Lemarié-Rieusset and Mayoufi extended the notion
of suitable solutions by modifying Definitions 13.4 and 13.5 in the following way:
Definition 13.7.
We call (H̃CKN ) the following set of hypotheses:
1. ⃗u, p and f⃗ are defined on a domain Ω ⊂ R × R3
2 1
2
2. on Ω, ⃗u belongs to L∞
t Lx ∩ Lt Ḣx :
Z
ZZ
⃗ ⊗ ⃗u|2 dt dx < +∞
|⃗u(t, x)|2 dx < +∞ and
|∇
sup
t∈R
(t,x)∈Ω
Ω
3. p belongs to D′ (Ω)
10/7
4. on Ω, f⃗ is a divergence free vector field in Lt,x (Ω)
5. ⃗u is a solution of the Navier–Stokes equations on Ω: div ⃗u = 0 and
⃗ u + f⃗ − ∇p
⃗ in D′ (Ω)
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
(13.53)
This is the same definition as for (HCKN ) (Definition 13.4), but for p: we no longer
assume that p belongs to Lqt 0 L1x (Ω) for some q0 > 1. It means that the distribution
⃗ ⊗ ⃗u|2 + 2⃗u · f⃗ − div((|⃗u|2 + 2p)⃗u)
µ = −∂t |⃗u|2 + ν∆|⃗u|2 − 2ν|∇
is no longer well defined on Ω.
In order to overcome this difficulty, Chamorro, Lemarié-Rieusset and Mayoufi proved
the following result:
Proposition 13.5.
R
R
Let (⃗u, p) satisfy (H̃CKN ). Let γ ∈ D(R) and let θ ∈ D(R3 ), with γ dt = θ dx = 1, and
let φϵ,α (t, x) = ϵ31α γ( αt )θ( xϵ ) (where α > 0 and ϵ > 0). The distributions ⃗u ∗ φϵ,α and p ∗ φϵ,α
(with convolution in both time and space variables) are well defined on O ⊂ Ω as soon as
we have d(Ō, R × R3 \ Ω) > 2(ϵ + α).
Moreover, the limit limϵ→0 limα→0 div((p ∗ φϵ,α )(⃗u ∗ φϵ,α )) is well defined in D′ (Ω) and
does not depend on the choices of θ and γ. Thus, the distribution
⃗ ⊗ ⃗u|2 + 2⃗u · f⃗ − div(|⃗u|2 ⃗u)
µ = −∂t |⃗u|2 + ν∆|⃗u|2 − 2ν|∇
− 2 lim lim div((p ∗ φϵ,α )(⃗u ∗ φϵ,α ))
ϵ→0 α→0
is well defined on Ω.
(13.54)
446
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, we may extend the notion of suitable solutions (Definition 13.5) to the notion of
dissipative solutions (where the term “dissipative” is borrowed from Duchon and Robert
[159]):
Dissipative solutions
Definition 13.8.
The solution ⃗u is dissipative if the distribution µ is a non-negative locally finite measure
on Ω.
With this definition, we may extend Caffarelli, Kohn and Nirenberg’s theorems [74]:
Theorem 13.8 gave a criterion for local Hölderianity, Theorem 13.9 gave an estimate of the
Hausdorff dimension of the singular set of a suitable solution. We have the following results
for dissipative solutions:
Caffarelli–Kohn–Nirenberg regularity criterion
Theorem 13.10.
Let Ω be a bounded domain of R × R3 . Let (⃗u, p) a weak solution on Ω of the Navier–
Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Assume that
• (⃗u, p, f⃗) satisfies the conditions (H̃CKN ): ⃗u ∈ L∞ L2 ∩ L2t Hx1 (Ω), p ∈ D′ (Ω),
div f⃗ = 0 and f⃗ ∈ L10/7 L10/7 (Ω)
• ⃗u is dissipative
• f⃗ ∈ L2 H 1 .
(A) There exists a positive constant ϵ∗ which depends only on ν and τ0 such that, if for
some (t0 , x0 ) ∈ Ω, we have
ZZ
1
⃗ ⊗ ⃗u|2 ds dx < ϵ∗
lim sup
|∇
r
2
2
r→0
(t0 −r ,t0 +r )×B(x0 ,r)
then ⃗u is bounded in a neighborhood of (t0 , x0 ).
(B) Let Σ be the smallest closed set in Q so that ⃗u is locally bounded on Q0 − Σ. Then
P 1 (Σ) = 0.
Proof. We are now going to prove Proposition 13.5 and Theorem 13.10. With no loss of
generality (due to the local character of the properties studied in Proposition 13.5 and
Theorem 13.10), we may assume that Ω = (a, b) × B, where B is an open ball in R3 .
Let ω
⃗ = curl ⃗u. We may write the Navier–Stokes equations as div ⃗u = 0 and
2
⃗ + |⃗u| ) + f⃗.
∂t ⃗u = ∆⃗u − ω
⃗ ∧ ⃗u − ∇(p
2
Partial Regularity Results for Weak Solutions
447
Let O be compactly embedded into Ω: Ō ⊂ Ω, and let ψ(t, x) = α(t)γ(x) a function in
D(Ω) such that ψ = 1 on a neighborhood of Ō. We define
⃗v = −
1⃗
∇ ∧ (ψ⃗
ω ).
∆
(13.55)
Note that ⃗v is defined on R × R3 . It can be seen locally (i.e., on O) as a perturbation of ⃗u:
indeed, we have
⃗ ∧ (ψ⃗u) − (∇ψ)
⃗
ψ⃗
ω=∇
∧ ⃗u
and
⃗ ∧ (∇
⃗ ∧ (ψ⃗u)) = −∆(ψ⃗u) + ∇(div(ψ⃗
⃗
⃗ u · ∇ψ)
⃗
∇
u)) = −∆(ψ⃗u) + ∇(⃗
so that
1 ⃗
⃗
⃗ ∧ (⃗u ∧ ∇ψ)
⃗
⃗.
∇(⃗u · ∇ψ)
+∇
= ψ⃗u + V
∆
On O, ψ⃗u = ⃗u; moreover, we write, for (t, x) ∈ O and α ∈ N3 ,
⃗v = ψ⃗u −
⃗ (t, x)| ≤ Cα
|∂xα V
⃗
|∇γ(y)|
|⃗u(t, y)| dy ≤ Cα,O ∥1B ⃗u(t, .)∥2
|x − y|4+|α|
Z
⃗ ∈ L∞ (O).
so that ∂xα V
t,x
We have
∂t⃗v = −
1⃗
1⃗
∇ ∧ ((∂t ψ)⃗
ω) − ∇
∧ (ψ∂t ω
⃗)
∆
∆
with
⃗ ∧ (⃗
⃗ ∧ f⃗.
∂t ω
⃗ = ν∆⃗
ω−∇
ω ∧ ⃗u) + ∇
First, we notice that
Now, we write
1 ⃗
∆∇
∧ ((∂t ψ)⃗
ω ) = 0 on O.
⃗ ∧ ⃗u
ψ∆⃗
ω =ψ∆∇
⃗ ∧ ⃗u) + (∆ψ)∇
⃗ ∧ ⃗u − 2
=∆(ψ ∇
X
⃗ ∧ ⃗u)
∂i ((∂i ψ)∇
i
⃗ ∧ ((∆ψ)⃗u) − (∇∆ψ)
⃗
=∆(ψ⃗
ω) + ∇
∧ ⃗u
X
X
⃗ ∧ ((∂i ψ)⃗u) + 2
⃗
−2
∂i ∇
∂i ((∂i ∇ψ)
∧ ⃗u)
i
i
⃗
=∆(ψ⃗
ω) + W
so that
−
1⃗
1⃗
⃗ ∧ (ψ⃗
⃗.
∇ ∧ (ψ∆⃗
ω ) = −∇
ω) − ∇
∧W
∆
∆
⃗ ∧ (ψ⃗
⃗ ∧ω
⃗ = ∆⃗v , as we have ∆V
⃗ = 0 on O.
On O, we have −∇
ω) = ∇
⃗ = ∆⃗u = ∆⃗v − ∆V
′
Thus, we find that, in D (O),
−
1⃗
⃗1
∇ ∧ (ψ∆⃗
ω ) = ∆⃗v + G
∆
⃗ 1 = 0 and G
⃗1 = − 1 ∇
⃗ ∧W
⃗ ∈ L∞
with div G
t,x (O).
∆
Similarly, we write (using div f⃗ = 0)
−
1⃗
1⃗
⃗ ∧ f⃗) =ψ f⃗ − 1 ∇(
⃗ f⃗ · ∇ψ)
⃗
⃗
∇ ∧ (ψ ∇
− ∇
∧ (f⃗ ∧ ∇ψ)
∆
∆
∆
448
The Navier–Stokes Problem in the 21st Century (2nd edition)
and we find that, in D′ (O),
−
1⃗
⃗ ∧ f⃗) = f⃗ + G
⃗2
∇ ∧ (ψ ∇
∆
⃗ 2 = 0 and G
⃗ 2 ∈ L2t L∞
with div G
x (O).
Finally, we write
⃗ ∧ (⃗
⃗ ∧ (⃗
⃗
ψ∇
ω ∧ ⃗u) =∇
ω ∧ ψ⃗u) − (∇ψ)
∧ (⃗
ω ∧ ⃗u)
so that
1⃗
⃗ ∧ (⃗
⃗ 3+G
⃗3
∇ ∧ (ψ ∇
ω ∧ ⃗u)) = −⃗
ω ∧ (ψ⃗u) − ∇P
∆
with
P3 =
1
div (⃗
ω ∧ ψ⃗u)
∆
and
⃗ ∧ ((∇ψ)
⃗
⃗3 = − 1 ∇
∧ (⃗
ω ∧ ⃗u)).
G
∆
⃗ 3 = 0 and G
⃗ 3 ∈ L2t L∞
u belongs to L2 L6 ∩L∞ L2 ⊂
On O, we have div G
x (O). Moreover, as ψ⃗
6 18/7
3/2 9/8
3/2 9/5
L L
, we have ω
⃗ ∧ ψ⃗u ∈ L L
and thus P3 ∈ L L . In particular, on O, we find
P3 ∈ L3/2 L3/2 (O).
⃗ ∧ (ψ⃗u). Recalling that ψ⃗u = ⃗v − V
⃗ , we write that
On O, we have ω
⃗ =∇
⃗ ∧ (ψ⃗u)) ∧ (ψ⃗u) = −(∇
⃗ ∧ ⃗v ) ∧ ⃗v + Z
⃗
−(∇
⃗ = (∇
⃗ ∧V
⃗ ) ∧ (ψ⃗u) + (∇
⃗ ∧ (ψ⃗u)) ∧ V
⃗ + (∇
⃗ ∧V
⃗)∧V
⃗ . On O, Z
⃗ belongs to L2 L2 , and
with Z
we write
⃗ =G
⃗ 4 − ∇P
⃗ 4
1O Z
⃗ 4 = P(1O Z),
⃗ so that div G
⃗ 4 = 0 and G
⃗ 4 ∈ L2 L2 , while P4 = − 1 div(1O Z)
⃗ ∈ L2 L6 .
with G
∆
1 ⃗
Thus far, we have obtained the following properties on ⃗v = − ∆
∇ ∧ (ψ⃗
ω ):
⃗ with V
⃗ ∈ L∞
ˆ on O, ⃗v = ⃗u + V
t Lipx (O)
2 1
2
v=0
ˆ ⃗v ∈ L∞
t Lx (O) ∩ Lt Hx (O) and div ⃗
ˆ ⃗v is a solution of the Navier–Stokes equations in D′ (O):
⃗ v − ∇P
⃗ +G
⃗
∂t⃗v = ν∆⃗v − ⃗v · ∇⃗
⃗ = νG
⃗ 1 + f⃗ + G
⃗2 + G
⃗3 + G
⃗ 4 , so that G
⃗ ∈ L2t L2x (O) and div G
⃗ = 0, and
with G
2
3/2 3/2
|⃗
v|
P = P3 + P4 − 2 , so that P ∈ Lt Lx (O).
The next step is a generalization of the formula of Duchon and Robert [159]. While in the
proof of Duchon and Robert’s theorem (Theorem 13.7), we used a mollifier θϵ (x) = ϵ13 θ( xϵ )
in the space variable and computed ∂t (|θϵ ∗ ⃗u|2 ), our proof of Proposition 13.5 will use a
mollifier φϵ,α in both time and space variables.
Partial Regularity Results for Weak Solutions
449
Let O′ be a relatively compact open subset of O and let ϵ, α be small enough, so that
⃗uϵ,α = (γα ⊗ θϵ ) ∗ ⃗u and ⃗v = (γα ⊗ θϵ ) ∗ ⃗v are well defined on O′ and involves only the values
of ⃗u and ⃗v in O. As ⃗uϵ,α is now a smooth function, we find that
∂t |⃗uϵ,α |2 = 2⃗uϵ,α .∂t ⃗uϵ,α
and thus
⃗ ⊗ ⃗uϵ,α |2 + 2⃗uϵ,α .(φϵ,α ∗ f⃗)
∂t |⃗uϵ,α |2 =ν∆|⃗uϵ,α |2 − 2ν|∇
− 2 div((p ∗ φϵ,α )⃗uϵ,α ) − 2⃗uϵ,α .φϵ,α ∗ div(⃗u ⊗ ⃗u)
We have similarly
⃗ ⊗ ⃗vϵ,α |2 + 2⃗vϵ,α .(φϵ,α ∗ G)
⃗
∂t |⃗vϵ,α |2 =ν∆|⃗vϵ,α |2 − 2ν|∇
− 2 div((P ∗ φϵ,α )⃗vϵ,α ) − 2⃗vϵ,α .φϵ,α ∗ div(⃗v ⊗ ⃗v )
Let ⃗uϵ = θϵ ∗⃗u (only convolution in the space variable is involved), so that ⃗uϵ,α = γα ∗⃗uϵ .
As α → 0+ , we have the strong convergence of ⃗uϵ,α to ⃗uϵ in L2 L2 (O′ ) and in L3 L3 (O′ ), of
⃗ ⊗ ⃗uϵ,α to ∇
⃗ ⊗ ⃗uϵ in L2 L2 (O′ ), of φϵ,α ∗ (⃗u ⊗ ⃗u) to θϵ ∗ (⃗u ⊗ ⃗u) in L3/2 L3/2 (O′ ), of φϵ,α ∗ f⃗
∇
2 2
⃗
to θϵ f in L L (O′ ) and the *-weak convergence of ⃗uϵ,α to ⃗uϵ in L∞ L2 so that we find
∂t |⃗uϵ |2 = lim+ ∂t |⃗uϵ,α |2
α→0
⃗ ⊗ ⃗uϵ |2 + 2⃗uϵ .(θϵ ∗ f⃗) − 2⃗uϵ .θϵ ∗ div(|⃗u ⊗ ⃗u)
=ν∆|⃗uϵ |2 − 2ν|∇
(13.56)
− 2 lim+ div((p ∗ φϵ,α )⃗uϵ,α )).
α→0
We now introduce
µϵ = 2⃗uϵ .θϵ ∗ div(⃗u ⊗ ⃗u) − div(|⃗u|2 ⃗u)
and
Mϵ = 2⃗vϵ .θϵ ∗ div(⃗v ⊗ ⃗v ) − div(|⃗v |2⃗v ).
⃗ ⊗ ⃗uϵ to ∇
⃗ ⊗ ⃗u in L2 L2 (O′ )
We have the strong convergence of ⃗uϵ to ⃗u in L2 L2 (O′ ), of ∇
3 3
′
2 1
′
2 2
′
⃗
⃗
and in L L (O ), of ⃗uϵ to ⃗u in L H (O ), of θϵ ∗ f to f in L L (O ) so that we find
∂t |⃗u|2 = lim+ ∂t |⃗uϵ |2
ϵ→0
⃗ ⊗ ⃗u|2 + 2⃗u · f⃗ − div(|⃗u|2 ⃗u)
=ν∆|⃗u|2 − 2ν|∇
(13.57)
− lim+ (µϵ + 2 lim+ div((p ∗ φϵ,α )⃗uϵ,α )).
ϵ→0
α→0
We have a similar result for ⃗vϵ , but with a better convergence: φϵ,α ∗P converges strongly
to θϵ ∗ P in L3/2 L3/2 (O′ ), and θϵ ∗ P converges strongly to P in L3/2 L3/2 (O′ ), so that
∂t |⃗v |2 = lim ∂t |⃗vϵ |2
ϵ→0+
⃗ ⊗ ⃗v |2 + 2⃗v · G
⃗ − div(|⃗v |2⃗v )
=ν∆|⃗v |2 − 2ν|∇
− 2 div(P⃗v ) − lim+ Mϵ .
ϵ→0
(13.58)
450
The Navier–Stokes Problem in the 21st Century (2nd edition)
We now rewrite µϵ and Mϵ as in the Theorem of Duchon and Robert. Let δy be defined
by δy h(t, x) = h(t, x − y) − h(t, x) and let Tϵ be the trilinear operator
Tϵ (⃗u, ⃗v , w)(t,
⃗
x) =
−
3 Z
X
∂k θϵ (y) δy uk (t, x) (δy ⃗v (t, x)δy w(t,
⃗ x)) dy
k=1
+2
3 Z
X
∂k θϵ (y) δy uk (t, x) (δy ⃗v (t, x).(θϵ ∗ w(t,
⃗ x) − w(t,
⃗ x))) dy
k=1
(notice that θϵ ∗ w(t,
⃗ x) − w(t,
⃗ x) =
R
θϵ (z)δz w(t,
⃗ x) dz). Duchon and Robert proved that
lim µϵ − Tϵ (⃗u, ⃗u, ⃗u) = 0
ϵ→0
and we have similarly
lim Mϵ − Tϵ (⃗v , ⃗v , ⃗v ) = 0
ϵ→0
in D′ (O′ ).
⃗ on O, so that
Now, we write ⃗v = ⃗u + V
⃗ , ⃗v , ⃗v ) + Tϵ (⃗u, V
⃗ , ⃗v ) + Tϵ (⃗u, ⃗u, V
⃗)
Tϵ (⃗v , ⃗v , ⃗v ) − Tϵ (⃗u, ⃗u, ⃗u) = Tϵ (V
This gives
ZZ
|Tϵ (⃗v , ⃗v , ⃗v ) − Tϵ (⃗u, ⃗u, ⃗u)| dt dx ≤
O′
⃗ (t, x)| Z Z Z
|δy V
1
C sup sup
(|δy ⃗u(t, x)|2 + |δy ⃗v (t, x)|2 ) dy dt dx
3
ϵ
ϵ
′
|y|<ϵ (t,x)∈O
O
|y|<ϵ
ZZ
⃗ ∥L∞ Lip sup
≤ C∥V
(|δy ⃗u(t, x)|2 + |δy ⃗v (t, x)|2 ) dt dx
|y|<ϵ
O′
→ϵ→0+ 0.
Thus, we find that
lim Mϵ − µϵ = 0.
ϵ→0+
As µ = limϵ→0+ Mϵ exists (due to equality (13.58)), we find that limϵ→0+ µϵ exists (and is equal to the same limit µ). Equality (13.57) then gives the existence of
limϵ→0+ limα→0+ div((p ∗ φϵ,α )⃗uϵ,α ): Proposition 13.5 is proved.
We now prove Theorem 13.10. Assumption that ⃗u is dissipative gives, using again equality (13.57), that µ = limϵ→0+ µϵ is a non-negative locally finite measure. Using equality
(13.58), we find that ⃗v is suitable.
Moreover, if Qr (t0 , x0 ) ⊂ O, we have
ZZ
1
⃗ ⊗V
⃗ |2 ds dx ≤ C∥V
⃗ ∥2L∞ Lip r4
|∇
r
2
2
(t0 −r ,t0 +r )×B(x0 ,r)
so that, for (t0 , x0 ) ∈ O,
ZZ
1
⃗ ⊗ ⃗v |2 ds dx =
lim sup
|∇
r→0 r
(t0 −r 2 ,t0 +r 2 )×B(x0 ,r)
ZZ
1
⃗ ⊗ ⃗u|2 ds dx.
lim sup
|∇
r→0 r
(t0 −r 2 ,t0 +r 2 )×B(x0 ,r)
Partial Regularity Results for Weak Solutions
451
⃗ that appears in the Navier–Stokes equations whose ⃗v is a solution
Recall that the force G
2 2
⃗
⃗ ∈ M10/7,2 . Let us assume that
satisfies G ∈ L L (O), so that 1O G
2
ZZ
1
⃗ ⊗ ⃗v |2 ds dx < ϵ∗
lim sup
|∇
2
2
r→0 r
(t0 −r ,t0 +r )×B(x0 ,r)
for some small enough ϵ∗ . As 2 < 5/2, we cannot apply Theorem 13.8 (Caffarelli, Kohn
and Nirenberg’s theorem) to ⃗v ; but, as 2 > 5/3, we may apply Lemma 13.4 (Kukavica’s
theorem) and find that, on a neighborhood of (t0 , x0 ), ⃗v belongs to M3,τ
2 for some τ > 5. As
3,τ
3,τ
⃗ ∈ L∞
⃗
1O V
,
we
have
1
V
∈
M
,
hence
⃗
u
belongs
to
M
on
a
neighborhood
of (t0 , x0 ).
O
t,x
2
2
2 1
⃗
Thus, as f ∈ L H , we can apply Theorem 13.3 (O’Leary’s theorem) and find that ⃗u is
bounded on a neighborhood of (t0 , x0 ). Point (A) of Theorem 13.10 is proved.
The proof of point (B) is similar to the proof of Theorem 13.9. Theorem 13.10 is proved.
As a final remark, we may check that no regularity in the t variable is provided for the
solution ⃗u in Theorem 13.10. Indeed, let us consider again Serrin’s example on page 406. ⃗u
is a solution of the Navier–Stokes equations on (0, 1) × B(0, 1) (with forcing term f⃗ = 0)
⃗
given by ⃗u = ζ(t)∇ψ,
where ψ is a harmonic function on R3 and ζ a bounded function on
2
R. The pressure p is given by p = − |⃗u2| − ∂t α ψ. Let us compute the distribution µ given
by Equation (13.54):
⃗ ⊗ ⃗u|2 − div(|⃗u|2 ⃗u)
µ = −∂t |⃗u|2 + ν∆|⃗u|2 − 2ν|∇
− 2 lim lim div((p ∗ φϵ,α )(⃗u ∗ φϵ,α ))
ϵ→0 α→0
⃗ ⊗ ⃗u|2 = 2ν⃗u.∆⃗u = 0, so that
As ∆⃗u = 0, we have ν∆|⃗u|2 − 2ν|∇
⃗ 2 ∂t (ζ(t)2 ) − div(|⃗u|2 ⃗u)
µ = − |∇ψ|
− 2 lim lim div((p ∗ φϵ,α )(⃗u ∗ φϵ,α ))
ϵ→0 α→0
1 x
1 t
2
2
2
2
⃗
⃗
= − |∇ψ| ∂t (ζ(t) ) + lim lim | 3 θ( ) ∗ ∇ψ| ∂t ( γ( ) ∗ ζ)
ϵ→0 α→0 ϵ
ϵ
α α
− div(|⃗u|2 ⃗u) + lim lim div((|⃗u|2 ∗ φϵ,α )(⃗u ∗ φϵ,α ))
ϵ→0 α→0
=0.
Thus, ⃗u is dissipative; however, ⃗u has no regularity at all with respect to t (if ζ is nowhere
continuous).
Chapter 14
A Theory of Uniformly Locally L2 Solutions
14.1
Uniformly Locally Square Integrable Solutions
We recall some basic results on uniformly locally square integrable solutions as they
were described in Basson [24] and Lemarié-Rieusset [313], with a slight modification: we
include forcing terms in the equations, as in Kikuchi and Seregin’s paper [261].
We are thus considering the equations

⃗
∂t ⃗u + div(⃗u ⊗ ⃗u) = ν∆⃗u + f⃗ − ∇p
(14.1)
div ⃗u = 0

⃗u(0, .) = ⃗u0
where ⃗u0 is a divergence-free uniformly locally square integrable vector field and f⃗ = div F,
where F is a uniformly square integrable tensor:
Z
|⃗u0 (y)|2 dy < +∞
sup
x∈Rd
and
|x−y|<1
Z 1Z
sup
x∈Rd
|F(s, y)|2 dy ds < +∞
|x−y|<1
0
and we are looking for a weak solution ⃗u on some (0, T ] × R3 (with 0 < T < 1).
Recall that in our definition of weak solution (Definition 6.13), a weak solution ⃗u of
Equations 14.1 on (0, T ) × R3 satisfies:
2
ˆ ⃗u ∈ (L∞
t Lx )uloc
⃗ ⊗ ⃗u ∈ (L2t L2x )uloc
ˆ ∇
⃗ = (Id − P) div(F − ⃗u ⊗ ⃗u)
ˆ p is locally L3/2 L3/2 and ∇p
2
2 2
A basic lemma on L2uloc , (L∞
t Lx )uloc and (Lt Lx )uloc is the following one:
Lemma 14.1.
Let f ∈ L1 (R3 ), g ∈ L2uloc and h ∈ (L2t L2x )uloc ((0, T ) × R3 ). Then:
• ∥f ∗ g∥L2uloc ≤ C∥f ∥1 ∥g∥L2uloc
• ∥f ∗ h∥(L2t L2x )uloc ≤ C∥f ∥1 ∥h∥(L2t L2x )uloc
2
2
• ∥f ∗ h∥(L∞
≤ C∥f ∥1 ∥h∥(L∞
t Lx )uloc
t Lx )uloc
Moreover, if α > 0, then
• ∥f ∗ g∥∞ ≤ Cα ∥(1 + |x|)
3+α
2
DOI: 10.1201/9781003042594-14
f ∥2 ∥g∥L2uloc
452
A Theory of Uniformly Locally L2 Solutions
• ∥f ∗ h∥L∞
≤ Cα ∥(1 + |x|)
t,x
• ∥f ∗
RT
0
3+α
2
453
2
f ∥2 ∥h∥(L∞
t Lx )uloc
√
3+α
h dt∥∞ ≤ Cα T ∥(1 + |x|) 2 f ∥2 ∥h∥(L2t L2x )uloc
P
Proof. Define fk = f (x)1[0,1]3 (x−k) and Qk = k+[0, 1]3 , k ∈ Z3 . Then k∈Z3 ∥fk ∥1 = ∥f ∥1
and
X
X
3+α
3+α
3+α
∥fk ∥2 ≤ Cα
(1 + |k|)− 2 ∥1Qk (1 + |x|) 2 f ∥2 ≤ Cα′ ∥(1 + |x|) 2 f ∥2 .
k∈Z3
k∈Z3
We have
Z
2
|fk ∗ g| dy ≤
∥fk ∥21
Z
|x−y|<1
|g(z)|2 dz
|x−k−z|<4
so that
∥f ∗ g∥L2uloc ≤
X
∥fk ∗ g∥L2uloc ≤ C
k∈Z3
X
∥fk ∥1 ∥g∥L2uloc = C∥f ∥1 ∥g∥L2uloc .
k∈Z3
2
Inequalities ∥f ∗ h∥(L2t L2x )uloc ≤ C∥f ∥1 ∥h∥(L2t L2x )uloc and ∥f ∗ h∥(L∞
t Lx )uloc
2
C∥f ∥1 ∥h∥(L∞
are
proved
in
a
similar
way.
t Lx )uloc
We have as well
Z
∥fk ∗ g∥2∞ ≤ sup ∥fk ∥22
|g(z)|2 dz
x∈R3
≤
|x−k−z|<4
so that
∥f ∗ g∥∞ ≤
X
∥fk ∗ g∥∞ ≤ C
k∈Z3
X
∥fk ∥2 ∥g∥L2uloc ≤ C ′ ∥(1 + |x|)
3+α
2
f ∥2 ∥g∥L2uloc .
k∈Z3
The estimates on f ∗ h will be proved in the same way.
We have as well estimates on L1uloc : ∥f ∗ g∥L1uloc ≤ C∥f ∥1 ∥g∥L1uloc . Thus, if H ∈ L1uloc ,
the equation
⃗ = (Id − P) div H
∇p
is well defined, since we have
⃗ = div H − ∇
⃗ 1
∇p
∆
3 X
3
X
∂i ∂j Hi,j
i=1 j=1
1
1
where the operator ∆
∂i ∂j ∂k is well defined on L1uloc : recall that G = 4π|x|
is the Green
function associated to −∆; then, if γ ∈ D satisfies γ = 1 on |x| < 1, if f0 = −∂i ∂j ∂k (γG)
and f1 = −∂i ∂j ∂k ((1 − γ)G), we have
1
∂i ∂j ∂k g = f0 ∗ g + f1 ∗ g
∆
with f0 ∈ E ′ is a compactly supported distribution (so that convolution with f0 is well
defined on D′ ) and f1 ∈ L1 (so that convolution with f1 is well defined on L1uloc ).
454
The Navier–Stokes Problem in the 21st Century (2nd edition)
Uniformly locally square integrable solutions
Theorem 14.1.
Let ⃗u0 ∈ L2uloc with div ⃗u0 = 0 and F ∈ (L2t L2x )uloc ((0, 1) × R3 ). Then there exists a
solution ⃗u to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.2)

⃗u(0, .) = ⃗u0
on (0, T ) × R3 with
1
T = min(1,
C0 ν(1 +
∥⃗
u0 ∥L2
uloc
ν
+
∥F∥(L2 L2 )
uloc
ν 3/2
)
)4
and
2
2 1
⃗u ∈ (L∞
t Lx )uloc ∩ (L Hx )uloc
Z
sup sup (
0<t<T
Z
sup (
x∈R3
0
T
x∈R3
|x−y|<1
|⃗u(t, y)|2 dy)1/2 ≤ 2(∥⃗u0 ∥L2uloc + C0
∥F∥(L2 L2 )uloc
√
)
ν
2 2
⃗ ⊗ ⃗u(s, y)|2 dy ds)1/2 ≤ √2 (∥⃗u0 ∥L2 + C0 ∥F∥(L√L )uloc )
|∇
uloc
ν
ν
|x−y|<1
Z
where the constant C0 does not depend on ν.
Moreover, this solution ⃗u is suitable: it satisfies in D′ the local energy inequality
2
|⃗u|2
|⃗u|2
⃗ ⊗ ⃗u|2 − div (p + |⃗u| )⃗u + ⃗u · f⃗
) ≤ ν∆(
) − ν|∇
(14.3)
∂t (
2
2
2
⃗ = (Id − P) div(F − ⃗u ⊗ ⃗u).
with f⃗ = div F and ∇p
Proof. Step 1: local existence for the mollified problem.
As for the proof of the existence of Leray
R solutions, we start with a mollification of the
non-linearity. We fix θ ∈ D(R3 ) with θ dx = 1 and we define, for ϵ > 0, θϵ = ϵ13 θ( xϵ ).
We then shall look for a solution of

⃗ u)
∂t ⃗u = ν∆⃗u + P(f⃗ − (θϵ ∗ ⃗u) · ∇⃗
(14.4)

⃗u(0, .) = ⃗u0
⃗ n by
As usual, we use Picard’s iterative scheme, and define inductively U
Z t
⃗ 0 = Wνt ∗ ⃗u0 +
⃗ n+1 = U
⃗ 0 − Bϵ ( U
⃗ n, U
⃗ n)
U
Wν(t−s) ∗ Pf⃗ ds and U
0
with
⃗,V
⃗)=
Bϵ (U
Z
t
⃗)·∇
⃗V
⃗ ) ds.
Wν(t−s) ∗ P((θϵ ∗ U
0
We shall show existence of a solution ⃗u on (0, Tϵ ) × R3 such that
2
2 1
⃗u ∈ Eϵ = {⃗u ∈ D′ ((0, Tϵ ) × R3 ) / div ⃗u = 0 and ⃗u ∈ (L∞
t Lx )uloc ∩ (L Hx )uloc }.
A Theory of Uniformly Locally L2 Solutions
455
Eϵ is a Banach space for the norm
! 12
Z
|⃗u(t, y)|2 dy
∥⃗u∥Eϵ = sup sup
0<t<Tϵ x∈R3
! 12
Z TZϵ
⃗ ⊗ ⃗u(s, y)|2 dy ds .
|∇
+ sup
x∈R3
|x−y|<1
|x−y|<1
0
⃗ 0 ∈ Eϵ : for x ∈ R3 , we split ⃗u0 in 1|x−y|<3 ⃗u0 (y) = ⃗u1,x (y)
It is easy to check that U
and 1|x−y|>3 ⃗u0 (y) = ⃗u2,x (y), and similarly F in 1|x−y|<3 F(t, y) = F1,x (t, y) and
1|x−y|>3 F2,x (t, y)(t, y); we then write:
ˆ ∥Wνt ∗ ⃗u1,x ∥L∞ L2 = ∥⃗u1,x ∥2 ≤ C∥⃗u0 ∥L2uloc and ∥Wνt ∗ ⃗u1,x ∥L2 Ḣ 1 = √12ν ∥⃗u1,x ∥2
R
νt
ˆ on |x − y| < 1, |Wνt ∗ ⃗u2,x (y)| ≤ C |x−z|>3 |x−z|
u0 (z)| dz ≤ CνTϵ ∥⃗u0 ∥L2uloc and
5 |⃗
R
1
⃗
|∇ ⊗ Wνt ∗ ⃗u2,x (y)| ≤ C |x−z|>3 |x−z|4 |⃗u0 (z)| dz ≤ C∥⃗u0 ∥L2uloc
Rt
ˆ ∥ 0 Wν(t−s) ∗ P div F1,x ds∥L∞ ((0,Tϵ ),L2 ) ≤√12ν ∥F1,x ∥L2 L2 ≤ √Cν ∥F∥(L2 L2 )uloc and
Rt
∥ 0 Wν(t−s) ∗ P div F1,x ds∥L2 ((0,Tϵ ),Ḣ 1 ) ≤ ν1 ∥F1,x ∥L2 L2
ˆ on |x − y| < 1, we have
t
Z
Z tZ
Wν(t−s) ∗ P div F2,x (s, .)(y) ds| ≤C
|
0
|x−z|>3
0
1
|F(s, z)| dz ds
|x − z|4
p
≤C Tϵ ∥F∥(L2 L2 )uloc
and
⃗ ⊗
|∇
Z
t
Z tZ
Wν(t−s) ∗ P div F2,x (s, .)(y) ds| ≤C
0
0
|x−z|>3
1
|F(s, z)| dz ds
|x − z|5
p
≤C Tϵ ∥F∥(L2 L2 )uloc
Thus, recalling that Tϵ < 1, we find that
∥Wνt ∗ ⃗u0 ∥Eϵ ≤ Cν ∥⃗u0 ∥L2uloc
and
Z
∥
(14.5)
t
Wν(t−s) ∗ P div F ds∥Eϵ ≤ Cν ∥F∥(L2 L2 )uloc
(14.6)
0
where the constant Cν depends only on ν.
Inequality (14.6) gives as well:
⃗,V
⃗ )∥E ≤Cν ∥(θϵ ∗ U
⃗)⊗V
⃗ ∥(L2 L2 )
∥Bϵ (U
ϵ
uloc
1/2
⃗ ∥L∞ L∞ ∥V
⃗ ∥L∞ L2
≤CCν Tϵ ∥θϵ ∗ U
uloc
⃗ ∥L∞ L2 ∥V
⃗
≤Tϵ1/2 C ′ ϵ−3/4 Cν ∥U
uloc
(14.7)
∥L∞ L2uloc
Thus, Picard’s algorithm will converge to a solution if Tϵ is small enough:
Tϵ < min(1,
ϵ3/2
)
Cν (∥⃗u0 ∥L2uloc + ∥F∥(L2 L2 )uloc )2
(where the constant Cν depends only on ν).
(14.8)
456
The Navier–Stokes Problem in the 21st Century (2nd edition)
Step 2: uniform existence time for the mollified problem.
The existence time Tϵ we found in Equation (14.8) goes to 0 as ϵ goes to 0; however,
if we want our approximation scheme to converge to a solution, we must find a time
of existence which does not depend on ϵ. From (14.8), we can see that, as long as
∥⃗u∥L2uloc remains bounded, we may extend the solution to a larger interval.
In order to control the size of ⃗u, we introduce a non-negative compactly supported
function φ0 ∈ D(R3 ) such that
X
φ0 (x − k) = 1
k∈Z3
and the set
B = {φx0 = φ0 (. − x0 ) / x0 ∈ R3 }
Then, we have
∥h∥L2uloc ≈ sup ∥hφ∥2 and ∥H∥(L2 L2 )uloc ≈ sup ∥Hφ∥L2 L2
φ∈B
We have
φ∈B
d
⃗ − (⃗u ∗ θϵ ) · ∇⃗
⃗ u|φ2 ⃗u⟩
∥φ⃗u∥22 = 2⟨ν∆⃗u + f⃗ − ∇p
dt
⃗ is well defined, so p is defined up to a function p(t) which
We have seen that ∇p
2
⃗
does not depend on x. In order to estimate ⟨∇p|φ
⃗u⟩ with φ = φx0 , we shall fix the
definition of p(t, x) on the support of φ in the following way: let R0 be such that the
support of φ0 is contained in the ball B(0, R0 ) and let K be the distribution kernel of
1 ⃗
⃗ 1 P3 P3 ∂i ∂j Hi,j ) = K ∗ H, then we define px (t, x) as
0
i=1
j=1
∆ (∇ ⊗ ∇): ∆ (
Z
1 ⃗
⃗
⊗ ∇)(1
(K(x − y) − K(x0 − y))H(t, y) dy
px0 (t, x) = (∇
B(x0 ,5R0 ) H) +
∆
|y−x0 |>5R0
with
H = F − (θϵ ∗ ⃗u) ⊗ ⃗u.
As H belongs to (L2 L2 )uloc , the singular integral operator
on the compactly supported function 1B(x0 ,5R0 ) H.
We write
ϖx0 =
πx0 =
1 ⃗
∆ (∇
⃗ is well defined
⊗ ∇)
1 ⃗
⃗
(∇ ⊗ ∇)(1
B(x0 ,5R0 ) F),
∆
1 ⃗
⃗
(∇ ⊗ ∇)(1
u) ⊗ ⃗u)
B(x0 ,5R0 ) (θϵ ∗ ⃗
∆
and, for |x − x0 | < R0 ,
Z
(K(x − y) − K(x0 − y))F(t, y) dy,
qx0 (t, x) =
|y−x0 |>5R0
Z
(K(x − y) − K(x0 − y))(θϵ ∗ ⃗u) ⊗ ⃗u)(t, y) dy
ρx0 (t, x) =
|y−x0 |>5R0
0
As div(⃗u ∗ θϵ ) = 0, we have (for Supp θ ⊂ B(0, R1 ) and ϵ < R
R1 )
Z
Z
⃗ u) · (φ2 ⃗u) dx = − (⃗u ∗ θϵ ) ⊗ ⃗u · (∇(φ
⃗ 2 ) ⊗ ⃗u) dx.
2 ((⃗u ∗ θϵ ) · ∇⃗
A Theory of Uniformly Locally L2 Solutions
457
0
Thus, we have (for Supp θ ⊂ B(0, R1 ) and ϵ < R
R1 )
Z
Z
d
⃗ ⊗ ⃗u) · ∇
⃗ ⊗ (φ2 ⃗u) dx − 2 F · ∇
⃗ ⊗ (φ2 ⃗u) dx
∥φ⃗u∥22 = − 2ν (∇
dt
Z
Z
⃗ 2 ) ⊗ ⃗u) dx.
+ 2 px0 div(φ2 ⃗u) dx + 2 (⃗u ∗ θϵ ) ⊗ ⃗u · (∇(φ
Z
Z
⃗ ⊗ ⃗u|2 dx + 4ν |φ∇
⃗ ⊗ ⃗u| |∇φ||⃗
⃗ u| dx
≤ − 2ν |φ∇
Z
Z
⃗
⃗ u| dx
+ 2 |φF||φ∇ ⊗ ⃗u| dx + 4 |φF| |∇φ||⃗
Z
Z
⃗ dx + 4 |⃗u ∗ θϵ ||⃗u|2 |φ||∇φ|
⃗ dx
+ 2 |px0 | |φ⃗u| |∇φ|
Z
Z
2
⃗
≤ − ν |φ∇ ⊗ ⃗u| dx + C1 ν
|⃗u|2 dx
|x−x0 |<R0
Z
Z
1
1
2
+ C1
|F| dx + C1
(|ϖx0 |2 + (|qx0 |2 ) dx
ν |x−x0 |<R0
ν |x−x0 |<R0
Z
Z
3/2
3/2
+ C1
(|πx0 | + |ρx0 | ) dx + C1
|⃗u|3 ds
|x−x0 |<R0
|x−x0 |<2R0
and finally we get
Z
Z
d
⃗ ⊗ ⃗u|2 dx + C1 ν
∥φ⃗u∥22 ≤ − ν |φ∇
|⃗u|2 dx
dt
|x−x0 |<R0
Z
Z
1
1
1
+C2
|F|2 dx + C2 (
|F(t, y)| dy)2
ν |x−x0 |<5R0
ν |y−x0 |>5R0 |x0 − y|4
Z
Z
1
2
3/2
+C2 (
|⃗u(t, y)| dy) + C2
|⃗u|3 dx
4
|y−x0 |>5R0 |x0 − y|
|x−x0 |<5R0
(14.9)
Let
α(t) = ∥⃗u∥L2uloc = sup ∥φ⃗u∥2
φ∈B
β(t) = ∥F∥(L2 L2 )uloc ((0,t)×R3 )
⃗ ⊗ ⃗u∥(L2 L2 ) ((0,t)×R3 )
γ(t) = ∥∇
uloc
δ(t) = ∥⃗u∥(L3 L3 )uloc ((0,t)×R3 )
Z tZ
= sup (
|φ(x)F(s, x)|2 dx ds)1/2
φ∈B
0
Z tZ
⃗ ⊗ ⃗u(s, x)|2 dx ds)1/2
= sup (
|φ(x)∇
φ∈B
0
Z tZ
= sup (
|φ(x)⃗u(s, x)|3 dx ds)1/3
φ∈B
0
Integrating our inequality (14.9) and using Lemma 14.1, we get (for t < min(T ∗ , 1),
where T ∗ is the maximal existence time)
Z tZ
⃗ ⊗ ⃗u|2 dx
∥φ⃗u∥22 + ν
|φ∇
0
Z t
1
≤∥φ⃗u0 ∥22 + C3 ν
α(s)2 ds + C3 β(t)2 + C3 δ(t)3 .
ν
0
We have
1/2
1/2
⃗ ⊗ (φ⃗u)∥ 2
∥φ⃗u∥L3 (dx) ≤ C∥∇
u∥L2 (dx) ≤ C
L (dx) ∥φ⃗
p
q
⃗ ⊗ ⃗u∥2
α(t) α(t) + ∥φ∇
458
The Navier–Stokes Problem in the 21st Century (2nd edition)
hence, for any η > 0,
⃗ ⊗ ⃗u∥22 ).
∥φ⃗u∥3L3 (dx) ≤ C(α(t)3 + η −3 α(t)6 + η∥φ∇
Hence, we get
Z tZ
⃗ ⊗ ⃗u|2 dx
|φ∇
Z t
≤ α(0)2 + C3 ν
α(s)2 ds + C4 ηγ(t)2
0
Z t
Z t
1
+C3 β(t)2 +C4 (
α(s)3 ds + η −3
α(s)6 ds)
ν
0
0
∥φ⃗u∥22
+ν
0
This gives in particular
νγ(t)2 ≤α(0)2 + C3 ν
Z
t
α(s)2 ds + C4 ηγ(t)2
Z t
Z t
1
+ C3 β(t)2 + C4 (
α(s)3 ds + η −3
α(s)6 ds)
ν
0
0
0
and, for η =
ν
2C4 ,
νγ(t)2 ≤2α(0)2 + 2C3 ν
Z
t
α(s)2 ds
0
Z t
Z t
1
2
3
−3
+ 2C3 β(t) + 2C4 (
α(s) ds + η
α(s)6 ds)
ν
0
0
Now, we write
∥φ⃗u∥22 ≤α(0)2 + C3 ν
Z
t
α(s)2 ds + C4 ηγ(t)2
0
Z t
Z t
1
2
3
−3
+ C3 β(t) + C4 (
α(s) ds + η
α(s)6 ds)
ν
0
0
Z t
≤2α(0)2 + 2C3 ν
α(s)2 ds
0
Z t
Z
1
2C4 −3 t
2
3
+ 2C3 β(t) + 2C4 (
α(s) ds + 2(
)
α(s)6 ds)
ν
ν
0
0
and finally
2
t
Z
2
α(s)2 ds
α(t) ≤2α(0) + C5 ν
0
1
1
+ 2C3 β(t)2 + C5 ( )−3
ν
ν
Let
r
B0 = ∥⃗u0 ∥L2uloc +
Z
t
α(s)6 ds
0
C3
∥F∥(L2 L2 )uloc ((0,1)×R3 ) )
ν
We have proved that (for t < 1)
∥⃗u(t, .)∥2L2
uloc
= α(t)2 ≤ 2B02 + C5 ν
Z
0
t
1
α(s)2 ds + C5 ( )−3
ν
Z
0
t
α(s)6 ds
A Theory of Uniformly Locally L2 Solutions
459
Thus ∥⃗u(t, .)∥L2uloc will remain bounded by 2B0 as long as t < 1, 4C5 νt < 1 and
64C5 B04 t < ν 3 .
It means that the existence time of the mild solution ⃗u may be estimated independently
from ϵ ∈ (0, 1): ⃗u exists at least on (0, T ∗ ), where
T ∗ = min(1,
1
ν3
q
,
).
4C5 ν 64C (∥⃗u ∥ 2 + C3 ∥F∥ 2 2
4
5
0 Luloc
(L L )uloc ((0,1)×R3 ) )
ν
(14.10)
Step 3: Weak convergence.
Let ⃗uϵ be the solution of the mollified problem (14.4). We have found a time T ∗ which
is independent from ϵ ∈ (0, 1) such that the solution ⃗uϵ exists on ((0, T ∗ ) × R3 ) and
satisfies
Z
p ∥F∥(L2 L2 )uloc
√
sup sup (
|⃗uϵ (t, y)|2 dy)1/2 ≤ 2(∥⃗u0 ∥L2uloc + C3
)
ν
0<t<T ∗ x∈R3 |x−y|<1
Z
sup (
x∈R3
0
T∗
p
2 2
⃗ ⊗ ⃗uϵ (s, y)|2 dy ds)1/2 ≤ √2 (∥⃗u0 ∥L2 + C3 ∥F∥(L√L )uloc )
|∇
uloc
ν
ν
|x−y|<1
Z
where C3 does not depend on ϵ.
From those energy estimates, we can see that, for every test function ϕ ∈ D′ ((0, T ∗ ) ×
R3 ), ϕ⃗uϵ remains bounded in L∞ L2 ∩ L2 Ḣ 1 . Moreover, we have
∂t ⃗uϵ = ν∆⃗uϵ + P div(F − (⃗uϵ ∗ θϵ ) ⊗ ⃗uϵ )
and we have seen that ⃗uϵ remains bounded in (L3 L3 )uloc ; thus, we can see that ϕ∂t ⃗uϵ
remains bounded in L3/2 H −3/2 .
We may then use the Rellich–Lions theorem (Theorem 12.1): we may find a sequence
ϵn → 0 and a function ⃗u ∈ (L∞ L2 )uloc ∩ (L2 Ḣ 1 )uloc such that:
ˆ ⃗u(ϵn ) is *-weakly convergent to ⃗u in (L∞ L2 )uloc and in (L2 Ḣ 1 )uloc
ˆ ⃗u(ϵn ) is strongly convergent to ⃗u in L2loc ((0, T ) × R3 ).
In order to show that the weak limit ⃗u satisfies
⃗ ⃗u),
∂t ⃗u = ν∆⃗u + P(f⃗ − ⃗u · ∇
we have only to check that we have the convergence in D′ of the non-linear term
P div((⃗uϵ ∗ θϵ ) ⊗ ⃗uϵ ) to P div(⃗u ⊗ ⃗u). As we know that ⃗uϵ is bounded in (L3 L3 )uloc
and is strongly convergent in (L2 L2 )loc , we see that (⃗uϵ ∗ θϵ ) ⊗ ⃗uϵ is bounded in
(L6/5 L6/5 )uloc and strongly convergent to ⃗u ⊗ ⃗u in (L6/5 L6/5 )loc . This is enough to
get the convergence of P div((⃗uϵ ∗ θϵ ) ⊗ ⃗uϵ ).
Step 4: Local energy estimates for the weak limit.
We now check that ⃗u is more precisely a suitable weak solution (i.e., fulfills the local
energy inequality). We work in the neighborhood B(x0 , R0 ) of a point x0 , and we
write
⃗ ⃗uϵ − ∇p
⃗ x ,ϵ
∂t ⃗uϵ = ν∆⃗uϵ + f⃗ − (⃗uϵ ∗ θϵ ) · ∇
0
with
1 ⃗
⃗
px0 ,ϵ (t, x) = (∇
⊗ ∇)(1
B(x0 ,5R0 ) Hϵ ) +
∆
Z
(K(x − y) − K(x0 − y))Hϵ (t, y) dy
|y−x0 |>5R0
460
The Navier–Stokes Problem in the 21st Century (2nd edition)
with
Hϵ = F − (θϵ ∗ ⃗uϵ ) ⊗ ⃗uϵ .
We then write
∂t (
|⃗uϵ |2
|⃗uϵ |2
⃗ ⊗ ⃗uϵ |2 + f⃗ · ⃗uϵ
) = ν∆(
) − ν|∇
2
2
|⃗uϵ |2
− div(
(⃗uϵ ∗ θϵ )) − div(px0 ,ϵ ⃗uϵ ).
2
We know that ⃗uϵn converge strongly to ⃗u in L2loc ((0, T )×R3 ); as the family is bounded
10/3 3/5
10/3 10/3
in (Lt Hx )uloc ⊂ (Lt Lx )uloc , we find that we have strong convergence in
3
3
Lloc ((0, T )×R ) as well. Thus, we have the following convergence results in D′ ((0, T )×
|⃗
u
|2
2
R3 ): ∂t |⃗uϵn |2 → ∂t |⃗u|2 , ∆|⃗uϵn |2 → ∆|⃗u|2 , div( ϵ2n (⃗uϵn ∗θϵn )) → div( |⃗u2| ⃗u) and ⃗u(ϵn ) ·
f⃗ → ⃗u·f⃗. Similarly, we find that px0 ,ϵn converges weakly to px0 in (L3/2 L3/2 )(B(x0 , R0 )
and the strong convergence of ⃗uϵn in (L3 L3 )loc gives the convergence of div(p⃗uϵn ) to
div(p⃗u).
Thus far, we have got that
∂t |⃗u|2 = ν∆|⃗u|2 − div((|⃗u|2 + 2p)⃗u) + 2⃗u · f⃗ − νT
with
⃗ ⊗ ⃗uϵ |2 .
T = lim 2|∇
n
ϵn →0
√
⃗ uϵ is weakly convergent
Let ϕ ∈ D ((0, T )×R ) be a non-negative function. As ϕ ∇⊗⃗
n
√
√
√
2 2
2
⃗
⃗
⃗ ⊗ ⃗uϵ ∥2 . Thus,
to ϕ ∇ ⊗ ⃗u in Lt Lx , we find that ∥ ϕ ∇ ⊗ ⃗u∥2 ≤ lim inf ϵn →0 ∥ ϕ ∇
n 2
we have
ZZ
⃗ ⊗ ⃗uϵ |2 ϕ(t, x) dt d
⟨T |ϕ⟩D′ ,D =2 lim
|∇
n
ϵn →0
ZZ
⃗ ⊗ ⃗u|2 ϕ(t, x) dt dx.
≥2
|∇
′
3
⃗ ⊗ ⃗u|2 + µ, where µ is a non-negative locally finite measure, and thus
Thus, T = 2|∇
⃗u is suitable.
The solution ⃗u we have constructed is continuous in (local) L2 norm at time t = 0:
Proposition 14.1.
The solution ⃗u constructed
in the proof of Theorem 14.1 satisfies: for every compact subset
R
K of R3 , limt→0+ K |⃗u(t, x) − ⃗u0 (t, x)|2 dx = 0.
Proof. First, we remark that the solution ⃗u satisfies ∂t ⃗u ∈ (L1t H −3/2 )uloc , so that t ∈
[0, T ) 7→ ⃗u(t, .) is continuous from [0, T ) to D′ . In particular, we have, for φ = φx0 ∈ B,
∥φ⃗u0 ∥2 ≤ lim inf ∥φ⃗u(t, .)∥2 .
t→0+
We must now estimate lim supt→0+ ∥φ⃗u(t, .)∥2 . Let γ ∈ C ∞ (R) be equal to 1 on (−∞, −1)
0
and to 0 on (−1/2, +∞). We define for t0 ∈ (0, T ∗ ) and η < t0 , γt0 ,η (t) = γ( t−t
η ). We have
∂t (φ2 γt0 ,η
|⃗uϵ |2
|⃗uϵ |2
⃗ ⊗ ⃗uϵ |2 + f⃗ · ⃗uϵ )
) = φ2 γt0 ,η (ν∆(
) − ν|∇
2
2
|⃗uϵ |2
|⃗uϵ |2
− φ2 γt0 ,η (div(
(⃗uϵ ∗ θϵ )) + div(px0 ,ϵ ⃗uϵ )) + φ2
∂t γt0 ,η
2
2
A Theory of Uniformly Locally L2 Solutions
461
so that
|⃗uϵ |2
∂t γt0 ,η dx dt =
2
Z
Z t0 Z
|⃗u0 |2
|⃗uϵ |2
⃗ ⊗ ⃗uϵ |2 + f⃗ · ⃗uϵ ) dx dt
φ
dx +
φ2 γt0 ,η (ν∆(
) − ν|∇
2
2
0
Z t0 Z
|⃗uϵ |2
−
φ2 γt0 ,η (div(
(⃗uϵ ∗ θϵ )) + div(px0 ,ϵ ⃗uϵ )) dx dt.
2
0
Z Z
−
φ2
If we let ϵn go
R ttoR 0, we have proven2 that every integral will converge in this equality, except
⃗ ⊗ ⃗uϵ | dx dt. But we have a control on this term:
the integral 0 0 φ2 γt0 ,η |∇
n
Z
t0
Z
⃗ ⊗ ⃗u|2 dx dt ≤ lim inf
φ γt0 ,η |∇
2
ϵn →0
0
Z
t0
Z
⃗ ⊗ ⃗uϵ |2 dx dt.
φ2 γt0 ,η |∇
n
0
Thus, we have
Z Z
Z
|⃗u|2
|⃗u0 |2
−
φ2
∂t γt0 ,η dx dt ≤ φ2
dx
2
2
Z t0 Z
|⃗u|2
⃗ ⊗ ⃗u|2 + f⃗ · ⃗u) dx dt
+
φ2 γt0 ,η (ν∆(
) − ν|∇
2
0
Z t0 Z
|⃗u|2
−
φ2 γt0 ,η (div((
+ px0 )⃗u) dx dt
2
0
R
If t0 is a Lebesgue point of the map t 7→ φ2 (x)|⃗u(t, x)|2 dx, we have
Z Z
Z
u|2
|⃗u(t0 , x)|2
2 |⃗
lim −
φ
∂t γt0 ,η dx dt = φ2 (x)
dx
η→0
2
2
and thus
Z
Z
|⃗u(t0 , x)|2
|⃗u0 |2
φ(x)2
dx ≤ φ2
dx
2
2
Z t0 Z
|⃗u|2
⃗ ⊗ ⃗u|2 + f⃗ · ⃗u) dx dt
+
φ2 (ν∆(
) − ν|∇
2
0
Z t0 Z
|⃗u|2
−
φ2 (div((
+ px0 )⃗u) dx dt
2
0
(14.11)
The right-hand side of inequality (14.11) is a continuous function of t0 , while the left-hand
side is a lower semi-continuous function of t0 ; thus, the inequality is fulfilled for every
t0 ∈ (0, T ). Letting t0 go to 0 proves that
lim sup ∥φ⃗u(t, .)∥2 ≤ ∥φ⃗u0 ∥2 .
t→0+
Thus, we have limt→0+ ∥φ⃗u(t, .)∥2 ≤ ∥φ⃗u0 ∥2 . As we have weak convergence of φ⃗u to φ⃗u0 in
L2 , we find that
lim+ ∥φ⃗u(t, .) − φ⃗u0 ∥2 = 0.
t→0
The meaning of Proposition 14.1 is that we have constructed a local version of a Leray
solution:
462
The Navier–Stokes Problem in the 21st Century (2nd edition)
Definition 14.1 (Local Leray solution).
Let ⃗u0 ∈ L2uloc with div ⃗u0 = 0 and F ∈ (L2t L2x )uloc ((0, T ) × R3 ). A weak solution ⃗u on
(0, T ) × R3 to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.12)

⃗u(0, .) = ⃗u0
is a local Leray solution if it satisfies the following requirements:
2
2 1
• ⃗u ∈ (L∞
t Lx )uloc ∩ (L Hx )uloc
• ⃗u is suitable
• for every compact subset K of R3 , limt→0+
14.2
R
K
|⃗u(t, x) − ⃗u0 (t, x)|2 dx = 0.
Local Inequalities for Local Leray Solutions
The local energy inequalities we have derived in the previous section are valid for all
local Leray solutions:
Local inequalities
Theorem 14.2.
Let ⃗u be a local Leray solution on (0, T ) × R3 to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.13)
⃗u(0, .) = ⃗u0

where ⃗u0 ∈ L2uloc with div ⃗u0 = 0 and F ∈ (L2t L2x )uloc ((0, T ) × R3 ).
Then, for a constant C0 which does not depend on ν, for
1
T0 = min(T, 1,
C0 ν(1 +
∥⃗
u0 ∥L2
uloc
ν
+
∥F∥(L2 L2 )
)
uloc
ν 3/2
)4
we have
Z
sup (
sup
0<t<T0
x∈R3
|x−y|<1
|⃗u(t, y)|2 dy)1/2 ≤ 2(∥⃗u0 ∥L2uloc + C0
∥F∥(L2 L2 )uloc
√
)
ν
and
Z
sup (
x∈R3
0
T0
2 2
⃗ ⊗ ⃗u(s, y)|2 dy ds)1/2 ≤ √2 (∥⃗u0 ∥L2 + C0 ∥F∥(L√L )uloc ).
|∇
uloc
ν
ν
|x−y|<1
Z
A Theory of Uniformly Locally L2 Solutions
463
Proof. We use the suitability of ⃗u and apply the local energy inequality to the test function
ϕ(s, x) = γt0 ,η (s)φ(x) (with φ ∈ B: φ(x) = φx0 (x) = φ0 (x − x0 )) (where γt0 ,η is defined
page 460). This gives
Z Z
Z
|⃗u|2
|⃗u0 |2
−
φ
∂t γt0 ,η dx dt ≤ φ
dx
2
2
Z t0 Z
|⃗u|2
⃗ ⊗ ⃗u|2 + f⃗ · ⃗u) dx dt
+
φγt0 ,η (ν∆(
) − ν|∇
2
0
Z t0 Z
|⃗u|2
−
φγt0 ,η (div((
+ px0 )⃗u) dx dt
2
0
R
2
Thus, we find again (letting η go to 0 for a Lebesgue point t0 of t 7→ φχ2R |⃗u2| dx, and
then using the lower semi-continuity of the same map to get the control on other times t0 )
that, for every t ∈ (0, T ):
Z
Z
|⃗u(t, x)|2
|⃗u0 |2
φ
dx ds ≤ φ
dx
2
2
Z tZ
|⃗u|2
⃗ ⊗ ⃗u|2 + f⃗ · ⃗u) dx ds
+
φ(ν∆(
) − ν|∇
2
0
Z tZ
|⃗u|2
−
φ div((
+ px0 )⃗u) dx ds
2
0
Z
Z
tZ
|⃗u0 |2
⃗ ⊗ ⃗u|2 dx ds
= φ
dx − ν
φ|∇
2
0
Z tZ
|⃗u|2
+ν
(∆φ)
dx ds
2
0
Z tZ
Z tZ
⃗
⃗ ⊗ ⃗u dx ds
−
φF · ∇ ⊗ ⃗u dx ds −
F · ∇φ
0
0
Z tZ
|⃗u|2
⃗
+
(⃗u · ∇φ)((
+ px0 ) dx ds
2
0
Defining again
α(t) = ∥⃗u∥L2uloc = sup ∥φ⃗u∥2
φ∈B
Z tZ
β(t) = ∥F∥(L2 L2 )uloc ((0,t)×R3 ) = sup (
φ∈B
⃗ ⊗ ⃗u∥(L2 L2 ) ((0,t)×R3 ) = sup (
γ(t) = ∥∇
uloc
φ∈B
0
Z tZ
⃗ ⊗ ⃗u(s, x)|2 dx ds)1/2
|φ(x)∇
0
Z tZ
δ(t) = ∥⃗u∥(L3 L3 )uloc ((0,t)×R3 ) = sup (
φ∈B
|φ(x)F(s, x)|2 dx ds)1/2
|φ(x)⃗u(s, x)|3 dx ds)1/2
0
we get (for t < min(T, 1)
∥φ⃗u∥22
Z tZ
+ν
⃗ ⊗ ⃗u|2 dx ≤∥φ⃗u0 ∥2 + C3 ν
|φ∇
2
0
Z
t
α(s)2 ds
0
1
+ C3 β(t)2 + C3 δ(t)3
ν
and we may conclude, following the lines on page 458.
464
The Navier–Stokes Problem in the 21st Century (2nd edition)
The same computations show that, when letting |x| go to +∞, the behavior of our
solution ⃗u depends only on the behavior of ⃗u0 and F near the infinity, and that the influence
of the small values of x may be easily controlled:
Asymptotic behavior of local Leray solutions
Theorem 14.3.
Let ⃗u be a local Leray solution on (0, T ) × R3 to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)

(14.14)
⃗u(0, .) = ⃗u0
where ⃗u0 ∈ L2uloc with div ⃗u0 = 0 and F ∈ (L2t L2x )uloc ((0, T ) × R3 ).
Let ω ∈ D(R3 ) be equal to 1 in a neighborhood of 0 and define χR (x) = 1 − ω(x/R).
Then there exists a positive constant CT so that for all 0 < t < T and all R > 1, we
have
r
1 + ln R
∥⃗u(t, .)χR ∥L2uloc ≤ CT (∥⃗u0 χR ∥L2uloc + ∥FχR ∥(L2 L2 )uloc +
).
(14.15)
R
⃗ ⊗ ⃗u∥(L2 L2 )
The constant CT depends only on ν, T , ∥F∥(L2 L2 )uloc , ∥∇
and
uloc
∞
2
∥⃗u∥(L L )uloc .
Proof. We use the suitability of ⃗u and apply the local energy inequality to the test function
ϕ(s, x) = γt0 ,η (s)φ(x)χ2R (x) (with φ ∈ B: φ(x) = φx0 (x) = φ0 (x − x0 )). This gives
|⃗u|2
∂t γt0 ,η dx dt ≤
2
Z
Z t0 Z
u0 |2
|⃗u|2
2 |⃗
⃗ ⊗ ⃗u|2 + f⃗ · ⃗u) dx dt
φχR
dx +
φχ2R γt0 ,η (ν∆(
) − ν|∇
2
2
0
Z t0 Z
|⃗u|2
−
φχ2R γt0 ,η (div((
+ px0 )⃗u) dx dt
2
0
Z Z
−
φχ2R
R
2
Thus, we find again (letting η go to 0 for a Lebesgue point t0 of t 7→ φχ2R |⃗u2| dx, and
then using the lower semi-continuity of the same map to get the control on other times t0 )
that, for every t ∈ (0, T ):
Z
Z
u(t, x)|2
|⃗u0 |2
2 |⃗
φχR
dx ds ≤ φχ2R
dx
2
2
Z tZ
|⃗u|2
⃗ ⊗ ⃗u|2 + f⃗ · ⃗u) dx ds
+
φχ2R (ν∆(
) − ν|∇
2
0
Z tZ
|⃗u|2
−
φχ2R (div((
+ px0 )⃗u) dx ds
2
0
A Theory of Uniformly Locally L2 Solutions
Z
Z tZ
u0 |2
2 |⃗
⃗ ⊗ ⃗u|2 dx ds
= φχR
dx − ν
φχ2R |∇
2
0
Z tZ
|⃗u|2
+ν
(∆φ)χ2R
dx ds
2
0
Z tZ
Z
tZ
u|2
2 |⃗
⃗ · ∇χ
⃗ R ) |⃗u|2 dx ds
+ν
φ∆(χR )
dx ds + ν
χR (∇φ
2
0
0
Z tZ
Z tZ
⃗ ⊗ ⃗u dx ds −
⃗ ⊗ ⃗u dx ds
−
φχ2R F · ∇
χ2R F · ∇φ
0
0
Z tZ
⃗ R ⊗ ⃗u dx ds
−2
φχR F · ∇χ
0
Z tZ
Z tZ
2
|⃗u|2
2
⃗
⃗ R )φχR (( |⃗u| + px ) dx ds
+
(⃗u · ∇φ)χ
((
+
p
)
dx
ds
+
2
(⃗u · ∇χ
x0
R
0
2
2
0
0
465
Let
α(t) = ∥⃗u∥L2uloc ,
β(t) = ∥F∥(L2 L2 )uloc ((0,t)×R3 )
⃗ ⊗ ⃗u∥(L2 L2 ) ((0,t)×R3 ) ,
γ(t) = ∥∇
uloc
δ(t) = ∥⃗u∥(L3 L3 )uloc ((0,t)×R3 )
and similarly
αR (t) = ∥χR ⃗u∥L2uloc ,
βR (t) = ∥χR F∥(L2 L2 )uloc ((0,t)×R3 )
⃗ ⊗ ⃗u∥(L2 L2 ) ((0,t)×R3 ) ,
γR (t) = ∥χR ∇
uloc
δR (t) = ∥χR ⃗u∥(L3 L3 )uloc ((0,t)×R3 )
We have
Z
Z tZ
⃗ ⊗ ⃗u|2 dx ds
φχ2R |⃗u(t, x)|2 dx ds + ν
φχ2R |∇
0
Z
Z t
Z
1 t
2
2
2
≤ φχR |⃗u0 | dx + C1 ν
αR (s) ds + C1 ν
α(s)2 ds
R 0
0
1
1
1
1
+ C1 βR (t)2 + C1 β(t)γ(t) + C1 ν β(t)2 + C1 δ(t)3
ν
R
R
R
Z tZ
2
⃗
+ C1 δ(t)δR (t)2 +
(⃗u · ∇φ)χ
R px0 dx ds
0
The last term, which includes px0 , must be carefully dealt with, as px0 is given by a nonlocal operator which is linear with respect to F and quadratic with respect to ⃗u. Recall that
φ is supported by the ball B(x0 , R0 ) and that, on the ball B(x0 , R0 ), we have
px0 = T1 (1|y−x0 |<5R0 (F − ⃗u ⊗ ⃗u)) + T2 (1|y−x0 |>5R0 (F − ⃗u ⊗ ⃗u))
where
T1 =
and
1 ⃗
⃗
(∇ ⊗ ∇)
∆
Z
T2 H =
(K(x, y) − K(x0 , y))H(y) dy.
466
The Navier–Stokes Problem in the 21st Century (2nd edition)
Let MχR be the pointwise multiplication by χR : MχR g = χR g. We write
Z tZ
2
⃗
(⃗u · ∇φ)χ
R px0 dx ds =
Z tZ
⃗
(⃗u · ∇φ)χ
R T1 (χR 1|y−x0 |<5R0 F) dx ds
Z tZ
⃗
(⃗u · ∇φ)χ
+
R [MχR , T1 ](1|y−x0 |<5R0 F) dx ds
0
Z tZ
⃗
(⃗u · ∇φ)χ
+
R T2 (χR 1|y−x0 |>5R0 F) dx ds
0
Z tZ
⃗
(⃗u · ∇φ)χ
+
R [MχR , T2 ](1|y−x0 |>5R0 F) dx ds
0
Z tZ
⃗
(⃗u · ∇φ)χ
u ⊗ ⃗u) dx ds
−
R T1 (χR 1|y−x0 |<R0 ⃗
0
Z tZ
⃗
(⃗u · ∇φ)χ
u ⊗ ⃗u) dx ds
−
R [MχR , T1 ](1|y−x0 |<R0 ⃗
0
Z tZ
⃗
−
u ⊗ ⃗u) dx ds
(⃗u · ∇φ)χ
R T2 (χR 1|y−x0 |>5R0 ⃗
0
Z tZ
⃗
−
u ⊗ ⃗u) dx ds
(⃗u · ∇φ)χ
R [MχR , T2 ](1|y−x0 |>5R0 ⃗
0
0
0
=I1 + · · · + I8
We already know how to control I1 , I3 , I5 and I7 :
Z t
1
|I1 | + |I3 | ≤C2 ν
αR (s)2 ds + C2 βR (t)2
ν
0
2
|I5 | + |I7 | ≤C2 δ(t)δR (t)
We have
Z tZ
Z
|I4 | ≤C2
|⃗u(x)|χR (x)
0
|x−x0 |<R0
|y−x|>4R0
Z tZ
Z
≤C3
|⃗u(x)|χR (x)
0
1
+ C3
R
≤C4
|x−x0 |<R0
Z tZ
|y−x|>R
|χR (x) − χR (y)|
|F(s, y)| dy dx ds
|x − y|4
1
|F(s, y)| dy dx ds
|x − y|4
Z
|⃗u(x)|χR (x)
0
+
|x−x0 |<R0
1 + ln (R/R0 )
(ν
R
Z
R>|y−x|>4R0
t
α(s)2 ds +
0
1
|F(s, y)| dy dx ds
|x − y|3
1
β(t)2 )
ν
Similarly, we have
1 + ln+ (R/R0 )
δ(t)3 .
R
The most difficult terms are I2 and I6 . They will be dealt with the help of Calderón’s lemma
on the commutator between a pseudo-differential operator of order 1 and the pointwise
multiplication with a Lipschitz function [313]: for 1 < p < +∞, we have for a Lipschitz and
H ∈ Lp ,
⃗ ⊗ T1 , Ma ]H∥p ≤ Cp ∥∇a∥
⃗ ∞ ∥H∥p .
∥[∇
|I8 | ≤ C5
A Theory of Uniformly Locally L2 Solutions
We thus write
Z tZ
0
467
Z tZ
⃗
⃗ R )χR T1 H dx ds
(⃗u · ∇φ)χR [MχR , T1 ]H dx ds = − 2
φ(⃗u · ∇χ
0
Z tZ
⃗ ⊗ T1 ]H dx ds
−
φχR ⃗u.[MχR , ∇
0
and get
|I2 | ≤ C6
1
(ν
R
Z
t
1
β(t)2 )
ν
α(s)2 ds +
0
and
|I6 | ≤ C6
1
δ(t)3 .
R
Summing up all those estimates, we get:
Z
Z tZ
2
2
⃗ ⊗ ⃗u|2 dx ds
φχR |⃗u(t, x)| dx ds + ν
φχ2R |∇
0
Z
1
≤ φχ2R |⃗u0 |2 dx + C7 βR (t)2
ν
Z t
1 + ln+ (R/R0 )
1
+ C7 ν
(ν
α(s)2 ds + β(t)2 )
R
ν
0
1
1 + ln+ (R/R0 )
β(t)γ(t) + C7
δ(t)3
R
R
Z t
+ C7 ν
αR (s)2 ds + C7 δ(t)δR (t)2
+ C7
0
⃗ ⊗ ⃗u∥(L2 L2 )
so that, for a constant DT (which depends only on ν, T , ∥F∥(L2 L2 )uloc , ∥∇
uloc
and ∥⃗u∥(L∞ L2 )uloc )
Z
φχ2R |⃗u(t, x)|2 dx ds + ν
Z tZ
⃗ ⊗ ⃗u|2 dx ds
φχ2R |∇
0
≤DT (∥χR ⃗u0 ∥2L2 + ∥χR F∥2(L2 L2 )uloc +
uloc
Z t
+ C7 ν
αR (s)2 ds + C7 δ(T )δR (t)2
1 + ln+ (R/R0 )
)
R
0
We then write
Z t
Z
2
δR (t) ≤ C8 (
αR (s) ds + γR (t)(
T
2
0
6
αR
(s) ds)1/6 )
0
and get
Z
φχ2R |⃗u(t, x)|2
Z tZ
dx ds + ν
⃗ ⊗ ⃗u|2 dx ds
φχ2R |∇
0
1 + ln+ (R/R0 )
)
uloc
R
Z t
8
+ (C7 (ν + C8 )T 2/3 + C72 C82 δ(T )2 )(
αR (s)6 ds)1/3
ν
0
ν
+ γR (t)2
2
≤DT (∥χR ⃗u0 ∥2L2
+ ∥χR F∥2(L2 L2 )uloc +
468
The Navier–Stokes Problem in the 21st Century (2nd edition)
+
and thus, writing ηR = ∥χR ⃗u0 ∥L2uloc + ∥χR F∥(L2 L2 )uloc + 1+ln R(R/R0 ) ,
Z t
ν
3
max(αR (t)2 , νγR (t)2 ) ≤ET (ηR
+
αR (s)6 ds)1/3 + γR (t)2
2
0
⃗ ⊗ ⃗u∥(L2 L2 )
( for a constant ET which depends only on ν, T , ∥F∥(L2 L2 )uloc , ∥∇
and
uloc
2
∥⃗u∥(L∞ L2 )uloc ). Finally, we can easily control νγR (t) and get
Z t
3
αR (s)6 ds)1/3
αR (t)2 ≤ 2ET (ηR
+
0
and thus
Z t
Z t
d
3
3
(ηR
+
αR (s)6 ds)1/3 ≤ (2ET )3 (ηR
+
αR (s)6 ds)1/3 .
dt
0
0
We then conclude by Grönwall’s lemma.
14.3
The Caffarelli, Kohn and Nirenberg ϵ–Regularity Criterion
We give here a variant of the regularity criterion of Caffarelli, Kohn and Nirenberg [74]
(Theorem 13.8).
Caffarelli–Kohn–Nirenberg ϵ–regularity criterion
Theorem 14.4.
Let q > 5/2. Let Ω be a domain of R × R3 . Let (⃗u, p) a weak solution on Ω of the
Navier–Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Assume that
• ⃗u ∈ L∞ L2 ∩ L2 Ḣ 1 (Ω)
3/2
3/2
• p ∈ Lt Lx (Ω)
• f⃗ ∈ Lqt Lqx (Ω)
• ⃗u is suitable: it satisfies in D′ the local energy inequality
|⃗u|2
|⃗u|2
|⃗u|2
2
⃗
∂t (
) ≤ ν∆(
) − ν|∇ ⊗ ⃗u| − div (p +
)⃗u + ⃗u · f⃗
2
2
2
Let r0 > 0 and Q0 = Qr0 (t0 , x0 ) = (t0 − r02 , t0 ) × B(x0 , r0 ). There exists
constants ϵ0 and C0 which depend only on ν and q (but not on x0 , t0 , r0 , ⃗u nor
that, if

0 ≤ λ ≤ ϵ0




RR
|⃗u|3 + |p|3/2 dx dt ≤ λ3 r02
Q0


 RR


|f⃗|q dx dt ≤ λ2q r05−3q
Q0
(14.16)
positive
f⃗) such
(14.17)
A Theory of Uniformly Locally L2 Solutions
469
then ⃗u is bounded on Q1 = (t0 − 14 r02 , t0 ) × B(x0 , r0 /2) and
sup
|⃗u(t, x)| ≤ C0 λ
(t,x)∈Q1
1
.
r0
Proof. The proof follows the lines of the proof of Theorem 13.8. For (t, x) ∈ Q3r0 /4 (t0 , x0 )
and 0 < r ≤ r0 /8, we define
R
ˆ Ur (t, x) = sups∈(t−r2 ,t) Br (t,x) |⃗u(s, y)|2 dx dy
ˆ Vr (t, x) =
RR
Qr (t,x)
⃗ ⊗ ⃗u(s, y)|2 ds dy
|∇
RR
ˆ Wr (t, x) = Qr (t,x) |⃗u(s, y)|3 ds dy
RR
ˆ Pr (t, x) = Qr (t,x) |p(s, y)|3/2 ds dy
ˆ Fr (t, x) =
RR
Qr (t,x)
|f⃗(s, y)|3/2 ds dy
We are going to estimate Ur , . . . with respect to Uρ , . . . for 0 < r < ρ/2 < r0 /16. We use
the suitability of ⃗u and apply the local energy inequality to a variant of the test function
ψ of Scheffer we defined on page 432 [426]. If ψ ∈ D(Qr0 (t0 , x0 )) with ψ ≥ 0, we have, for
τ ∈ (t0 − r02 , t0 )
Z
Z
Z
2
⃗ ⊗ ⃗u(s, y)|2 dy ds
ψ(τ, y)|⃗u(τ, y)| dy+2ν
ψ(s, y)|∇
s<τ
Z
Z
≤
(∂t ψ(s, y) + ν∆ψ(s, y))|⃗u(s, y)|2 dy ds
Z
Zs<τ
(14.18)
⃗
+
(|⃗u(s, y)|2 + p(s, y))⃗u(s, y) · ∇ψ(s,
y) dy ds
s<τ
Z
Z
+2
ψ(s, y)⃗u(s, y) · f⃗(s, y) dy ds
s<τ
The choice of ψ is then the following one: we choose a non-negative function ω ∈ D(R × R3 )
such that ω is supported in (−1, 1) × B(0, 3/4) and is equal to 1 on (−1/4, 1/4) × B(0, 1/2),
a non-negative smooth function θ on R that is equal to 1 on (−∞, τ1 ) and to 0 on (τ2 , +∞)
for some τ < τ1 < τ2 < t and we define
ψ(s, y) = r3 ω(
s−t
s−t y−x
,
)θ( 2 )H(r2 + t − s, x − y)
ρ2
ρ
r
where 0 < r ≤ ρ/2 ≤ r0 /16 and H(t, x) = Wνt (x). We then obtain
ZZ
r3
max(Ur (t, x), 2νVr (t, x)) ≤ C
|⃗u(s, y)|2 dy ds
5
ρ
Q (t,x)
Z Zρ
1
|⃗u(s, y)|3 dy ds
+C
r
Q (t,x)
ZZ ρ
1
+C
|p(s, y)|3/2 dy ds
Qρ (t,x) r
ZZ
1/2
+ Cr
|f⃗(s, y)|3/2 dy ds
Qρ (t,x)
470
The Navier–Stokes Problem in the 21st Century (2nd edition)
As
ZZ
(
|⃗u(s, y)|3 dy ds)1/3 ≤Cr1/6 (Ur (t, x) + Vr (t, x))1/2
Qr (t,x)
and
ZZ
|⃗u(s, y)|2 dy ds ≤ Cρ5/3 Wρ2/3 ,
Qρ (t,x)
we obtain
r
1
1
Wr (t, x) ≤C( )5 Wρ (t, x) + C Wρ (t, x)3/2 + C Pρ (t, x)3/2 + Cr5/4 Fρ (t, x)3/2
ρ
r
r
If wr =
1
r 2 Wr ,
pr =
1
r 2 Pr
and fr =
1
F ,
r 1/2 r
we get
r
ρ
ρ
ρ
wr (t, x) ≤ C1 (( )3 wρ (t, x) + ( )3 wρ (t, x)3/2 + ( )3 pρ (t, x)3/2 + ( )3/4 fρ3/2 )
ρ
r
r
r
(14.19)
where the constant C1 does not depend on r nor ρ.
We now turn our attention to the pressure. We introduce a function θ ∈ D(R3 ) with
θ = 1 on B(0, 13/16) and with Supp θ ⊂ B(0, 15/16) and we define ζρ,t,x (y) = θ( y−x
ρ ). For
the sake of simplicity, we write ζ for ζρ,t,x .
On (t − ρ2 , t) × B(x, 3ρ/4), we have ζp = p. From
∆p = div f⃗ −
3 X
3
X
∂i ∂j (ui uj ),
i=1 j=1
we get
∆(ζp) = ζ div f⃗ − ζ
3 X
3
X
i=1 j=1
ui uj + 2
3
X
∂j (p∂j ζ) − p∆ζ
j=1
and we may write ζ(y)p(s, y) = pρ,t,x (s, y) + qρ,t,,x (s, y) + ϖρ,t,x with

P3 P3

qρ,t,x =

j=1
l=1 G ∗ (ζ∂j ∂l (uj ul ))




P3
pρ,t,x = −2 j=1 ∂j G ∗ ((∂j ζ)p) + G ∗ ((∆ζ)p)





ϖ
⃗ − G div(ζ f⃗)
G ∗ (f⃗ · ∇ζ)
ρ,t,x =
We have
qρ,t,x =
3 X
3
X
G ∗ (∂j ∂l (ζuj ul )) − 2G ∗ ((∂j ζ)∂l (uj ul )) − G ∗ ((∂j ∂l ζ)uj ul ).
j=1 l=1
We have, on Qr (t, x),
|2G ∗ ((∂j ζ)∂l (uj ul )) + G ∗ ((∂j ∂l ζ)uj ul )| ≤ CM1Qρ (t,x) uj ul
so that
∥qρ,t,x ∥L3/2 (Qr (t,x)) ≤ C∥⃗u∥2L3 (Qρ (t,x)) .
On Qr (t, x), we have
1
|pρ,t,x (s, y)| ≤ 3
ρ
Z
Z
1
|p(s, z)| dz ≤ C 2 (
|p(s, z)|3/2 dz)2/3
ρ
B(x,ρ)
B(x,ρ)
A Theory of Uniformly Locally L2 Solutions
so that
∥pρ,t,x ∥L3/2 (Qr (t,x)) ≤ C
471
r2
∥p∥L3/2 (Qρ (t,x)) .
ρ2
Finally, we find
∥ϖρ,t,x (s, .)∥L3/2 (B(x,r)) ≤ Cr∥ϖρ,t,x (s, .)∥L3 (B(x,r)) ≤ C ′ r∥f⃗(s, .)∥L3/2 (B(x,ρ))
so that
∥ϖρ,t,x (s, .)∥L3/2 (Qr (t,x)) ≤ Cr∥f⃗(s, .)∥L3/2 (Qρ (t,x)) .
We thus obtain
r
Pr (t, x) ≤ C(Wρ (t, x) + ( )3 Pρ (t, x) + r3/2 Fρ (t, x))
ρ
Dividing by r2 , we obtain
r
pr (t, x) ≤ C2 (wρ (t, x) + ( )3 pρ (t, x) + fρ (t, x))
ρ
(14.20)
where the constant C2 does not depend on r nor ρ.
We shall now consider a sequence ρn = κn r0 /8, where κ ∈ (0, 1/2) will be fixed below.
Let
χn (t, x) = wρn (t, x) + ηpρn (t, x)
where η > 0 will be fixed below as well. From (14.19) and (14.20), we get
χn+1 (t, x) ≤ max(C1 , C2 )κ3 χn (t, x)
+ ηC2 χn (t, x)
+ C1 κ−3 (1 + η −3/2 )χn (t, x)3/2
(14.21)
+ C1 κ−3/4 fρ3/2
+ ηC2 fρn (t, x)
n
We then fix κ such that max(C1 , C2 )κ3 ≤ 1/5 and η such that ηC2 ≤ 1/5. If α is such that
C1 κ−3 (1 + η −3/2 )α1/2 ≤
1
1
and C1 κ−3/4 α1/2 ≤ ,
5
5
then we find that the inequalities
χ0 (t, x) ≤ α
and
sup fρ (t, x) ≤ α
ρ<r0 /8
imply that
χn (t, x) ≤ α
for every n ∈ N. Remark that we have
ZZ
1
χ0 (t, x) ≤ C 2
|⃗u|3 + η|p(t, x)|3/2 ds dy ≤ C(1 + η)λ3
r0
Q0
and
3
5
(3− q )
3/2
3/2
sup fρ (t, x) ≤ C∥f⃗∥L5/3 (Q0 ) ≤ C ′ ∥f⃗∥Lq (Q0 ) r02
≤ C ′ λ3 .
ρ<r0 /8
We have proved the following lemma:
472
The Navier–Stokes Problem in the 21st Century (2nd edition)
Lemma 14.2.
Let q ≥ 5/3. Let Ω be a domain of R × R3 . Let (⃗u, p) a weak solution on Ω of the Navier–
Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Assume that
• ⃗u ∈ L∞ L2 ∩ L2 Ḣ 1 (Ω)
3/2
3/2
• p ∈ Lt Lx (Ω)
• f⃗ ∈ Lqt Lqx (Ω)
• ⃗u is suitable
Let r0 > 0 and Q0 = Qr0 (t0 , x0 ) = (t0 − r02 , t0 ) × B(x0 , r0 ). There exist positive constants
ϵ1 and C3 which depend only on ν and q (but not on x0 , t0 , r0 ,⃗u nor f⃗) such that, if

0 ≤ λ ≤ ϵ1




RR
|⃗u|3 + |p|3/2 dx dt ≤ λ3 r02
Q0



 RR

|f⃗|q dx dt ≤ λ2q r05−3q
Q0
then, for every (t, x) ∈ Q3r0 /4 (t0 , x0 ) and 0 < r ≤ r0 /8, we have
ZZ
|⃗u|3 + |p|3/2 dy ds ≤ C3 λ3 r2 .
Qr (t,x)
Then, we follow again the lines of Kukavica’s paper [286] and we shall prove that, if
q > 5/3, 1Q3r0 /4 (t0 ,x0 ) ⃗u belongs to a parabolic Morrey space M3,τ
2 with τ > 5. We introduce
the reduced quantities
5
3
αr (t, x, τ ) = r5(−1+ τ ) Wr (t, x) = r3(−1+ τ ) wr (t, x)
3
5
βr (t, x, τ ) = r5(−1+ τ ) Pr (t, x) = r3(−1+ τ ) pr (t, x)
and
7
15
5
γr (t, x, τ ) = r− 2 + τ Fr (t, x) = r3(−1+ τ ) fr (t, x)
5
Multiplying (14.19) and (14.20) by r3(−1+ τ ) , we find
15
ρ
r 15
αr (t, x, τ ) ≤C1 (( ) τ αρ (t, x, τ ) + ( )6− τ αρ (t, x, τ )wρ (t, x)1/2 )
ρ
r
ρ 6− 15
ρ 5 5
+C1 (( ) τ βρ (t, x, τ )pρ (t, x)1/2 + ( )3( 4 − τ )γρ (t, x, τ )fρ (t, x)1/2 )
r
r
(14.22)
and
15
15
r 15
ρ
ρ
βr (t, x, τ ) ≤ C2 (( )3− τ αρ (t, x, τ ) + ( ) τ βρ (t, x, τ ) + ( )3− τ γρ (t, x, τ ))
r
ρ
r
(14.23)
Recalling that, due to Lemma 14.2, we have (if λ ≤ ϵ1 ), wr ≤ C3 λ3 and pr ≤ C3 λ3 , and
that fr (t, x) ≤ C4 λ3 by assumption. We shall now consider a sequence ρn = κn r0 /8, where
κ ∈ (0, 1/2) will be fixed below. Let
χn (t, x, τ ) = αρn (t, x, τ ) + ηβρn (t, x, τ )
A Theory of Uniformly Locally L2 Solutions
473
where η > 0 will be fixed below as well. From (14.22) and (14.23), we get
15
χn+1 (t, x, τ ) ≤ max(C1 , C2 )κ τ χn (t, x, τ )
15
+ ηC2 κ τ
+ C1 κ
−3
15
τ −6
χn (t, x, τ )
(14.24)
(1 + η −1 )(C3 λ3 )1/2 χn (t, x, τ )
5
15
+ C1 κ3( τ −1) (C4 λ3 )1/2 γρn (t, x, τ ) + ηC2 κ τ
We then fix κ such that max(C1 , C2 )κ
small enough to grant that
15
C1 κ τ
−6
15
τ
(1 + η −1 )(C3 λ3 )1/2 ≤
−3
γρn (t, x, τ )
15
≤ 1/5 and η such that ηC2 κ τ
−3
≤ 1/5. If λ is
5
1
1
and C1 κ3( τ −1) (C4 λ3 )1/2 ≤
5
5
we find that
χn (t, x, τ ) ≤ max(χ0 (t, x), sup γρ (t, x, τ ))
ρ<r0 /8
for every n ∈ N.
Remark that we have
χ0 (t, x, τ ) ≤ C
and for
1
σ
=
2
τ
ZZ
1
5(1− 3 )
r0 τ
|⃗u|3 + η|p(t, x)|3/2 ds dy ≤ C
Q0
1
3(1− 5 )
r0 τ
(1 + η)λ3
+ 15 ,
3/2
sup γρ (t, x, τ ) ≤ C∥f⃗∥Lσ (Q0 ) ;
ρ<r0 /8
thus, if τ is chosen with
2
τ
+
1
5
≥ 1q , we find
15
2
1
ρ<r0 /8
1
1
( + − )
3/2
sup γρ (t, x, τ ) ≤ C∥f⃗∥Lq (Q0 ) r02 τ 5 q ≤ C ′
3(1− 5 )
r0 τ
λ3 .
We have proved the following lemma:
Lemma 14.3.
Let q > 5/3 and τ > 5 with τ2 + 15 ≥ 1q . Let Ω be a domain of R × R3 . Let (⃗u, p) a weak
solution on Ω of the Navier–Stokes equations
⃗ u + f⃗ − ∇p,
⃗
∂t ⃗u = ν∆⃗u − ⃗u · ∇⃗
div ⃗u = 0.
Assume that
• ⃗u ∈ L∞ L2 ∩ L2 Ḣ 1 (Ω)
3/2
3/2
• p ∈ Lt Lx (Ω)
• f⃗ ∈ Lqt Lqx (Ω)
• ⃗u is suitable
Let r0 > 0 and Q0 = Qr0 (t0 , x0 ) = (t0 − r02 , t0 ) × B(x0 , r0 ). There exist positive constants
ϵ2 and C5 which depend only on ν, q and τ (but not on x0 , t0 , r0 , ⃗u nor f⃗) such that, if

0 ≤ λ ≤ ϵ2




RR
|⃗u|3 + |p|3/2 dx dt ≤ λ3 r02
Q0




 RR
|f⃗|q dx dt ≤ λ2q r05−3q
Q0
474
The Navier–Stokes Problem in the 21st Century (2nd edition)
3/2,τ /2
then, 1Q3r0 /4 (t0 ,x0 ) ⃗u ∈ M3,τ
and 1Q3r0 /4 (t0 ,x0 ) p ∈ M2
2
with
−1+ τ5
∥1Q3r0 /4 (t0 ,x0 ) ⃗u∥M3,τ ≤ C5 λr0
2
and
2(−1+ τ5 )
∥1Q3r0 /4 (t0 ,x0 ) p∥M3/2,τ /2 ≤ C5 λ2 r0
2
The next move is to use the subcritical estimates on ⃗u and p⃗ to bootstrap those regularity
estimates to higher regularity estimates. Indeed, let ϕ be a smooth function on (−∞, 0] × R3
such that ϕ is equal to 1 on (−1, 0) × B(0, 1) and to 0 outside of (−(3/2)2 , 0) × B(0, 3/2) .
We define
4(t − t0 ) 2(x − x0 )
ψ(t, x) = ϕ(
,
).
r02
r0
Assume now that q > 5/2. Thus, we may choose τ in Lemma 14.3 such that
τ > 10.
Let ⃗v = ψ ⃗u. We have
∂t⃗v = ν∆⃗v + ⃗g −
3
X
∂j⃗hj
j=1
with
⃗ u + p∇ψ
⃗ + ψ f⃗
⃗g = ∂t ψ ⃗u + ν∆ψ ⃗u + (⃗u · ∇ψ)⃗
and
hj,l = 2∂j ψ ul + ψuj ul + pψ δj,l .
3/2,τ /2
We find that hj,l ∈ M2
with
−2+ 10
τ
∥hj,l ∥M3/2,τ /2 ≤ C6 λr0
2
3/2,τ /4
and, since τ < 4q, that ⃗g ∈ M2
with
−3+ 20
τ
∥⃗g ∥M3/2,τ /4 ≤ C6 λr0
2
α
Let C be the (homogeneous) space of parabolic Hölderian functions of exponent α ∈ (0, 1):
∥F ∥C α =
|F (t, x) − F (s, y)|
.
1/2 + |x − y|)α
(t,x)̸=(s,y) (|t − s|
sup
Rt
α
Applying Proposition 13.4, we find that 1t>t0 −r02 t0 −r2 Wν(t−s) ∗ ⃗g ds belongs to Ct,x
with
0
R
t
β
20
10
α = 2 − τ and that 1t>t0 −r02 t0 −r2 Wν(t−s) ∗ ∂j⃗hj ds belongs to Ct,x with β = 1 − τ . As ⃗v
0
−3+ 20
τ
r0α +
is equal to 0 when |x − x0 | > r02 , we find that ⃗v is bounded on QR0 (t0 , x0 ) by Cλr0
−2+ 10
τ
r0β
Cr0
= C7 λr0−1 . The theorem is proved.
A Theory of Uniformly Locally L2 Solutions
475
Combining Theorems 14.2, 14.3 and 14.4, we get the following corollary:
Inequalities in the L∞ norm for local Leray solutions
Theorem 14.5.
Let ⃗u be a local Leray solution on (0, T ) × R3 to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.25)
⃗u(0, .) = ⃗u0

where ⃗u0 ∈ L2uloc with div ⃗u0 = 0 and F ∈ (L2t L2x )uloc ((0, T ) × R3 ). Assume moreover
that
R
• limx→+∞ |x−y|<1 |⃗u0 (y)|2 dy = 0
• limx→+∞
RT R
0
|x−y|<1
|F(s, y)|2 dy ds = 0
• f⃗ = div F satisfies the following requirements:
– for |x| > 1: 1|x|>1 f⃗ belongs to (Lq Lqt,x )uloc for some q > 5/2 and
RT R
limx→+∞ 0 |x−y|<1 |f⃗(s, y)|q dy ds = 0
– for |x| ≤ 1: for some β > 0, tβ 1|x|<1 f⃗ ∈ L2 ((0, T ), L2 )
Then, for a constant C0 which does not depend on ν, for
1
T0 = min(T, 1,
C0 ν(1 +
we have
Z
∥⃗
u0 ∥L2
uloc
ν
+
∥F∥(L2 L2 )
uloc
ν 3/2
)
)4
T0
∥⃗u(t, .)∥∞ dt < +∞
T0 /2
Proof. Recall that, on the neighborhood of x0 , the pressure can be defined as
Z
1 ⃗
⃗
px0 (t, x) = (∇ ⊗ ∇)(1B(x0 ,5R0 ) H) +
(K(x − y) − K(x0 − y))H(t, y) dy
∆
|y−x0 |>5R0
where
H = F − ⃗u ⊗ ⃗u
⃗ Due to Theorem 14.2, we have a uniform
and K is the distribution kenel of
⊗ ∇).
control of ⃗u on (0, T0 ), where T0 is given by
1 ⃗
∆ (∇
1
T0 = min(T, 1,
C0 ν(1 +
∥⃗
u0 ∥L2
uloc
ν
+
∥F∥(L2 L2 )
uloc
ν 3/2
)
)4
(where the constant C0 does not depend on ν). This control is the following one:
Z
∥F∥(L2 L2 )uloc
√
sup sup (
|⃗u(t, y)|2 dy)1/2 ≤ 2(∥⃗u0 ∥L2uloc + C0
)
3
ν
0<t<T0 x∈R
|x−y|<1
476
The Navier–Stokes Problem in the 21st Century (2nd edition)
and
T0
Z
sup (
x∈R3
2 2
⃗ ⊗ ⃗u(s, y)|2 dy ds)1/2 ≤ √2 (∥⃗u0 ∥L2 + C0 ∥F∥(L√L )uloc ).
|∇
uloc
ν
ν
|x−y|<1
Z
0
Then (the proof of) Theorem 14.3 gives that, for R > 1,
Z
Z T0 Z
sup sup (
|⃗u(t, y)|2 dy)1/2 + sup (
0<t<T0 |x|>R
|x|>R
|x−y|<1
⃗ ⊗ ⃗u(s, y)|2 dy ds)1/2
|∇
|x−y|<1
0
r
Z
2
≤ CT ( sup (
1/2
|⃗u0 (y)| dy)
|x|>R/2
+
|x−y|<1
T0
Z
+ sup (
|x|>R/2
0
Z
1 + ln R
R
|F(s, y)|2 dy ds)1/2 )
|x−y|<1
where the constant CT depends on ν, T , ∥⃗u0 ∥L2uloc and ∥F∥(L2 L2 )uloc , but not on R. As
T0 ≤ 1, we find
Z T0 Z
sup
|⃗u(t, y)|3 dy ds
|x|>R
|x−y|<1
0
≤C
CT3
r
Z
( sup (
|x|>R/2
2
3/2
|⃗u0 (y)| dy)
+(
|x−y|<1
Z
+ sup (
|x|>R/2
T0
Z
1 + ln R 3
)
R
|F(s, y)|2 dy ds)3/2 )
|x−y|<1
0
RT R
Now, we estimate sup|x|>R 0 0 |x−y|<1 |px (t, y)|3/2 dy ds. We assume that R >
⃗ ⊗ ∇)(1
⃗
max(2, 5R0 ). For qx = 1 (∇
B(x,5R ) H), we find
∆
Z
T0
Z
0
3/2
|qx (t, y)|
Z
Z
|⃗u(s, y)|3 + |F(s, y)|3/2 ds dy
dy ds ≤ C
|x−y|<1
0
0
|x−y|<5R0
0
so that
Z
T0
Z
|qx (t, y)|3/2 dy ds
sup
|x|>R
|x−y|<1
0
r
Z
2
≤ CT ( sup (
|x|>R/2
|⃗u0 (y)| dy)
|x−y|<1
Z
+ sup (
|x|>R/2
For ϖx (t, y) =
R
and thus, since
Z
0
T0 Z
0
T0
Z
3/2
+(
1 + ln R 3
)
R
|F(s, y)|2 dy ds)3/2 ).
|x−y|<1
(K(y−z)−K(x−z))H(t, z) dz, we write (for |x| > R and |x−y| < 1)
Z
1
|ϖx (t, y)| ≤ C
|H(t, z)| dz.
4
|z−x|>5R0 |z − x|
|z−x|>5R0
R
1
|z|>5R0 (|z| ln |z|)3
dz < +∞,
Z
(ln(|x − z|))3/2 T0
|ϖx (t, y)| dy dt ≤C
|H(t, z)|3/2 dz dt
|z − x|9/2
|x−y|<1
|z−x|>5R0
0
Z
Z
X
(ln |k|)3/2 T0
≤
|H(t, z)|3/2 dz dt
9/2
|k|
3
√
0
z∈x+k+[0,1]
3
3/2
k∈Z ,|k|≥5R0 − 3
Z
A Theory of Uniformly Locally L2 Solutions
477
and splitting the last sum between |k| > R/4 and |k| < R/4, we find
Z
T0
Z
|ϖx (t, y)|3/2 dy dt
|x−y|<1
0
≤C
ln R
R
3/2
Z
( sup (
x∈R3
|⃗u0 (y)|2 dy)3/2
|x−y|<1
T0
Z
+ sup (
x∈R3
Z
|F(s, y)|2 dy ds)3/2 )
|x−y|<1
0
r
Z
+CT ( sup (
|x|>R/2
2
3/2
|⃗u0 (y)| dy)
+
|x−y|<1
T0
Z
+ sup (
|x|>R/2
Z
1 + ln R
R
!3
|F(s, y)|2 dy ds)3/2 ).
|x−y|<1
0
√
Thus, we find that we have, writing Qx0 = (0, T0 ) × B(x0 , T0 ),
ZZ
lim
|⃗u(s, y)|3 ds dy = 0
x0 →+∞
Qx0
ZZ
lim
x0 →+∞
|px (s, y)|3/2 ds dy = 0
Qx0
ZZ
lim
x0 →+∞
|f⃗(s, y)|q ds dy = 0
Qx0
Then, by applying Theorem 14.4, we find that there exists some R > 0 such that for
T0 /2 < t < T0 and |x| > R, we have |⃗u(t, x)| ≤ Cν √1T .
0
It remains to evaluate |⃗u(t, x)| when |x| < R. We fix ψ ∈ D which is equal to 1 on
B(0, 3R) and we write, for T0 /2 < t < T0 ,
⃗u =Wν(t−T0 /2) ∗ (ψ⃗u(T0 /2, .)) + Wνt ∗ ((1 − ψ)⃗u(T0 /2, .))
Z t
Z t
+
Wν(t−s) ∗ P div(ψF) ds +
Wν(t−s) ∗ P div((1 − ψ)F) ds
T0 /2
Z
T0 /2
t
Z
−
t
Wν(t−s) ∗ P div(ψ(⃗u ⊗ ⃗u)) ds −
T0 /2
Wν(t−s) ∗ P div((1 − ψ)(⃗u ⊗ ⃗u) ds
T0 /2
For x ∈ B(0, R), we find
|⃗u(t, x)| ≤
√
Z
∥Wν(t−T0 /2) ∗(ψ⃗u(T0 /2, .))∥L∞ (dx) + C
Z
|x−y|>2R
Z t Z
t
νt
|⃗u(T0 /2, y)| dy
|x − y|4
Wν(t−s) ∗P div(ψF) ds∥L∞ (dx) + C
+|
T0 /2
|F(s, y)|
T0 /2
Z
|x−y|>2R
t
+∥
Wν(t−s) ∗ P div(ψ(⃗u ⊗ ⃗u)) ds∥L∞ (dx)
T0 /2
Z t
Z
|⃗u ⊗ ⃗u(s, y)|
+C
T0 /2
|x−y|>2R
dy ds
|x − y|4
dy ds
|x − y|4
478
The Navier–Stokes Problem in the 21st Century (2nd edition)
The integration at the large (|x − y| > 2R) is easy to control, and we have
√
Z
νt
|⃗u(T0 /2, y)| dy ≤CR ∥⃗u(T0 /2, .)∥L2uloc
|x
−
y|4
|x−y|>2R
Z t Z
1
|F(s, y)| dy ds ≤CR ∥F∥(L2t L2x )uloc
4
T0 /2 |x−y|>2R |x − y|
Z t Z
dy ds
|⃗u ⊗ ⃗u(s, y)|
≤CR ∥⃗u∥2L∞ L2
uloc
|x
− y|4
T0 /2 |x−y|>2R
Moreover, we have
∥Wν(t−T0 /2) ∗ (ψ⃗u(T0 /2, .))∥L∞ (dx) ≤ C(νt)−3/4 ∥ψ⃗u(T0 /2, .)∥2
and
Z
T0
Z
∥
t
Z
T0
Wν(t−s) ∗ P div(ψ(⃗u ⊗ ⃗u)) ds∥L∞ (dx) dt ≤ C
T0 /2
0
T0 /2
∥ div(ψ(⃗u ⊗ ⃗u))∥Ḃ −1/2 dt
2,1
where
∥ div(ψ(⃗u ⊗ ⃗u))∥Ḃ −1/2 ≤ C∥⃗u∥2H 1 (B(0,5R) .
2,1
⃗
Finally, we have div(ψF) = ∇ψ.F
+ ψ f⃗ ∈ L2 ((T0 /2, T0 ), L2 ), so that
Z
t
3/2
Wν(t−s) ∗ P div(ψF) ds ∈ L4 ((T0 /2, T0 ), Ḃ2,1 ) ⊂ L4 ((T0 /2, T0 ), L∞ ).
T0 /2
Thus, we have 1B(0,5R) ⃗u ∈ L1 ((T0 /2, T0 ), L∞ ).
Using Theorem 14.4, we may give a more quantitative statement:
Quantitative inequalities for the L∞ norm of local Leray solutions
Theorem 14.6.
Let ⃗u be a local Leray solution on (0, T ) × R3 to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)

(14.26)
⃗u(0, .) = ⃗u0
where ⃗u0 ∈ L2uloc with div ⃗u0 = 0 and F ∈ (L2t L2x )uloc ((0, T ) × R3 ). Assume moreover
that
1
• |⃗u0 (x)| ≤ C0 |x|
1
• |F(t, x)| ≤ C0 (√νt+|x|)
2
1
• | div F(t, x)| ≤ C0 (√νt+|x|)
3
A Theory of Uniformly Locally L2 Solutions
479
Then, for constants T0 > 0, R1 > 1 and C1 which depend only on ν, T and C0 , we
have
s
|x|
sup
sup
|⃗u(t, x)| ≤ C1
ln
|x|
|x|>R1 T0 /2<t<T0
and
Z
T0
∥⃗u(t, .)∥∞ dt ≤ C1 .
T0 /2
Proof. Remark that |x|1 2 ∈ L1uloc . Thus, ⃗u0 ∈ L2uloc , and its norm is controlled by C0 .
Similarly, we have
Z +∞ Z
Z
Z
dy dt
1 +∞
dt
dy
√
√
=
2
4
4
ν
|y|
(
νt
+
|y|)
(
t
+
1)
0
|x−y|<1
0
|x−y|<1
so that F ∈ (L2t L2x )uloc and its norm is controlled by C0 . It means that we have a control of
⃗ ⊗ ⃗u∥(L2 L2 )
the norms ∥⃗u∥L∞ L2uloc and ∥∇
on a band (0, T0 ) × R3 , where T0 depends only
uloc
on ν, T and C0 (Theorem 14.2).
Now, we check that ⃗u and F fulfill the assumptions of Theorem 14.5:
ˆ for |x| > 2,
Z
|⃗u0 (y)|2 dy ≤ CC02
|x−y|<1
ˆ for |x| > 2,
T
Z
Z
1
→x→∞ 0
|x|2
|F(s, y)|2 dy ds ≤ CC02
|x−y|<1
0
√
1
ˆ | t div F(t, x)| ≤ C0 (√νt+|x|)
2 , so that
by C0 .
√
1
→x→∞ 0
|x|2
t div F ∈ (L2t L2x )uloc and its norm is controlled
ˆ for |x| > 1: |1|x|>1 div F(t, x)| ≤ C0 |x|−3 ; the function 1|x|>1 |x|−3 belongs to Lq for
evert q > 1. Moreover, for |x| > 2,
T
Z
0
Z
|x−y|<1
| div F(s, y)|q dy ds ≤ CC0q |x|−3q →x→∞ 0
Moreover, the proof of Theorem 14.5 gives us the following estimates for |x| > 5R0 (for
constants C2 , C3 which depend only on ν, T , C0 (and q):
T0
Z
T0
|⃗u(t, y)|3 dy ds ≤C2 (|x|−3 +
Z
3/2
|px (t, y)|
Z
T
dy ds ≤C2 (|x|
Z
+
|x−y|<1
ln |x|
|x|
3/2
ln |x|
|x|
3/2
−3q
≤
T05
C3
) ≤ T05
)≤
s
q
| div F(s, y)| dy ds ≤C2 |x|
0
−3
|x−y|<1
0
|x−y|<1
0
Z
Z
ln |x|
|x|
T05
!2q
s
ln |x|
|x|
s
ln |x|
|x|
C3
C3
!3
!3
480
The Navier–Stokes Problem in the 21st Century (2nd edition)
q
|x|
< ϵ0 , Theorem 14.4 gives that, for
Then, if R1 is large enough to ensure that C3 ln|x|
|x| > R1 , we have, for T0 /2 < t < T0 ,
s
ln |x| 1
|⃗u(t, x)| ≤ C4 (C3
) .
|x| T0
Thus, the theorem is proved.
14.4
A Weak-Strong Uniqueness Result
In this section, we generalize the von Wahl weak-strong uniqueness theorem [494] (see
Proposition 12.3), replacing the L2 Leray solutions by L2uloc suitable solutions:
Weak-strong uniqueness
Theorem 14.7.
Let ⃗u1 , ⃗u2 be two local Leray solutions on (0, T ) × R3 to the same problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.27)
⃗u(0, .) = ⃗u0

where ⃗u0 ∈ L2uloc with div ⃗u0 = 0 and F ∈ (L2t L2x )uloc ((0, T ) × R3 ).
Assume moreover that ⃗u1 can be written as ⃗u1 = ⃗u3 + ⃗u4 with ⃗u3 ∈ L2 ((0, T ), L∞ )
and ⃗u4 ∈ L∞ ((0, T ), V̄ 1 ) (where V̄ 1 = M(H 1 7→ L2 )) and that
sup ∥⃗u4 (t, .)∥V̄ 1 < ϵ0 ν
0<t<T
where ϵ0 is a small positive constant which does not depend on ν, ⃗u0 , F, T , ⃗u1 nor ⃗u2 .
Then ⃗u1 = ⃗u2 .
Proof. Let pi be the pressure associated to ⃗ui . We have, due to the suitability of ⃗u2 and the
regularity of ⃗u1 ,

2
2
2
⃗ ⊗ ⃗u1 |2 − div((p1 + |⃗u1 | )⃗u1 ) + ⃗u1 · div F

∂t ( |⃗u21 | ) =
ν∆( |⃗u21 | ) − ν|∇

2






⃗ ⊗ ⃗u1 ).(∇
⃗ ⊗ ⃗u2 ) − div(p2 ⃗u1 + p1 · ⃗u2 )
∂t (⃗u1 · ⃗u2 ) =
ν∆(⃗u1 · ⃗u2 ) − 2ν(∇
⃗
⃗ 1 ) + (⃗u1 + ⃗u2 ) · div F

−⃗u1 .(⃗u2 · ∇⃗u2 ) − ⃗u2 .(⃗u1 · ∇u






2
2
2

⃗ ⊗ ⃗u2 |2 − div((p2 + |⃗u2 | )⃗u2 ) + ⃗u2 · div F − µ
∂t ( |⃗u22 | ) = ν∆( |⃗u22 | ) − ν|∇
2
where µ is some non-negative locally finite measure.
Let w
⃗ = ⃗u1 − ⃗u2 and q = p1 − p2 . We obtain
∂t (
|w|
⃗2
|w|
⃗2
⃗ ⊗ w|
) =ν∆(
) − ν|∇
⃗ 2 − div(q w)
⃗ −A−µ
2
2
A Theory of Uniformly Locally L2 Solutions
481
with
|⃗u1 |2
|⃗u2 |2
⃗ u2 ) + ⃗u2 .(⃗u1 · ∇u
⃗ 1)
⃗u1 +
⃗u2 ) − (⃗u1 .(⃗u2 · ∇⃗
2
2
1
= div |⃗u1 |2 ⃗u1 + |⃗u2 |⃗u2 − (⃗u1 · ⃗u2 )(⃗u1 + ⃗u2 )
2
1
⃗ u1 ) + ⃗u1 .(⃗u1 · ∇u
⃗ 2 ) − ⃗u1 .(⃗u2 · ∇⃗
⃗ u2 ) − ⃗u2 .(⃗u1 · ∇u
⃗ 1)
+
⃗u2 .(⃗u2 · ∇⃗
2
1
1
⃗ u1 ) − ⃗u1 .(w
⃗ w)
= div |w|
⃗ 2 ⃗u1 + (⃗u1 · w)
⃗ w
⃗ − |w|
⃗ 2w
⃗ +
w.(
⃗ w
⃗ · ∇⃗
⃗ ·∇
⃗
2
2
1
⃗ w).
= div |w|
⃗ 2 ⃗u1 + 2(⃗u1 · w)
⃗ w
⃗ − |w|
⃗ 2w
⃗ − ⃗u1 .(w
⃗ ·∇
⃗
2
A = div(
We then follow the lines of the proof of Theorem 14.2. For φ(x) = φx0 (x) = φ0 (x − x0 ),
we write
Z
Z tZ
Z tZ
|w(t,
⃗ x)|2
|w|
⃗2
⃗ ⊗ w|
φ
dx ds ≤ − ν
φ|∇
⃗ 2 dx ds + ν
(∆φ)
dx ds
2
2
0
0
Z tZ
|w|
⃗2
⃗
+
(w
⃗ · ∇φ)(q
⃗ · ⃗u1 −
) dx ds
x0 + w
2
0
Z tZ
Z
Z
t
⃗2
⃗ |w|
⃗ w)
+
(⃗u1 · ∇φ)
dx ds −
φ⃗u1 .(w
⃗ ·∇
⃗ dx ds
2
0
0
where qx0 is defined as where
qx0 (t, x) =
1 ⃗
⃗
(∇ ⊗ ∇)(1
B(x0 ,5R0 ) H) +
∆
Z
(K(x − y) − K(x0 − y))H(t, y) dy
|y−x0 |>5R0
and
H = ⃗u1 ⊗ ⃗u1 − ⃗u2 ⊗ ⃗u2 = w
⃗ ⊗ ⃗u1 + ⃗u1 ⊗ w
⃗ −w
⃗ ⊗ w.
⃗
We then write
qx0 = R1,x0 + R2,x0 + S1,x0 + S2,x0
with
1 ⃗
⃗
⊗ ∇)(1
⃗ ⊗ w)
⃗
R1,x0 = − (∇
B(x0 ,5R0 ) w
Z ∆
R2,x0 =
(K(x − y) − K(x0 − y))w
⃗ ⊗ w(t,
⃗ y) dy
|y−x0 |>5R0
1 ⃗
⃗
S1,x0 = − (∇
⊗ ∇)(1
u1 ⊗ w
⃗ +w
⃗ ⊗ ⃗u1 ))
B(x0 ,5R0 ) (⃗
∆
Z
S1,x0 =
(K(x − y) − K(x0 − y))(⃗u1 ⊗ w
⃗ +w
⃗ ⊗ ⃗u1 )(t, y) dy
|y−x0 |>5R0
Defining again
α(t) = ∥w∥
⃗ L2uloc = sup ∥φw∥
⃗ 2
φ∈B
⃗ ⊗ w∥
γ(t) = ∥∇
⃗ (L2 L2 )uloc ((0,t)×R3 ) = sup (
φ∈B
Z tZ
0
Z tZ
δ(t) = ∥w∥
⃗ (L3 L3 )uloc ((0,t)×R3 ) = sup (
φ∈B
⃗ ⊗ w(s,
|φ(x)∇
⃗ x)|2 dx ds)1/2
0
|φ(x)w(s,
⃗ x)|3 dx ds)1/2
482
The Navier–Stokes Problem in the 21st Century (2nd edition)
we find
Z tZ
Z t
|w|
⃗2
ν
(∆φ)
dx ds ≤Cν
α2 (s) ds
2
0
0
Z tZ
⃗2
⃗ |w|
−
(w
⃗ · ∇φ)
dx ds ≤Cδ(t)3
2
0
Z tZ
3
⃗
(w
⃗ · ∇φ)(R
1,x0 + R2,x0 ) dx ds ≤Cδ(t)
0
while
Z tZ
⃗
(w
⃗ · ∇φ)(⃗
u1 · w)
⃗ dx ds ≤C
0
Z
t
∥⃗u3 (s, .)∥∞ α(s)2 ds
s
0
γ(t)2
+ C∥⃗u4 ∥L∞ V̄ 1 γ(t)
t
Z
α(s)2 ds
+
0
Z tZ
⃗
(w
⃗ · ∇φ)(S
1,x0 + S2,x0 ) dx ds ≤C
0
Z
t
∥⃗u3 (s, .)∥∞ α(s)2 ds
s
0
+ C∥⃗u4 ∥L∞ V̄ 1 γ(t) γ(t)2 +
t
Z
α(s)2 ds
0
Z tZ
2
⃗
⃗ |w|
(⃗u1 · ∇φ)
dx ds ≤C
2
Z
t
∥⃗u3 (s, .)∥∞ α(s)2 ds
Z t
2
+ C∥⃗u4 ∥L∞ V̄ 1 (γ(t) +
α(s)2 ds)
0
s
Z tZ
Z t
⃗ w)
−
φ⃗u1 .(w
⃗ ·∇
⃗ dx ds ≤Cγ(t)
∥⃗u3 (s, .)∥2∞ α(s)2 ds
0
0
0
0
s
+ C∥⃗u4 ∥L∞ V̄ 1 γ(t)
γ(t)2
t
Z
α(s)2 ds
+
0
Writing
Z t
Z t
δ(t)2 ≤ C(
α(s)2 ds + γ(t)(
α6 (s) ds)1/6 ),
0
0
we obtain (for every η > 0, with constants that do not depend on η)
Z tZ
|w(t,
⃗ x)|2
⃗ ⊗ w|
dx ds + ν
φ|∇
⃗ 2 dx ds
2
0
Z t
Z t
Z t
2
2
3/2
3/2
≤ C0 ν
α (s) ds + C0 (
α (s) ds) + C0 γ(t) (
α6 (s) ds)1/4
0
0
0
s
Z t
Z t
+C0
∥⃗u3 (s, .)∥∞ α(s)2 ds + C0 γ(t)
∥⃗u3 (s, .)∥2∞ α(s)2 ds
Z
φ
0
0
2
Z
+C0 ∥⃗u4 ∥L∞ V̄ 1 (γ(t) +
0
s
t
2
α(s) ds) + C0 ∥⃗u4 ∥L∞ V̄ 1 γ(t) γ(t)2 +
Z
0
t
α(s)2 ds
A Theory of Uniformly Locally L2 Solutions
Z t
Z t
2
≤C1 ν
α (s) ds + C1 (
α2 (s) ds)3/2
0
0
Z t
+ C1 ηγ(t)2 + C1 η −3
α6 (s) ds
0
Z t
+ C1
∥⃗u3 (s, .)∥∞ α(s)2 ds
0
Z t
+ C1 η −1
∥⃗u3 (s, .)∥2∞ α(s)2 ds + C1 ηγ(t)2
0
Z t
+ C1 ∥⃗u4 ∥L∞ V̄ 1 (γ(t)2 +
α(s)2 ds)
483
0
and thus
Z t
Z t
2
3/2
−3
max(α(t) , νγ(t) ) ≤C1 ν
α (s) ds + C1 (
α (s) ds) + C1 η
α6 (s) ds
0
0
0
Z t
Z t
2
+ C1
∥⃗u3 (s, .)∥∞ α(s) ds + C1 ∥⃗u4 ∥L∞ V̄ 1
α(s)2 ds
0
0
Z t
−1
+ C1 η
∥⃗u3 (s, .)∥2∞ α(s)2 ds
2
Z
2
t
2
0
+ C1 (2η + ∥⃗u4 ∥L∞ V̄ 1 )γ(t)2
Thus, if η is chosen such that C1 η < 14 ν and if C1 ϵ0 < 14 , we find, for a constant Cν which
depends on ν,
Z t
Z t
Z t
6
2
6
3
6
3/2
α(t) ≤Cν t
α(s) ds + Cν t (
α(s) ds) + Cν (
α(s)6 ds)6
0
0
0
Z t
Z t
2
+ Cν (
∥⃗u3 (s, .)∥∞ ds)
∥⃗u3 (s, .)∥∞ α(s)6 ds
0
0
Z t
+ Cν ∥⃗u4 ∥3L∞ V 1 t2
α(s)6 ds
0
Z t
Z t
+ Cν (
∥⃗u3 (s, .)∥2∞ ds)2
∥⃗u3 (s, .)∥2∞ α(s)6 ds
0
As long as
Rt
0
0
α(s)6 ds < 1, we find that
6
Z
α(t) ≤ Cν
t
A(s)α(s)6 ds
0
with
A(s) = T 2 + T 3 + 1 + T ∥⃗u3 ∥2L2 L∞ ∥⃗u3 (s, .)∥∞ + T 2 ∥⃗u4 ∥3L∞ V̄ 1 + ∥⃗u3 ∥2L2 L∞ ∥⃗u3 (s, .)∥2∞ .
We then conclude by Grönwall’s lemma that α
⃗ = 0, i.e. ⃗u1 = ⃗u2 .
14.5
Global Existence for Local Leray Solutions
In this section, we show how to turn the local existence result of Theorem 14.1 into a
global existence result, assuming the initial data and the forcing term vanish at infinity.
484
The Navier–Stokes Problem in the 21st Century (2nd edition)
Definition 14.2. [Vanishing at infinity functions]
Let φ0 ∈ D(R3 ), φ0 ≥ 0, such that
X
φ0 (x − k) = 1
k∈Z3
and let
B = {φx0 = φ0 (. − x0 ) / x0 ∈ R3 }
Then define Lpuloc (R3 ) for 1 ≤ p < +∞ by
∥h∥Lpuloc = sup ∥hφ∥p .
φ∈B
We define E p , the space of functions in Lpuloc that vanish at infinity, by
f ∈ E p ⇔ f ∈ Lpuloc and
lim ∥f φx0 ∥p = 0.
x0 →∞
Lemma 14.4.
D(R3 ) is dense in E p .
p
p
⊂
be the space of compactly supported functions in E p . Then Ecomp
Proof. Let Ecomp
p
p
3
p
L ⊂ E continuous embeddings). As D(R ) is dense in LP, we just have to check that
p
is dense in E p . But this is obvious, since limN →∞ ∥f |k|>N φ0 (x − k)∥Lpuloc = 0 for
Ecomp
p
f ∈E .
Lemma 14.5.
p
For 1 ≤ p < +∞, let Eσp be the space of divergence-free vector fields in (E p )3 and let Ecomp,σ
p 3
be the space of compactly supported functions divergence-free vector fields in (E ) . Then
p
is dense in Eσp .
Ecomp,σ
Proof. This is proved via a decomposition on a divergence-free wavelet basis . Let us recall
⃗ i (1 ≤ i ≤ 3),
the results of [309, 313]: there exists compactly supported C 1 vector fields ϕ
∗
∗
⃗
⃗
⃗
ϕi (1 ≤ i ≤ 3), ψl (1 ≤ l ≤ 14) and ψl (1 ≤ l ≤ 14) so that
⃗ i generate a bi-orthogonal multi-resolution analysis V
⃗j of
• the scaling functions ϕ
2
3 3
∗
⃗ generate the dual multi-resolution analysis V
⃗j
(L (R )) while ϕ
i
• If P⃗j is the associated projection operator
P⃗j (f⃗) =
3
XX
⃗ ∗ ⟩ϕ
⃗ i,j,k
⟨f⃗|ϕ
i,j,k
k∈Z3 i=1
⃗ i,j,k (x) = 23j/2 ϕ
⃗ i (2j x − k) and ϕ
⃗ ∗ (x) = 23j/2 ϕ
⃗ ∗ (2j x − k) and if f⃗ ∈ L1
with ϕ
i
i,j,k
loc
with div f⃗ = 0, then div P⃗j (f⃗) = 0. In particular, if f⃗ ∈ Eσp , then P⃗j (f⃗) ∈ Eσp ,
limj→+∞ ∥P⃗j (f⃗) − f⃗∥Lpuloc = 0 and limj→−∞ ∥P⃗j (f⃗)∥Lpuloc = 0
⃗l = 0 (1 ≤ l ≤ 14) and, when f⃗ ∈ E p , we have
• div ψ
σ
(P⃗j+1 − P⃗j )(f⃗) =
14
XX
⃗ ∗ ⟩ψ
⃗i,j,k
⟨f⃗|ψ
i,j,k
k∈Z3 i=1
A Theory of Uniformly Locally L2 Solutions
485
⃗i,j,k (x) = 23j/2 ψ
⃗i (2j x − k) and ψ
⃗ ∗ (x) = 23j/2 ψ
⃗ ∗ (2j x − k). In particular, the
with ψ
i
i,j,k
⃗ defined by
operator Π
⃗ f⃗) =
Π(
14
XXX
⃗ ∗ ⟩ψ
⃗i,j,k
⟨f⃗|ψ
i,j,k
j∈Z k∈Z3 i=1
is a Calderón–Zygmund operators and is a bounded projection operator from (L2 )3
onto the space L2σ of divergence-free square integrable vector fields; but this not an
⃗ ∗ ̸= 0) and Π
⃗ ̸= P.
orthogonal projection operator (div ψ
i
• for f⃗ ∈ Eσp and j ∈ Z, we have
14
X X
⃗ ∗ ⟩ψ
⃗i,j,| ∥Lp = 0
⟨f⃗|ψ
i,j,k
uloc
lim ∥
N →+∞
|k|>N i=1
Thus, we have

lim
lim ∥f − P⃗J+1 (f⃗)∥Lpuloc +
J→+∞ N →+∞
J
X
j=−J
∥
14
X X
⃗ ∗ ⟩ψ
⃗i,j,k ∥Lp
⟨f⃗|ψ
i,j,k
uloc
|k|>N i=1
+∥P⃗−J (f⃗)∥Lpuloc = 0
and thus
lim
J
14
X
X X
⃗ ∗ ⟩ψ
⃗i,j,k ∥Lp = 0.
⟨f⃗|ψ
i,j,k
uloc
lim ∥f⃗ −
J→+∞ N →+∞
The lemma is proved, since
j=−J |k|≤N i=1
PJ
j=−J
P
|k|≤N
P14
⃗ ⃗∗
⃗
i=1 ⟨f |ψi,j,k ⟩ψi,j,k
p
belongs to Ecomp,σ
.
We give a definition forces that vanish at infinity similar to Definition 14.2:
Definition 14.3 (Vanishing at infinity forces).
Let φ0 ∈ D(R3 ), φ0 ≥ 0, such that
X
φ0 (x − k) = 1
k∈Z3
and let
B = {φx0 = φ0 (. − x0 ) / x0 ∈ R3 }
Then define (Lp H s )uloc ((0, T ) × R3 ), for 1 ≤ p < +∞ and s ∈ R, by
∥f ∥(Lp H s )uloc = sup ∥f φ∥Lp H s .
φ∈B
We define F s,p , the space of functions in (Lp H s )uloc that vanish at infinity, by
f ∈ F s,p ⇔ f ∈ (Lp H s )uloc and
lim ∥f φx0 ∥Lp H s = 0.
x0 →∞
Lemma 14.6.
Lp H s is dense in F s,p .
Proof. Just check that lim ∥f
N →∞
X
|k|>N
φ0 (x − k)∥(Lp H s )uloc = 0 for f ∈ F s,p .
486
The Navier–Stokes Problem in the 21st Century (2nd edition)
We may now discuss the existence of global solutions in L2uloc . Existence of global weak
solutions, generalizing the result of Leray for ⃗u0 ∈ (L2 )3 was first established for ⃗u0 ∈ (Lp )3
(2 ≤ p < ∞) by C. Calderón [77] and later by Lemarié-Rieusset [310]. The case ⃗u0 ∈ L2uloc
in absence of force was discussed by Lemarié-Rieusset in [313].
Global uniformly locally square integrable solutions
Theorem 14.8.
Let ⃗u0 ∈ L2uloc with div ⃗u0 = 0, vanishing at infinity: ⃗u0 ∈ Eσ2 . Let
\
\
F∈
F 2,3 ((0, T ) × R3 ) ⊂
(L3 H 2 )uloc ((0, T ) × R3 ).
0<T <+∞
0<T <+∞
Then there exists a global suitable weak solution ⃗u to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.28)
⃗u(0, .) = ⃗u0

on (0, +∞) × R3 such that
\
2
3
2 1
3
⃗u ∈
(L∞
t Lx )uloc ((0, T ) × R ) ∩ (L Hx )uloc ((0, T ) × R ).
0<T <+∞
Proof. Step 1: the local solution with initial data in Eσ2 .
By Theorem 14.1, we know that there exists a local Leray solution ⃗u1 defined on an
interval (0, T1 ). By Theorem 14.3, we even know that ⃗u1 ∈ L∞ ((0, T1 ), E 2 ). Moreover,
we know by Theorem 14.5 that ⃗u1 ∈ L1 (T1 /2, T1 ), L∞ ). As L∞ ∩ E2 ⊂ E3 with
2/3
1/3
∥⃗u1 ∥L3uloc ≤ ∥⃗u1 ∥L2 ∥⃗u1 ∥∞ , we find that ⃗u1 ∈ L3 ((T1 /2, T1 ), E 3 ).
uloc
Moreover, for almost every t ∈ (0, T1 ), we have
lim ∥⃗u1 (t, .) − ⃗u1 (s, .)∥L2uloc = 0.
s→t,s>t
(14.29)
Indeed, as ⃗u1 is suitable, we use the local energy inequality o get that for all φ ∈ B
and for all Lebesgue points t, τ of ∥⃗uφ∥2 with t < τ , we have the inequality
Z τ
2
⃗ ⊗ ⃗u(s, .))φ(x)∥22 ds ≤
∥⃗u(τ, .)φ(x)∥2 + 2ν
∥∇
t
∥⃗u(t, .)φ(x)∥22 + ν
τ
ZZ
|⃗u|2 ∆(φ2 (x)) dx ds +
t
τ
ZZ
⃗
(|⃗u|2 + 2(p − pφ ))(⃗u.∇)φ
ds dx.
t
Then, we use the weak continuity of t → ⃗u(t, .) to conclude that this inequality is valid
for almost all t and for all τ > t. Thus, for all φ ∈ B and for all Lebesgue points t of
∥⃗uφ∥2 , we have limτ >t,τ →t ∥(⃗u(τ ) − ⃗u(t))φ∥2 = 0. Moreover, Theorem 14.3 implies a
good uniform (in the time variable) control of the decay of ∥⃗u(τ )φ0 (x − k)∥2 when k
goes to ∞, whereas we have a good control of ∥(⃗u(τ ) − ⃗u(t))φ0 (x − k)∥2 on the points
tPwhich are Lebesgue points for all ∥⃗uφ0 (x − k)∥2 , k ∈ Z3 ; hence, for almost all t. Since
k φ0 (x − k) = 1, this gives (14.29).
A Theory of Uniformly Locally L2 Solutions
487
Step 2: the local solution with initial data in Eσ3 .
We now consider a time t1 ∈ (T1 /2, T1 ) such that ⃗u1 (t1 , .) ∈ E 3 and such that
lim
s→t1 ,s>t1
∥⃗u1 (t1 , .) − ⃗u1 (s, .)∥L2uloc = 0.
In particular, ⃗u1 is a local Leray solution on (t1 , T1 ) of the Cauchy problem for the
Navier—Stokes equations with initial value ⃗u1 (t1 , .). We are going to associate to this
Cauchy problem two other local Leray solutions.
First, we remark that τ ≥ 0 7→ eντ ∆ ⃗u1 (t1 , .) is continuous from [0, +∞) to E 3 and
sup ∥eντ ∆ ⃗u1 (t1 , .)∥L3uloc ≤ ∥⃗u1 (t1 , .)∥L3uloc .
0<τ <1
Moreover,
sup
√
0<τ <1
τ ∥eντ ∆ ⃗u1 (t1 , .)∥∞ ≤ Cν ∥⃗u1 (t1 , .)∥L3uloc
and
lim
√
τ →0
τ ∥eντ ∆ ⃗u1 (t1 , .)∥∞ = 0.
In particular, τ 1/4 eντ ∆ ⃗u(t1 , .) ∈ L∞ ((0, 1), E 6 ) and
lim τ 1/4 ∥eντ ∆ ⃗u1 (t1 , .)∥L6uloc = 0.
τ →0
Finally, as L3uloc ⊂ L2uloc , we have
∥eντ ∆ ⃗u1 (t1 , .)∥(L2 H 1 )uloc ((0,1)×R3 ) ≤ Cν ∥⃗u1 (t1 , .)∥L3uloc .
R
⃗ = t eν(t−s)∆ P div F(s, .) ds for t1 < t < t1 + 1. We fix ψ0 ∈ D
Secondly, we write W
t1
such that ψ is non-negative and is identically equal to 1 on a neighborhood of the
support of φ0 (ψ0 (x) = 1 when the distance of x to the support of φ0 is no more than
5). Writing
⃗ (t, .)|
φ0 (x − k)|W
Z t
Z tZ
ν(t−s)∆
≤|
e
P div(ψ0 (. − k)F(s, .)) ds| + C
t1
t1
≤C ′
Z
t
t1
+ C′
|x−y|>5
1
|F(s, y)| dy
|x − y|4
1
∥ψ0 (−k)F(s, .)∥H 2 ds
ν(t − s)
Z t
X
1
∥φ0 (. − j)F(s, .)∥H 2 ds
|j − k|4 t1
3
p
j∈Z ,j̸=k
≤Cν′′ ∥F∥(L3 H 2 )uloc ((t1 ,t1 +1)×R3 ) .
⃗ belongs to L∞ ((t1 , t1 + 1) × R3 ) and that we have
Thus, we find that W
⃗ (t1 + τ, .)∥L3 ≤ C sup ∥W
⃗ (t1 + τ, .)∥∞ ≤ Cν ∥F∥(L3 H 2 ) ((t ,t +1)×R3 )
sup ∥W
1 1
uloc
uloc
0<τ <1
0<τ <1
and
lim
τ →0
√
⃗ (t1 + τ, .)∥∞ = 0.
τ ∥W
488
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ belongs to C([t1 , t1 + 1], E 3 ) (due to the density of L3 H 2 in F 2,3 ).
More precisely, W
Finally we check that
⃗ ⊗W
⃗|
φ0 (x − k)|∇
Z t
Z tZ
ν(t−s)∆ ⃗
≤|
e
∇ ⊗ P div(ψ0 (. − k)F(s, .)) ds| + C
t1
t1
|x−y|>5
1
|F(s, y)| dy
|x − y|5
so that
⃗ ∥(L2 H 1 ) ((t ,t +1)×R3 ) ≤ Cν ∥F∥L2 L2 ((t ,t +1)×R3 ) .
∥W
1 1
uloc
1 1
uloc
Those estimates allow us to get a mild solution ⃗v1 in C([t1 , t1 + τ1 ], E 3 ) of the Cauchy
problem for the Navier—Stokes equations with initial value ⃗u1 (t1 , .), for τ1 small
enough. Indeed, we look for ⃗v1 as a solution of
⃗ − B(⃗v1 , ⃗v1 )
⃗v1 (t, .) = eν(t−t1 )∆ ⃗u1 (t1 , .) + W
where
Z
t
eν(t−s)∆ P div(⃗v ⊗ w)
⃗ ds.
B(⃗v , w)
⃗ =
t1
We have the estimates (uniformly in t for t1 < t < t1 + τ1 ≤ t1 + 1)
Z
t
1
⃗ .)∥∞ ds
∥⃗v (s, .)∥L6uloc ∥w(s,
3/4
t1 (t − s)
p
|s − t1 |1/4 ∥⃗v (s, .)∥L6uloc
sup
(s − t1 ∥w(s,
⃗ .)∥∞
∥B(⃗v , w)(t,
⃗
.)∥∞ ≤Cν
≤ C0,ν √
1
t − t1
sup
t1 <s<t1 +τ1
t1 <s<t1 +τ1
and
Z
t
1
∥⃗v (s, .)∥L6uloc ∥w(s,
⃗ .)∥∞ ds
(t
−
s)1/2
t1
p
|s − t1 |1/4 ∥⃗v (s, .)∥L6uloc
sup
(s − t1 ∥w(s,
⃗ .)∥∞ .
∥B(⃗v , w)(t,
⃗
.)∥L6uloc ≤Cν
≤ C0,ν
1
|t − t1 |1/4
sup
t1 <s<t1 +τ1
t1 <s<t1 +τ1
This gives the existence of a mild solution ⃗v1 on [t1 , t1 + τ1 ], provided that
√
sup
0<τ <τ1
τ ∥eντ ∆ (t1 , .)∥∞ + sup |τ |1/4 ∥eντ ∆ (t1 , .)∥L6uloc <
0<τ <τ1
1
8C0,ν
and
sup
√
0<τ <τ1
⃗ (t1 + τ, .)∥∞ + sup |τ |1/4 ∥W
⃗ (t1 + τ, .)∥L6 <
τ ∥W
uloc
0<τ <τ1
1
.
8C0,ν
Moreover, this solution ⃗v1 belongs to L∞ ([t1 , t1 + τ1 ], E 3 ), since
Z
t
∥B(⃗v , w)(t,
⃗
.)∥E 3 ≤Cν
t1
≤ Cν′
sup
t1 <s<t1 +τ1
1
∥⃗v (s, .)∥E 3 ∥w(s,
⃗ .)∥E 3 ds
(t − s)1/2
|s − t1 |1/4 ∥⃗v (s, .)∥E 6
sup
t1 <s<t1 +τ1
|s − t1 |1/4 ∥w(s,
⃗ .)∥E 6 .
It belongs more precisely to C([t1 , t1 + τ1 ], E 3 ): for the continuity of B(⃗v1 , ⃗v1 ) in E 3 ,
apply Theorem 8.1.
A Theory of Uniformly Locally L2 Solutions
489
We remark finally that ⃗v1 belongs to (L2 H 1 )uloc ((t1 , t1 + τ1 ) × R3 ), since
∥B(⃗v1 , ⃗v1 )∥(L2 H 1 )uloc ((t1 ,t1 +1)×R3 ) ≤Cν ∥⃗v1 ⊗ ⃗v1 ∥L2 L2uloc ((t1 ,t1 +1)×R3 )
≤
√
Cν′ τ1
sup
t1 <s<t1 +τ1
|s − t1 |1/4 ∥⃗v1 (s, .)∥E 6
sup
t1 <s<t1 +τ1
∥⃗v1 (s, .)∥E 3 .
As ⃗v1 is regular enough, being locally L4 L4 , it satisfies the energy equality, thus ⃗v1 is
a local Leray solution as well.
We now construct a third local Leray solution w
⃗ 1 on [t1 , t1 + 1]. We remark that we
have the inequality
∥f g∥2 ≤ C∥f ∥E 3 ∥g∥H 1
(14.30)
since
X
∥f g∥22 ≈
∥φ0 (x − k)f g∥22 ,
k∈Z3
∥g∥2H 1 ≈
X
∥φ0 (x − k)g∥2H 1
k∈Z3
and
∥φ0 (x − k)f g∥2 ≤ C∥ψ0 (x − k)f ∥3 ∥φ0 (x − k)g∥H 1 .
We then decompose ⃗u1 (t1 , .) in ⃗u1 (t1 , .) = α
⃗ 1,t1 + β⃗1,t1 with α
⃗ 1,t1 small in Eσ3 and
3
⃗
β1,t1 ∈ Ecomp,σ , and we decompose F in F = G1 + H1 with G1 small in F 2,3 ((t1 , t1 +
1) × R3 ) and H1 ∈ L3 ((t1 , t1 + 1), H 2 ). If α
⃗ 1,t1 and G1 are small enough, we have a
(small) solution α
⃗ 1 in C([t1 , t1 + 1], E 3 ) of the equations
(
∂t α
⃗ 1 = ν∆⃗
α1 + P div(G1 − α
⃗1 ⊗ α
⃗ 1 ),
α
⃗ 1 (t1 , .) = α
⃗ 1,t1 .
In order to construct w
⃗ 1 , we are going to construct β⃗1 = w
⃗1 − α
⃗ 1 . Thus, we require
⃗
β1 to be solution of the problem
(
∂t β⃗1 = ν∆β⃗1 + P div(H1 − β⃗1 ⊗ β⃗1 − α
⃗ 1 ⊗ β⃗1 − β⃗1 ⊗ α
⃗ 1 ),
(14.31)
⃗
⃗
β1 (t1 , .) = β1,t .
1
This is solved through the Leray mollification. We take θ ∈ D(R3 ) with
and define, for ϵ > 0, θϵ (x) = ϵ13 θ( xϵ ). We replace the system (14.31) with
R
θ dx = 1
(
⃗1,ϵ ⊗ α
∂t β⃗1,ϵ = ν∆β⃗1,ϵ + P div(H1 − (θϵ ∗ β⃗1,ϵ ) ⊗ β⃗1,ϵ − α
⃗ 1 ⊗ β⃗1,ϵ − β
⃗ 1 ),
⃗
⃗
β1,ϵ (t1 , .) = β1,t .
(14.32)
1
⃗ ∈ L2 ((t1 , t1 + 1), H 1 ) and V
⃗ =
For U
Rt
t1
⃗ −U
⃗ ⊗α
eν(t−s)∆ P div(⃗
α1 ⊗ U
⃗ 1 ) ds, we have
C
⃗ ∥L∞ ((t ,t +1),L2 ) ≤ √2 ∥U
⃗ ⊗α
⃗ ∥L2 H 1
∥V
⃗ 1 ∥L2 L2 ≤ √ ∥⃗
α1 ∥L∞ E 3 ∥U
1 1
ν
ν
and
C
⃗ ⊗V
⃗ ∥L2 ((t ,t +1),L2 ) ≤ 2 ∥U
⃗ ⊗α
⃗ ∥L2 H 1 .
∥∇
⃗ 1 ∥L2 L2 ≤ ∥⃗
α1 ∥L∞ E 3 ∥U
1 1
ν
ν
As β⃗1,t1 ∈ L2 and H1 ∈ L2 ((t1 , t1 + 1)), Leray’s formalism gives a solution β⃗1,ϵ ∈
C([t1 , t1 + τ1,ϵ ], L2 ) ∩ L2 ([t1 , t1 + τ1,ϵ ], L2 ) for a small interval [t1 , t1 + τ1,ϵ ] whose size
490
The Navier–Stokes Problem in the 21st Century (2nd edition)
depends on ϵ, on H and on ∥β⃗1,t1 ∥2 , and this solution is then extended to [t1 , t1 + 1]
as the L2 norm of β⃗1,ϵ can easily be controlled by the equality
Z
Z
⃗1,ϵ · ∇
⃗ ⊗ β⃗1,ϵ ∥22 = 2 α
⃗ β⃗1,ϵ ) dx − 2 H1 · (∇
⃗ ⊗ β⃗1,ϵ ) dx
∂t ∥β⃗1,ϵ ∥22 + 2ν∥∇
⃗ 1 · (β
and thus
⃗1,ϵ ∥2 +C∥∇⊗
⃗1,ϵ ∥2 (∥α1 ∥E 3 ∥β⃗1,ϵ ∥2 +∥H1 ∥2 ).
⃗ β⃗1,ϵ ∥22 ≤ C∥α1 ∥E 3 ∥∇⊗
⃗ β
⃗ β
∂t ∥β⃗1,ϵ ∥22 +2ν∥∇⊗
2
if α
⃗ 1 is small enough, we have C∥⃗
α1 ∥L∞ E3 < ν and the L2 norm of β⃗1,ϵ is well
controlled.
Thus, we have a solution w
⃗ 1,ϵ = α
⃗ 1 + β⃗1,ϵ of the problem
(
⃗1,ϵ ⊗ β⃗1,ϵ ),
∂t w
⃗ 1,ϵ = ν∆w
⃗ 1,ϵ + P div(F1 − w
⃗ 1,ϵ ⊗ w
⃗ 1,ϵ − (θϵ ∗ β⃗1,ϵ ) ⊗ β⃗1,ϵ + β
w
⃗ 1,ϵ (t1 , .) = ⃗u1 (t1 , .).
As w
⃗ 1,ϵ is controlled in (L2 H 1 )uloc and ∂t w
⃗ 1,ϵ is controlled in (L2 H −2 )uloc , we can
apply the Rellich–Lions theorem (Theorem 12.1): we may find a sequence ϵn → 0 and
a function w
⃗ 1 ∈ (L∞ L2 )uloc ∩ (L2 H 1 )uloc such that:
ˆ w
⃗ 1,ϵn is *-weakly convergent to w
⃗ 1 in (L∞ L2 )uloc and in (L2 H 1 )uloc
ˆ w
⃗ 1,ϵn is strongly convergent to w
⃗ 1 in L2loc ((0, T ) × R3 ).
It is then easy to check that w
⃗ 1 is a solution of
(
∂t w
⃗ 1 = ν∆w
⃗ 1 + P div(F − w
⃗1 ⊗ w
⃗ 1 ),
w
⃗ 1 (t1 , .) = ⃗u1 (t1 , .).
We then follow the same lines as for the proof of Theorem 14.1 and find that w
⃗ 1 is
suitable:
|w
⃗ 1 |2
|w
⃗ 1 |2
|w
⃗ 1 |2
⃗ ⊗w
) ≤ ν∆(
) − ν|∇
⃗ 1 |2 − div (p1 +
)w
⃗1 + w
⃗ 1 · div F (14.33)
∂t (
2
2
2
⃗ 1 = (Id − P) div(F − w
with ∇p
⃗1 ⊗ w
⃗ 1 ). Another useful property of w
⃗ is that
w
⃗ −α
⃗ 1 ∈ L∞ L2 ∩ L2 HH 1 ⊂ L4 L3 , so that w
⃗ 1 ∈ L4 E 3 .
Thus, on for t∗1 = min(t1 + τ1 , T1 ), we have three local Leray solutions ⃗u1 , ⃗v1 and w
⃗1
on (t1 , t∗1 ) × R3 , to the same problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.34)

⃗u(t1 , .) = ⃗u1 (t1 , .)
where ⃗u1 (t1 , .) ∈ L2uloc with div ⃗u1 (t1 , .) = 0 and F ∈ (L2t L2x )uloc ((0, T ) × R3 ). As
⃗v1 ∈ C([t1 , t∗1 ], E 3 ), it can be written as ⃗v1 = ⃗v1,1 +⃗v1,2 with ⃗v1,1 ∈ L2 ((t1 , t∗1 ), L∞ ) and
⃗v1,2 ∈ L∞ ((0, T ), E 3 ) with ∥⃗v1,2 (t, .)∥E 3 small enough. By the strong-weak uniqueness
theorem (Theorem 14.7), we have ⃗u1 = ⃗v1 on (t1 , t∗1 ), and similarly w
⃗ 1 = ⃗v1 on (t1 , t∗1 ).
Let T2 = t1 + 1. We get a local Leray solution ⃗u2 for the Navier—Stokes problem on
(0, T2 ) × R3 by defining ⃗u2 (t, x) = ⃗u1 (t, x) for 0 < t < T1 and ⃗u2 (t, x) = w
⃗ 1 (t, x) for
t1 < t < t1 + 1.
A Theory of Uniformly Locally L2 Solutions
491
Step 3: the global solution. Assume that, for some N ≥ 2, we have a local Leray
solution ⃗uN for the Navier–Stokes problem on (0, TN ) × R3 with TN > 1, such that
⃗uN ∈ L3 ((0, TN ), E 3 ). We consider a time tN ∈ (TN − 21 , TN ) such that ⃗uN (tN , .) ∈ E 3
and such that
∥⃗uN (tN , .) − ⃗uN (s, .)∥L2uloc = 0.
lim
s→tN ,s>tN
so that ⃗uN is a local Leray solution on (tN , TN ) of the Cauchy problem for the Navier–
Stokes equations with initial value ⃗uN (tN , .). We associate to this Cauchy problem two
other local Leray solutions, a solution ⃗vN such that ⃗vN ∈ ([tN , tN + τN ), E 3 ) and a
solution w
⃗ N defined on [tN , tN + 1].For t∗N = min(TN , tN + τN ), using the strongweak uniqueness theorem, we find that ⃗uN = ⃗vN = w
⃗ N on (tN , t∗N ). Thus, we have
a solution ⃗uN +1 on (0, TN +1 , with TN +1 = tN + 1, with ⃗uN +1 (t, x) = ⃗uN (t, x) for
0 < t < TN and ⃗uN +1 (t, x) = w
⃗ N (t, x) for tN < t < tN + 1.
As TN +1 ≥ TN + 21 , we have limN →+∞ TN = +∞. We then have a global solution by
defining ⃗u(t, x) = ⃗u2 (t, x) on (0, T2 ), and ⃗u(t, x) = ⃗uN +1 (t, x) on (TN , TN +1 ].
14.6
Weighted Estimates
Local Leray solutions to the Navier–Stokes equations allowed Jia and Šverák [245] to
construct in 2014 self-similar solutions for large (homogeneous of degree -1) smooth data.
Their result has been extended in 2016 by Lemarié-Rieusset [319] to solutions for rough
locally square integrable data. We remark that an homogeneous (of degree -1) and locally
square integrable data is automatically uniformly locally L2 .
Recently, Bradshaw and Tsai [57] and Chae and Wolf [100] considered the case of solutions which are self-similar according to a discrete subgroup of dilations. Those solutions
are related to an initial data which is self-similar only for a discrete group of dilations;
in contrast to the case of self-similar solutions for all dilations, such an initial data, when
locally L2 , is not necessarily uniformly locally L2 , therefore their results are no consequence
of the theory of local Leray solutions.
In this section, we follow Fernández-Dalgo and Lemarié-Rieusset [173] and construct
an alternative theory to obtain infinite-energy global weak solutions for large initial data,
which include the discretely self-similar locally square integrable data. More specifically, we
consider the weight
1
Φ(x) =
1 + |x|2
and the space
L2Φ = L2 (Φ dx).
(The construction by Bradshaw,
Z Kukavica and Tsai is very similar [56] in the sightly more
1
general condition lim
|⃗u0 (x)|2 dx = 0. We prefer to work in the space L2Φ ,
R→+∞ R2 B(0,R)
as it is a Hilbert space, so that results are easier to state.) In this context, we adapt the
definition of local Leray solution into the following one:
Definition 14.4 (Weighted Leray solution).
1
Let ⃗u0 ∈ L2Φ (where Φ(x) = 1+|x|
u0 = 0 and F ∈ L2 ((0, T ), L2Φ (R3 )). A weak
2 ) with div ⃗
492
The Navier–Stokes Problem in the 21st Century (2nd edition)
solution ⃗u on (0, T ) × R3 to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.35)
⃗u(0, .) = ⃗u0

is a weighted Leray solution if it satisfies the following requirements:
• ⃗u ∈ L∞ ((0, T ), L2Φ )
⃗ ⊗ ⃗u ∈ L2 ((0, T ), L2 )
• ∇
Φ
• ⃗u is suitable
R
• limt→0+ |⃗u(t, x) − ⃗u0 (t, x)|2 Φ(x) dx = 0.
Theorem 14.1 becomes:
Weighted square integrable solutions
Theorem 14.9.
1
Let ⃗u0 ∈ L2Φ (where Φ(x) = 1+|x|
u0 = 0 and F ∈ L2 ((0, T ), L2Φ (R3 )). Then,
2 ) with div ⃗
if T < +∞, there exists a weighted Leray solution ⃗u to the problem

∂t ⃗u = ν∆⃗u + P div(F − ⃗u ⊗ ⃗u)
(14.36)

⃗u(0, .) = ⃗u0
on (0, T ) × R3 .
If T = +∞, there exists a global solution ⃗u which is a weighted Leray solution on
every bounded interval (0, T0 ).
Proof. The proof of the theorem in the case of L2Φ wil be simpler than the proof in the
case of L2uloc , as we may use more easily the Riesz transforms, which are bounded on
1
1
4/3
L2 (Φ) = L2 ( 1+|x|
( (1+|x|12 )4/3 dx) (since the weight (1+|x|
2 dx) and on L
2 )γ belongs to the
3
Muckenhoupt class Ap (R ) for every γ ∈ [0, 3/2) and every p ∈ (1, +∞) [215]).
Step 1: local existence.
In contrast with the proof for Leray solutions in the L2 case or for local Leray solutions
in the case of L2uloc , it is useless to start with a mollification of the non-linearity. The
mollified equation (14.4), i.e.

⃗ u)
∂t ⃗u = ν∆⃗u + P(div F − (θϵ ∗ ⃗u) · ∇⃗
⃗u(0, .) = ⃗u0

cannot be solved by Picard’s iterative scheme as the operator
Z t
⃗,V
⃗)=
⃗)⊗V
⃗ ) ds
Bϵ ( U
Wν(t−s) ∗ P div((θϵ ∗ U
0
2
∞
⃗
is not bounded on L∞
t LΦ : the mollified drift θϵ ∗ U will not belong to Lt,x .
A Theory of Uniformly Locally L2 Solutions
493
The simplest way to get a solution is by approximating the problem with the problem
in L2 ; we take a function θ ∈ D(R3 ) which is equal to 1 on B(0, 1) and, for R > 1, we
x
define ⃗u0,R = P(θ( R
)⃗u0 ) and consider the Cauchy problem

x
⃗ uR )
∂t ⃗uR = ν∆⃗uR + P(div(θ( R
)F) − ⃗uR · ∇⃗

⃗uR (0, .) = ⃗u0,R
As ⃗u0,R belongs to L2 , we know, by Theorems 12.2 and 13.6, that we may find a
suitable weak Leray solution ⃗uR defined on (0, T ). We are going to estimate ⃗uR and
⃗ ⊗ ⃗uR in L2 independently from R and then let R go to +∞.
∇
Φ
In order to control ⃗uR , we shall use the local energy inequality (inequality (14.3)). Let
0 < t0 < T . We use the test function
Ω(t, x) = Φ(x)θϵ (t)φ2 (x/S)
(14.37)
where φ ∈ D(R3 ) satisfies φ(x) = 1 on B(0, 1), and where
θϵ (t) = α(
t−ϵ
t − t0 + 2ϵ
) − α(
)
ϵ
ϵ
with α a smooth non-decreasing function on R such that α(s) = 0 when s ≤ 1 and
α(s) = 1 for s ≥ 2, S > 0, 0 < ϵ < t0 /3.
Integrating inequality (14.3) against Ω(t, x), we find that:
−
1
2
Z
0
T
.
θϵ′ (t)∥⃗uR (t, .)φ( )∥2L2 dt ≤
Φ
S
Z T
Z
ν
x
θϵ (t)( |⃗uR (t, x)|2 ∆(φ2 )( )Φ(x) dx) dt
2S 2 0
S
Z
Z
ν T
x
⃗ uR (t, x) · ∇φ(
⃗ x ) dx) dt
+
θϵ (t)( φ( )Φ(x)∇⃗
S 0
S
S
Z
Z
ν T
x
+
θϵ (t)( |⃗uR (t, x)|2 φ2 ( )∆Φ(x) dx) dt
2 0
S
Z T
x ⃗
−ν
θϵ (t)∥φ( )(∇
⊗ ⃗uR )∥2L2 dt
Φ
S
0
Z T
Z
2
x
1
⃗ x ) dx) dt
+
θϵ (t)( Φ(x)φ( )(pR + |⃗uR |2 )⃗uR · ∇φ(
S 0
S
2
S
Z T
Z
x
1
⃗
+
θϵ (t)( φ2 ( )(pR + |⃗uR |2 )⃗uR · ∇Φ(x)
dx) dt
S
2
0
Z T
Z
x ⃗
−
θϵ (t)( Φ(x)φ2 ( ) (∇
⊗ ⃗uR ) · F dx) dt
S
0
Z
Z
2 T
x ⃗ x
−
θϵ (t)( Φ(x)φ( ) (∇φ(
) ⊗ ⃗uR ) · F dx) dt
S 0
S
S
Z T
Z
x ⃗
−
θϵ (t)( φ2 ( ) (∇Φ(x)
⊗ ⃗uR ) · F dx) dt
S
0
494
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗
Noticing that |∇Φ|
≤ 2Φ3/2 ≤ 2Φ and |∆Φ| ≤ 6Φ3/2 ≤ 6Φ, and letting S go to ∞,
we get
Z
Z
Z
1 T ′
ν T
2
−
θ (t)∥⃗uR (t, .)∥L2 dt ≤
θϵ (t)( |⃗uR (t, x)|2 ∆Φ(x) dx) dt
Φ
2 0 ϵ
2 0
Z T
⃗ ⊗ ⃗uR ∥2 2 dt
−ν
θϵ (t)∥∇
L
Φ
0
T
Z
Z
1
⃗
+
θϵ (t)( (pR + |⃗uR |2 )⃗uR · ∇Φ(x)
dx) dt
2
0
Z T
Z
⃗ ⊗ ⃗uR ) · F dx) dt
−
θϵ (t)( Φ(x) (∇
0
T
Z
Z
⃗
θϵ (t)( (∇Φ(x)
⊗ ⃗uR ) · F dx) dt
−
0
T
Z
θϵ (t)∥⃗uR (t, .)∥2L2 dt
≤3ν
Φ
0
T
Z
⃗ ⊗ ⃗uR ∥2 2 dt
θϵ (t)∥∇
L
−ν
Φ
0
Z
T
Z
+2
0
T
Z
Z
+
1
θϵ (t)(|pR | + |⃗uR |2 )|⃗uR |Φ3/2 (x) dx dt
2
⃗ ⊗ ⃗uR | |F| dx dt
θϵ (t)Φ(x)|∇
0
Z
T
Z
θϵ (t)|⃗uR |Φ(x) dx dt
+2
0
If t0 is a Lebesgue point of t 7→ ∥⃗uR (t, .)∥2L2 (and as 0 is a continuity point of t 7→
Φ
∥⃗uR (t, .)∥2L2 ), we find that
Φ
Z t0
Z t0
⃗ ⊗ ⃗uR ∥2 2 dt + 6ν
∥⃗uR (t0 , .)∥2L2 ≤∥⃗u0,R ∥2L2 − 2ν
∥∇
∥⃗uR ∥2L2 dt
LΦ
Φ
Φ
Φ
0
0
Z t0
Z t0
⃗ ⊗ ⃗uR ∥L2 ∥F∥L2 dt + 4
+2
∥∇
∥⃗uR ∥L2Φ ∥F∥L2Φ dt
Φ
Φ
0
0
Z t0 Z
1
+4
(|pR | + |⃗uR |2 )|⃗uR |Φ3/2 (x) dx dt
2
0
(14.38)
This inequality is thus satisfied for almost every t0 , and even for every t0 as t 7→ ⃗uR (t, .)
is weakly continuous from [0, T ) to L2 , hence from [0, T ) to L2Φ .
We then write
∆pR +
3 X
3
X
∂i ∂j (ui,R uj,R ) =
i=1 j=1
so that
pR = qR − ϖ =
3 X
3
X
3 X
3
X
∂i ∂j Fi,j
i=1 j=1
Ri Rj (ui,R uj,R ) −
i=1 j=1
As the Riesz transforms are bounded on L2Φ , we find
∥ϖ∥L2Φ ≤ C∥F∥L2Φ .
3 X
3
X
i=1 j=1
Ri Rj Fi,j .
A Theory of Uniformly Locally L2 Solutions
Moreover,
√
495
Φ⃗uR is controlled in H 1 , hence in L3 :
√
√
√
1/2 ⃗
1/2
∥ Φ⃗uR ∥3 ≤C∥ Φ⃗uR ∥2 ∥∇
⊗ ( Φ⃗uR )∥2
√
√
√
1/2
1/2
⃗ ⊗ ⃗uR ∥1/2 )
≤C∥ Φ⃗uR ∥2 (∥ Φ⃗uR ∥2 + ∥ Φ∇
2
1/2
1/2
⃗ ⊗ ⃗uR ∥1/2
=C∥⃗uR ∥L2 (∥⃗uR ∥L2 + ∥∇
).
L2
Φ
Φ
Φ
In particular, Φui,R uj,R is controlled in L3/2 , hence ui,R uj,R is controlled in
L3/2 (Φ3/2 ). But this control cannot be transferred to a control on qR as the Riesz
transforms are not bounded on L3/2 (Φ3/2 ).
√
8
4/3
Instead, we shall use the control of Φ⃗uR in L 3 to control Φq√
(as the Riesz
R in L
4/3
4/3
transforms are bounded on L (Φ dx)) and the control of Φ⃗uR in L4 to control
R t0 R
|qR ||⃗uR |Φ3/2 dx dt. We have
0
√
√
√
5/8
3/8
∥ Φ⃗uR ∥8/3 ≤∥ Φ⃗uR ∥2 ∥ Φ⃗uR ∥6
√
√
5/8 ⃗
3/8
≤C∥ Φ⃗uR ∥2 ∥∇
⊗ ( Φ⃗uR )∥2
5/8
3/8
3/8
⃗ ⊗ ⃗uR ∥ 2 )
≤C∥⃗uR ∥L2 (∥⃗uR ∥L2 + ∥∇
L
Φ
Φ
Φ
and
√
√
√
1/4
3/4
∥ Φ⃗uR ∥4 ≤∥ Φ⃗uR ∥2 ∥ Φ⃗uR ∥6
1/4
3/4
3/4
⃗ ⊗ ⃗uR ∥ 2 ).
≤C∥⃗uR ∥L2 (∥⃗uR ∥L2 + ∥∇
L
Φ
Φ
Φ
Thus far, (14.38) becomes
Z t0
Z t0
⃗ ⊗ ⃗uR ∥2 2 dt + 6ν
∥⃗uR (t0 , .)∥2L2 ≤∥⃗u0,R ∥2L2 − 2ν
∥∇
∥⃗uR ∥2L2 dt
LΦ
Φ
Φ
Φ
0
0
Z t0
Z t0
⃗ ⊗ ⃗uR ∥L2 ∥F∥L2 dt + C1
+ C0
∥∇
∥⃗uR ∥L2Φ ∥F∥L2Φ dt
Φ
Φ
0
0
Z t0
Z t0
3/2 ⃗
3/2
+ C2
∥⃗uR ∥3L2 dt + C3
∥⃗uR ∥L2 ∥∇
⊗ ⃗uR ∥L2 dt
Φ
0
Φ
0
Φ
and finally
∥⃗uR (t0 , .)∥2L2
Φ
25
+ ν
4
Z
+ C2
Z
0
t0
Z
≤∥⃗u0,R ∥2L2
Φ
⃗ ⊗ ⃗uR ∥2 2 dt
∥∇
L
−ν
Φ
0
t0
∥⃗uR ∥2L2
Φ
0
t0
∥⃗uR ∥3L2
Φ
dt +
2C02
C4
dt + 33
4ν
+ C12
ν
Z
0
Z
t0
∥F∥2L2 dt
(14.39)
Φ
0
t0
∥⃗uR ∥6L2 dt.
Φ
As the Riesz transforms are bounded on L2Φ , we know that ∥⃗u0,R ∥L2Φ ≤ A0 ∥⃗u0 ∥L2Φ .
Let
Z
2C02 + C12 T
2
2
2
A1 = A0 ∥⃗u0 ∥L2 +
∥F∥2L2 dt.
(14.40)
Φ
Φ
ν
0
From (14.39), we see that we have
∥⃗uR (t0 , .)∥2L2 + ν
Φ
Z
0
t0
⃗ ⊗ ⃗uR ∥2 2 dt ≤ 4A21
∥∇
L
Φ
(14.41)
496
The Navier–Stokes Problem in the 21st Century (2nd edition)
as long as
t0 (25ν + 8C2 A1 +
16C34 4
A1 ) ≤ 3.
ν3
A4
As 8C2 A1 ≤ 6ν + 2C24 ν 31 , we see that, on (0, T0 ) with
3
T0 =
31ν +
16C34 +2C24 4
A1
ν3
,
inequality (16.23) is fulfilled.
As the control (16.23) does not depend on R, we may apply the Rellich–Lions theorem
(Theorem 12.1) and get a sequence Rk such that ⃗uRk is strongly convergent to a limit ⃗u
⃗
in L2loc ([0, T0 )×R3 ). Moreover, ⃗uRk is weakly-* convergent to ⃗u in L∞ ((0, T0 ), L2Φ ), ∇⊗
2
2
⃗
⃗uRk is weakly convergent to ∇ ⊗ ⃗u in L ((0, T0 ), LΦ ) and pRk = qRk − ϖ is convergent
to p = q − ϖ with weak convergence of qRk to q in L8/3 ((0, T0 ), L4/3 (Φ4/3 dx)). In
particular, we find that ⃗u is solution to
⃗ u = ν∆⃗u − ∇p
⃗ + div F
∂t ⃗u + ⃗u · ∇⃗
and
div ⃗u = 0
3
on (0, T0 ) × R . Moreover, we know by Theorem 6.2, that the limit ⃗u (as a weak limit
of uniformly controlled suitable solutions) satisfies the local Leray energy inequality
(i.e. ⃗u is suitable).
As the Riesz transforms are bounded on L2Φ , we have as well the strong convergence
of ⃗u0,R to ⃗u0 in L2 . For every φ ∈ D(R3 ), φ⃗uRk belongs to L2 ((0, T0 ), H 1 ) and φ∂t ⃗uRk
belongs to L2 ((0, T0 ), H −3/2 ), so that (from Lemma (6.1)), we can represent ⃗uRk as
Z t
⃗uRk (t, .) = ⃗u0,Rk +
∂t ⃗urk (s, .) ds
0
(so that φ⃗uRk ∈ C([0, T0 ], H −3/2 )). Moreover, φ∂t ⃗uRk is bounded in L2 ((0, T0 ), H −3/2 ),
hence the weak convergence of ⃗uRk to ⃗u gives the weak convergence of φ∂t ⃗uRk to φ∂t ⃗u
in L2 ((0, T0 ), H −3/2 ), and then the weak convergence of φ⃗uRk (t, .) to φ⃗u(t, .) in H −3/2 ;
Rt
finally, we get the weak convergence of ⃗uRk (t, .) to ⃗u(t, .) = ⃗u0 + 0 ∂t ⃗u(s, .) ds in L2Φ
as ⃗uRk (t, .) is bounded in L2Φ .
For fixed t, we thus have the weak convergence of
⃗ ⊗ ⃗uR (s, .))
(⃗uRk (t, .), 10<s<t ∇
k
in L2Φ × L2 ((0, t), L2Φ ) and thus
⃗ ⊗ ⃗u∥2 2
∥⃗u(t, .)∥2L2 + ν∥∇
L ((0,t),L2 ))
Φ
Φ
⃗ ⊗ ⃗uR ∥2 2
≤ lim inf ∥⃗uRk (t, .)∥2L2 + 2ν∥∇
k L ((0,t),L2 ))
k→+∞
Φ
Φ
16C34 4
≤ ∥⃗u0 ∥2L2 + tA21 (25ν + 8C2 A1 +
A1 ).
Φ
ν3
Rt
Thus, lim supt→0 ∥⃗u(t, .)∥L2Φ ≤ ∥⃗u0 ∥L2Φ . As t 7→ ⃗u(t, .) = ⃗u0 + 0 ∂t ⃗u(s, .) ds is continuous from [0, T0 ] to D′ and is bounded in L2Φ , it is weakly continuous to L2Φ , and ⃗u0
is the weak limit of ⃗u(t, .) ass t decreases to 0. Thus, ∥⃗u0 ∥L2Φ ≤ lim inf t→0 ∥⃗u(t, .)∥L2Φ .
A Theory of Uniformly Locally L2 Solutions
497
Hence, we have ∥⃗u0 ∥L2Φ = limt→0 ∥⃗u(t, .)∥L2Φ , and this turns the weak convergence in
L2Φ into strong convergence:
lim inf ∥⃗u(t, .) − ⃗u0 ∥L2Φ = 0.
t→0
We have proved the existence of a weighted Leray solution on (0, T0 ).
Step 2: size estimates for weighted Leray solutions.
We have shown the existence of a weighted Leray solution ⃗u on (0, T0 ), where
3
T0 =
31ν +
16C34 +2C24
ν3
A20 ∥⃗u0 ∥2L2 +
Φ
2C02 +C12
ν
RT
0
2 ,
∥F∥2L2 dt
Φ
which is of the order of magnitude
T0 ≈ C5
ν5
ν 6 + ν 2 ∥⃗u0 ∥4L2 + (
Φ
RT
0
∥F∥2L2 dt)2
.
Φ
Now, if ⃗v is another weighted Leray solution on (0, T0 ) (with associated pressure q),
⃗ ⊗⃗v in L2 ((0, T0 ), L2 )
we show that we have a control of ⃗v in L∞ ((0, T0 ), L2Φ ) and of ∇
Φ
In order to control ⃗v , we shall use again the local energy inequality (inequality (14.3)).
Let 0 < t0 < T0 . We use the test function Ω(t, x) = Φ(x)θϵ (t)φ2 (x/S), where φ ∈
D(R3 ) satisfies φ(x) = 1 on B(0, 1), and where
t−ϵ
t − t0 + 2ϵ
) − α(
)
ϵ
ϵ
with α a smooth non-decreasing function on R such that α(s) = 0 when s ≤ 1 and
α(s) = 1 for s ≥ 2, S > 0, 0 < ϵ < t0 /3.
θϵ (t) = α(
Integrating inequality (14.3) against Ω(t, x), we find that:
Z
1 T ′
.
−
θϵ (t)∥⃗v (t, .)φ( )∥2L2 dt ≤
Φ
2 0
S
Z T
Z
ν
x
θ
(t)(
|⃗v (t, x)|2 ∆(φ2 )( )Φ(x) dx) dt
ϵ
2S 2 0
S
Z
Z
ν T
x
⃗ v (t, x) · ∇φ(
⃗ x ) dx) dt
+
θϵ (t)( φ( )Φ(x)∇⃗
S 0
S
S
Z T
Z
ν
x
+
θϵ (t)( |⃗v (t, x)|2 φ2 ( )∆Φ(x) dx) dt
2 0
S
Z T
x ⃗
−ν
θϵ (t)∥φ( )(∇ ⊗ ⃗v )∥2L2 dt
Φ
S
0
Z T
Z
2
x
1
⃗ x ) dx) dt
+
θϵ (t)( Φ(x)φ( )(q + |⃗v |2 )⃗v · ∇φ(
S 0
S
2
S
Z T
Z
x
1
⃗
+
θϵ (t)( φ2 ( )(q + |⃗v |2 )⃗v · ∇Φ(x)
dx) dt
S
2
0
Z T
Z
x ⃗
−
θϵ (t)( Φ(x)φ2 ( ) (∇
⊗ ⃗v ) · F dx) dt
S
0
Z
Z
2 T
x ⃗ x
−
θϵ (t)( Φ(x)φ( ) (∇φ(
) ⊗ ⃗v ) · F dx) dt
S 0
S
S
Z T
Z
x ⃗
−
θϵ (t)( φ2 ( ) (∇Φ(x)
⊗ ⃗v ) · F dx) dt.
S
0
498
The Navier–Stokes Problem in the 21st Century (2nd edition)
Letting S go to ∞, we get
−
1
2
Z
0
T
θϵ′ (t)∥⃗v (t, .)∥2L2 dt ≤
Φ
T
Z
ν
2
Z
θϵ (t)(
0
T
Z
⃗ ⊗ ⃗v ∥2 2 dt
θϵ (t)∥∇
L
−ν
Φ
0
Z
+
|⃗v (t, x)|2 ∆Φ(x) dx) dt
T
Z
1
⃗
θϵ (t)( (q + |⃗v |2 )⃗v · ∇Φ(x)
dx) dt
2
0
Z T
Z
⃗ ⊗ ⃗v ) · F dx) dt
−
θϵ (t)( Φ(x) (∇
0
T
Z
Z
−
θϵ (t)(
⃗
(∇Φ(x)
⊗ ⃗v ) · F dx) dt
0
T
Z
θϵ (t)∥⃗v (t, .)∥2L2 dt
≤3ν
Φ
0
T
Z
⃗ ⊗ ⃗v ∥2 2 dt
θϵ (t)∥∇
L
−ν
Φ
0
Z
T
Z
1
θϵ (t)(|q| + |⃗v |2 )|⃗v |Φ3/2 (x) dx dt
2
+2
0
T
Z
Z
+
⃗ ⊗ ⃗v | |F| dx dt
θϵ (t)Φ(x)|∇
0
Z
T
Z
θϵ (t)|⃗v |Φ(x) |F| dx dt.
+2
0
If t0 is a Lebesgue point of t 7→ ∥⃗v (t, .)∥2L2 (and as 0 is a continuity point of t 7→
Φ
∥⃗v (t, .)∥2L2 ), we find that
Φ
Z t0
Z t0
⃗ ⊗ ⃗v ∥2 2 dt + 6ν
∥⃗v (t0 , .)∥2L2 ≤∥⃗u0 ∥2L2 − 2ν
∥∇
∥⃗v ∥2L2 dt
LΦ
Φ
Φ
Φ
0
0
Z t0
Z t0
⃗ ⊗ ⃗v ∥L2 ∥F∥L2 dt + 4
+2
∥∇
∥⃗v ∥L2Φ ∥F∥L2Φ dt
Φ
Φ
0
0
Z t0 Z
1
+4
(|q| + |⃗v |2 )|⃗v |Φ3/2 (x) dx dt
2
0
(14.42)
This inequality is thus satisfied for almost every t0 , and even for every t0 as t 7→ ⃗v (t, .)
is weakly continuous from [0, T0 ) to L2Φ .
We then write
∆q +
3 X
3
X
∂i ∂j (vi vj ) =
i=1 j=1
so that
q = ϖ1 − ϖ2 =
3 X
3
X
3 X
3
X
∂i ∂j Fi,j
i=1 j=1
Ri Rj (vi vj ) −
i=1 j=1
We have the inequalities
∥ϖ2 ∥L2Φ ≤ C∥F∥L2Φ
3 X
3
X
i=1 j=1
Ri Rj Fi,j .
A Theory of Uniformly Locally L2 Solutions
499
and
∥ϖ1 ∥L4/3 ≤ C∥⃗v ∥2L8/3 .
Φ4/3
Φ4/3
Moreover, we have
√
5/8
3/8
⃗ ⊗ ⃗v ∥3/8
∥ Φ⃗v ∥8/3 ≤ C∥⃗v ∥L2 (∥⃗v ∥L2 + ∥∇
)
L2
Φ
and
Φ
Φ
√
1/4
3/4
⃗ ⊗ ⃗v ∥3/4
∥ Φ⃗v ∥4 ≤ C∥⃗v ∥L2 (∥⃗v ∥L2 + ∥∇
).
L2
Φ
Φ
Φ
Thus far, (14.42) becomes
∥⃗v (t0 , .)∥2L2
Φ
Z
≤∥⃗u0 ∥2L2
Φ
−ν
Z t0
t0
⃗ ⊗ ⃗v ∥2 2 dt
∥∇
L
Φ
0
Z
2C02 + C12 t0
dt +
∥F∥2L2 dt
Φ
ν
0
0
t0
4 Z t0
C
∥⃗v ∥3L2 dt + 33
∥⃗v ∥6L2 dt.
Φ
Φ
4ν 0
0
25
+ ν
4
Z
+ C2
∥⃗v ∥2L2
Φ
(14.43)
with the same constants C0 , C1 , C2 , C3 as in inequality (16.23).
This gives that the solution ⃗v satisfies on (0, T0 ) the inequality
Z t
2
⃗ ⊗ ⃗v ∥2 2 ds
∥⃗v (t, .)∥L2 + ν
∥∇
L
Φ
Φ
0
≤
4(A20 ∥⃗u0 ∥2L2
Φ
2C02 + C12
+
ν
Z
0
(14.44)
T
∥F∥2L2 dt)2 .
Φ
Thus, the control of ⃗v does not depend on the specific solution ⃗v but is the same as
the control on ⃗u.
Step 3: global existence.
With no loss of generality, we may assume that F ∈ L2 ((0, +∞), L2Φ (R3 )). (If F is
defined only on (0, T ) × R3 , we extend F by 0 for t > T .)
In order to prove global existence, we use the scaling properties of the Navier–Stokes
equations: if ⃗v is a solution on (0, T ) (with associated pressure q) of the Cauchy
problem with initial value ⃗v0 and forcing term f⃗ = div G, then for λ > 0, λ⃗v (λ2 t, λx) is
a solution on (0, λ−2 T ) (with associated pressure λ2 q(λ2 t, λx)) of the Cauchy problem
with initial value λ⃗v0 (λx) and forcing term div Gλ , where Gλ = λ2 G(λ2 t, λx).
Thus, we consider λ > 1 and for n ∈ N we consider the Cauchy problem with initial
value ⃗v0,n = λn ⃗u0 (λn x) and forcing tensor Fn = λ2n F(λ2n t, λn x). If we find a solution
⃗vn on (0, Tn ), then we have a solution ⃗un = λ−n⃗vn (λ−2n t, λ−n x), defined on (0, λ2n Tn ),
of the Cauchy problem with initial value ⃗u0 and forcing tensor F.
We know that we have a weighted Leray solution ⃗vn on (0, Tn ) with
Tn ≈ C5
ν5
ν 6 + ν 2 ∥⃗v0,n ∥4L2 + (
Φ
R +∞
0
∥Fn ∥2L2 dt)2
.
Φ
This gives a weighted Leray solution ⃗un on (0, λ2n Tn ). We easily check that
lim λ2n Tn = +∞.
n→+∞
500
The Navier–Stokes Problem in the 21st Century (2nd edition)
Indeed, we have
Z
Z
λn (1 + |y|2 )
2
2 −n
−n
∥⃗v0,n ∥L2 = |⃗u0 (y)| λ Φ(λ y) dy = |⃗u0 (y)|2 Φ(y) 2n
dy = o(λn )
Φ
(λ + |y|2 )
and
+∞
Z
∥Fn ∥2L2 dt =
Φ
0
+∞
Z
Z
|F(t, y)|2 λ−n Φ(λ−n y) dy
0
so that
+∞
Z
0
k
∥Fn ∥2L2 dt =
Φ
2k
Z
+∞
Z
|F(t, y)|2 Φ(y)
0
λn (1 + |y|2 )
dy = o(λn ).
(λ2n + |y|2 )
k
Moreover, λ ⃗un (λ t, λ x) is a weighted Leray solution, defined on (0, λ2(n−k) Tn ), of
the Cauchy problem with initial value ⃗u0,k and forcing tensor Fk . Thus, if λ2(n−k) Tn ≥
Tk , we have a control on (0, Tk ) given by inequality (14.44)
Z t
k
2k
k
2
⃗ ⊗ ⃗un )(λ2k s, λk .)∥2 2 ds
∥λ ⃗un (λ t, λ .)∥L2 + ν
λ4k ∥(∇
LΦ
Φ
0
(14.45)
2
2 Z +∞
2C0 + C1
2
2
2
2
≤4(A0 ∥⃗u0,k ∥L2 +
∥Fk ∥L2 dt) .
Φ
Φ
ν
0
This gives a control for ⃗un on (0, λ2k Tk ), writing Φk (x) = Φ(λ−k x),
Z t
⃗ ⊗ ⃗un (s, .)∥2 2 ds
∥⃗un (t, .)∥2L2 + ν
∥∇
LΦ
Φk
k
0
Z
2
2C0 + C12 +∞
≤4(A20 ∥⃗u0 ∥2L2 +
∥F∥2L2 dt)2 .
Φk
Φk
ν
0
This can be rewritten in terms of the weight Φ, as Φ ≤ Φk ≤ λ2k Φ,
Z t
⃗ ⊗ ⃗un (s, .)∥2 2 ds
∥⃗un (t, .)∥2 2 + ν
∥∇
LΦ
LΦ
0
≤4λ2k (A20 ∥⃗u0 ∥2L2 +
Φ
2C02 + C12
ν
Z
0
(14.46)
+∞
∥F∥2L2 dt)2 .
Φ
As the control (14.46) does not depend on n, we may again apply the Rellich–Lions
theorem (Theorem 12.1) and get a sequence nj such that ⃗unj is strongly convergent to
a limit ⃗u in L2loc ([0, λ2k Tk ) × R3 ). Using Cantor’s diagonal process, we get a sequence
nq such that ⃗unq is strongly convergent to a limit ⃗u in L2loc ([0, +∞)×R3 ). By Theorem
6.2, the limit ⃗u satisfies the local Leray energy inequality (i.e. ⃗u is suitable) and is a
weighted Leray solution on every bounded interval (0, T ).
14.7
A Stability Estimate
When comparing two weighted weak Leray solutions, we find a problem as the interaction
of the solutions may grow too fast at infinity. Thus, we must assume that one of the solutions
remains bounded. We will prove the following stability estimate:
A Theory of Uniformly Locally L2 Solutions
501
Lemma 14.7.
Let ⃗u1 , ⃗u2 be two weighted Leray solutions on (0, T ) × R3 to the problems

⃗ ui − ∇p
⃗ i
∂t ⃗ui = ν∆⃗ui − ⃗ui · ∇⃗

(14.47)
⃗ui (0, .) = ⃗u0,i
1
where ⃗u0,i ∈ L2 ( 1+|x|
u0,i = 0.
2 dx) with div ⃗
Assume moreover that ⃗u1 ∈ L2 ((0, T ), L∞ ). Then, we have, for every t0 ∈ (0, T ),
∥⃗u1 (t0 , .) − ⃗u2 (t0 , .)∥2L2
Φ
≤e
sup0<s<t (∥⃗
u1 (s,.)∥ 2 +∥⃗
u2 (s,.)∥ 2 )4
0
L
L
Φ
Φ
Ct0 (ν+
ν3
) C
e
R t0
0
∥⃗
u 1 ∥2
∞
ν
(14.48)
dt
∥⃗u1 (0, .)
− ⃗u2 (0, .)∥2L2 ,
Φ
where the constant C does not depend on ⃗u1 , ⃗u2 , ν nor T .
Proof. Let pi be the pressure associated to ⃗ui . We have, due to the suitability of ⃗u2 and the
regularity of ⃗u1 ,

2
2
2
⃗ ⊗ ⃗u1 |2 − div((p1 + |⃗u1 | )⃗u1 )

∂t ( |⃗u21 | ) =
ν∆( |⃗u21 | ) − ν|∇

2






⃗ ⊗ ⃗u1 ).(∇
⃗ ⊗ ⃗u2 ) − div(p2 ⃗u1 + p1 · ⃗u2 )
∂t (⃗u1 · ⃗u2 ) = ν∆(⃗u1 · ⃗u2 ) − 2ν(∇
⃗
⃗ 1)

−⃗u1 .(⃗u2 · ∇⃗u2 ) − ⃗u2 .(⃗u1 · ∇u






2
2
2

⃗ ⊗ ⃗u2 |2 − div((p2 + |⃗u2 | )⃗u2 ) − µ
∂t ( |⃗u22 | ) =
ν∆( |⃗u22 | ) − ν|∇
2
where µ is some non-negative locally finite measure.
Let w
⃗ = ⃗u1 − ⃗u2 and q = p1 − p2 . We obtain
∂t (
|w|
⃗2
|w|
⃗2
⃗ ⊗ w|
) =ν∆(
) − ν|∇
⃗ 2 − div(q w)
⃗ −A−µ
2
2
with
|⃗u1 |2
|⃗u2 |2
⃗ u2 ) + ⃗u2 .(⃗u1 · ∇u
⃗ 1)
⃗u1 +
⃗u2 ) − (⃗u1 .(⃗u2 · ∇⃗
2
2
1
= div |⃗u1 |2 ⃗u1 + |⃗u2 |⃗u2 − (⃗u1 · ⃗u2 )(⃗u1 + ⃗u2 )
2
1
⃗ u1 ) + ⃗u1 .(⃗u1 · ∇u
⃗ 2 ) − ⃗u1 .(⃗u2 · ∇⃗
⃗ u2 ) − ⃗u2 .(⃗u1 · ∇u
⃗ 1)
+
⃗u2 .(⃗u2 · ∇⃗
2
1
1
⃗ u1 ) − ⃗u1 .(w
⃗ w)
= div |w|
⃗ 2 ⃗u1 + (⃗u1 · w)
⃗ w
⃗ − |w|
⃗ 2w
⃗ +
w.(
⃗ w
⃗ · ∇⃗
⃗ ·∇
⃗
2
2
1
⃗ w).
= div |w|
⃗ 2 ⃗u1 + 2(⃗u1 · w)
⃗ w
⃗ − |w|
⃗ 2w
⃗ − ⃗u1 .(w
⃗ ·∇
⃗
2
A = div(
We use again the test function Ω(t, x) = Φ(x)θϵ (t)φ2 (x/S) described in equation (14.37)
RR
⃗ 2
and compute
Ω(t, x)∂t ( |w|
2 ) dt dx.
502
The Navier–Stokes Problem in the 21st Century (2nd edition)
Integrating inequality (14.3) against Ω(t, x), we find that:
1
−
2
Z
T
.
θϵ′ (t)∥w(t,
⃗ .)φ( )∥2L2 dt ≤
Φ
S
Z T
Z
ν
x
θϵ (t)( |w(t,
⃗ x)|2 ∆(φ2 )( )Φ(x) dx) dt
2S 2 0
S
Z
Z
ν T
x
⃗ w(t,
⃗ x ) dx) dt
+
θϵ (t)( φ( )Φ(x)∇
⃗ x) · ∇φ(
S 0
S
S
Z T
Z
ν
x
+
θϵ (t)( |w(t,
⃗ x)|2 φ2 ( )∆Φ(x) dx) dt
2 0
S
Z T
x ⃗
−ν
θϵ (t)∥φ( )(∇
⊗ w)∥
⃗ 2L2 dt
Φ
S
0
Z T
Z
2
x
1 2
⃗ x ) dx) dt
+
θϵ (t)( Φ(x)φ( )(q + ⃗u1 · w
⃗ − |w|
⃗ )w
⃗ · ∇φ(
S 0
S
2
S
Z T
Z
x
1 2
⃗
+
θϵ (t)( φ2 ( )(q + ⃗u1 · w
⃗ − |w|
⃗ )w
⃗ · ∇Φ(x)
dx) dt
S
2
0
Z
Z
2 T
x 1 2
⃗ x ) dx) dt
+
θϵ (t)( Φ(x)φ( )( |w|
⃗ )⃗u1 · ∇φ(
S 0
S 2
S
Z T
Z
x 1 2
⃗
+
θϵ (t)( φ2 ( )( |w|
⃗ )⃗u1 · ∇Φ(x)
dx) dt
S
2
0
Z T
Z
x
⃗ w)
+
θϵ (t)( Φ(x)φ2 ( ) ⃗u1 .(w
⃗ ·∇
⃗ dx) dt
S
0
0
⃗
Noticing that |∇Φ|
≤ 2Φ3/2 ≤ 2Φ and |∆Φ| ≤ 6Φ2 ≤ 6Φ, and letting S go to ∞, we get
−
1
2
Z
0
T
θϵ′ (t)∥⃗uR (t, .)∥2L2 dt ≤3ν
T
Z
Φ
θϵ (t)∥w(t,
⃗ .)∥2L2 dt
Φ
0
T
Z
⃗ ⊗ w∥
θϵ (t)∥∇
⃗ 2L2 dt
−ν
Φ
0
T
Z
Z
T
Z
+
0
+
1
2
1 2
θϵ (t)(|q| + |⃗u1 · w|
⃗ + |w|
⃗ )|w|Φ
⃗ 3/2 (x) dx dt
2
Z
θϵ (t)|w|
⃗ 2 |⃗u1 |Φ3/2 (x) dx dt
0
Z
+
0
T
Z
⃗ w)
θϵ (t)Φ(x) dx ⃗u1 .(w
⃗ ·∇
⃗ dt.
A Theory of Uniformly Locally L2 Solutions
503
If t0 is a Lebesgue point of t 7→ ∥w(t,
⃗ .)∥2L2 (and as 0 is a continuity point of t 7→
Φ
∥w(t,
⃗ .)∥2L2 ), we find that
Φ
∥w(t
⃗ 0 , .)∥2L2
Φ
t0
Z
+2
Z
+
0
t0
Z
t0
Z
≤∥w(0,
⃗ .)∥2L2
Φ
− 2ν
⃗ ⊗ w∥
∥∇
⃗ 2L2 dt + 6ν
Φ
0
Z
0
t0
∥w∥
⃗ 2L2 dt
Φ
1 2
(|q| + |⃗u1 · w|
⃗ + |w|
⃗ )|w|Φ
⃗ 3/2 (x) dx dt
2
Z
2
(14.49)
3/2
|w|
⃗ |⃗u1 |Φ (x) dx dt
Z t0 Z
⃗ w)
+2
Φ(x) dx ⃗u1 .(w
⃗ ·∇
⃗ dt.
0
0
This inequality is thus satisfied for almost every t0 , and even for every t0 as t 7→ w(t,
⃗ .) is
weakly continuous from [0, T ) to L2Φ .
We then write
∆q =
3 X
3
X
∂i ∂j (u2,i u2,j − u1,i u1,j ) =
i=1 j=1
3 X
3
X
∂i ∂j j(wi wj − u1,i wj − wi u1,j )
i=1 j=1
so that
q = ϖ0 − ϖ1 =
3 X
3
X
Ri Rj (wi wj ) −
i=1 j=1
3 X
3
X
Ri Rj (u1,i wj + wi u1,j ).
i=1 j=1
As the Riesz transforms are bounded on L2Φ , we find
∥ϖ1 ∥L2Φ ≤ C∥⃗u1 ∥∞ ∥w∥
⃗ L2Φ .
On the other hand we know that
√
5/4
3/4
⃗ ⊗ w∥3/4
∥Φϖ0 ∥4/3 ≤ C∥ Φw∥
⃗ 24/3 ≤ C ′ ∥w∥
⃗ L2 (∥w∥
⃗ L2 + ∥∇
).
L2
Φ
Φ
Φ
Finally, we get the inequality
Z t0
Z t0
⃗ ⊗ w∥
∥w(t
⃗ 0 , .)∥2L2 ≤∥w(0,
⃗ .)∥2L2 − 2ν
∥∇
⃗ 2L2 dt + 6ν
∥w∥
⃗ 2L2 dt
Φ
Φ
Φ
Φ
0
0
Z t0
+C
∥⃗u1 ∥∞ ∥w∥
⃗ 2L2 dt
Φ
0
Z t0 Z
⃗ ⊗ w∥
+C
∥⃗u1 ∥∞ ∥w∥
⃗ L2Φ ∥∇
⃗ L2Φ dt
0
Z t0
Z t0
3/2 ⃗
3/2
3
+C
∥w∥
⃗ L2 dt + C
∥w∥
⃗ L2 ∥∇
⊗ w∥
⃗ L2 dt
Φ
0
Φ
0
(14.50)
Φ
and thus
∥w(t
⃗ 0 , .)∥2L2 ≤∥w(0,
⃗ .)∥2L2 − ν
Φ
Φ
Z
+C
0
t0
Z
0
t0
⃗ ⊗ w∥
∥∇
⃗ 2L2 dt
∥⃗u1 ∥2∞
(ν +
+
ν
Φ
∥⃗u1 ∥4L2
Φ
ν3
+
∥⃗u2 ∥4L2
Φ
ν3
(14.51)
)∥w∥
⃗ 2L2
Φ
dt.
504
The Navier–Stokes Problem in the 21st Century (2nd edition)
This gives, for every t0 ∈ (0, T ),
∥w(t
⃗ 0 , .)∥2L2 ≤ e
Ct0 (ν+ ν13 sup0<s<t0 (∥⃗
u1 (s,.)∥L2 +∥⃗
u2 (s,.)∥L2 )4 ) C
Φ
Φ
e
∥⃗
u 1 ∥2
∞
ν
R t0
0
Φ
dt
∥w(0,
⃗ .)∥2L2 .
Φ
(14.52)
Of course, this stability estimate is much more easy to get when we consider weak Leray
solutions, as the terms involving the pressures disappear in the energy balance:
Lemma 14.8.
Let ⃗u1 , ⃗u2 be two weak Leray solutions on (0, T ) × R3 to the problems

⃗ ui − ∇p
⃗ i
∂t ⃗ui = ν∆⃗ui − ⃗ui · ∇⃗
(14.53)
⃗ui (0, .) = ⃗u0,i

2
where ⃗u0,i ∈ L with div ⃗u0,i = 0.
Assume moreover that ⃗u1 ∈ L2 ((0, T ), L∞ ). Then, we have, for every t0 ∈ (0, T ),
Z t0
2
⃗ ⊗ (⃗u1 − ⃗u2 )∥22 ds
∥⃗u1 (t0 , .) − ⃗u2 (t0 , .)∥2 + 2ν
∥∇
0
(14.54)
Z Z t0
2
⃗
≤ ∥⃗u1 (0, .) − ⃗u2 (0, .)∥2 + 2
⃗u1 · ((⃗u1 − ⃗u2 ) · ∇(⃗u1 − ⃗u2 )) dx ds.
0
14.8
Barker’s Theorem on Weak-Strong Uniqueness
Let us recall the weak-strong uniqueness criterion (Theorem 12.4) given by Prodi and
Serrin [406, 435] for Leray solutions of the Navier–Stokes problem

⃗
⃗

∂t ⃗u + ⃗u · ∇⃗u = ∆⃗u − ∇p
div ⃗u = 0


⃗u(0, .) = ⃗u0
where ⃗u0 is a square-integrable divergence-free vector field on the space R3 : if the NavierStokes equations have a solution ⃗u on (0, T ) such that
⃗u ∈ Lpt Lqx with
2 3
+ ≤ 1 and 2 < p < +∞
p q
then, if ⃗v is a Leray solution with the same initial value ⃗u0 , we have ⃗u = ⃗v on (0, T ). Let
us remark that the existence of such a solution ⃗u restricts the range of the initial value ⃗u0 :
when 2 < p < +∞, existence of a time T > 0 and of a solution ⃗u ∈ Lpt Lqx is equivalent to
−2
the fact that ⃗u0 belongs to the Besov space Bq,pp
A natural endpoint case for this criterion is the assumption that ⃗u0 ∈ L2 ∩ bmo−1 , or
−1
more precisely to L2 ∩ bmo−1
grants existence of a mild
0 , where the restriction to bmo0
solution, due to the Koch and Tataru theorem [266].
Proposition 14.2.
For 0 < T < ∞, define
∥⃗u∥XT = sup
0<t<T
√
t∥⃗u(t, .)∥∞ +
sup
0<t<T,x0 ∈R3
−3/2
Z tZ
(t
0
√
B(x0 , t)
|⃗u(s, y)|2 dy ds)1/2 .
A Theory of Uniformly Locally L2 Solutions
505
Then ⃗u0 ∈ bmo−1 if and only if (et∆ ⃗u0 )0<t<T ∈ XT (with equivalence of the norms
∥⃗u0 ∥bmo−1 and ∥et∆ ⃗u0 ∥XT ).
Koch and Tataru’s theorem is then the following one:
Theorem 14.10.
There exists C0 (which does not depend on T ) such that, if ⃗u and ⃗v are defined on (0, T )×R3 ,
then
∥B(⃗u, ⃗v )∥XT ≤ C0 ∥⃗u∥XT ∥⃗v ∥XT ,
R t (t−s)∆
where B(⃗u, ⃗v ) = 0 e
P div(⃗u ⊗ ⃗v ) ds.
Corollary 14.1.
Let ⃗u0 ∈ bmo−1 with div ⃗u0 = 0. If ∥et∆ ⃗u0 ∥XT < 4C1 0 , then the integral Navier–Stokes
equations have a solution on (0, T ) such that ∥⃗u∥XT ≤ 2∥et∆ ⃗u0 ∥XT .
This is the unique solution such that ∥⃗u∥XT ≤ 2C1 0 .
This Corollary grants local existence of a solution for the Navier–Stokes equations when
the initial value belongs to the space bmo−1
0 :
Definition 14.5.
u ∈ bmo−1 and limT →0 ∥et∆ ⃗u0 ∥XT = 0.
⃗u0 ∈ bmo−1
0 if ⃗
Let us remark that the initial values for the Prodi–Serrin criterion satisfy ⃗u0 ∈ L2 ∩
−1+ 3
Bq,p q
with 1 < p < +∞ and p2 + 3q ≤ 1, hence belong to L2 ∩ bmo−1
0 . However, there is
no weak-strong uniqueness result of Leray weak solutions for initial values in L2 ∩ bmo−1
0 .
−1+ 2
q
Barker noticed that L2 ∩ bmo−1
0 ⊂ B∞,q , while the Prodi-Serrin criterion requires a higher
−1+ 3
regularity (⃗u0 ∈ B∞,q q ). Barker’s theorem [18] states that weak-strong uniqueness holds
2
s
with only a slight improvement in regularity (⃗u0 ∈ L2 ∩ bmo−1
0 ∩ B∞,q with s > −1 + q ).
−1
2
It is easy to check that, if 0 < s < 1 − q , if ⃗u0 ∈ bmo0 , and if ⃗u is the mild solution
−s
is equivalent to
with ∥⃗u∥XT ≤ 2C1 0 , then ⃗u0 ∈ Bq,∞
2
⃗u ∈ L s ,∞ ((0, T ), Lq )
or to
sup ts/2 ∥⃗u(t, .)∥q < +∞.
0<t<T
Lemarié-Rieusset [324] proved a generalization of Barker’s result by relaxing the integrability requirement by a weighted integrability assumption (and restricting weak-strong
uniqueness to suitable Leray solutions):
Theorem 14.11.
Let ⃗u0 be a divergence-free vector field with ⃗u0 ∈ L2 ∩ bmo−1
0 . Assume moreover that the
mild solution ⃗u of the Navier–Stokes equations with initial value ⃗u0 such that ∥⃗u∥XT <
1
2C0 is such that
sup ts/2 ∥⃗u∥Lq ( 1 N dx) < +∞
0<t<T
with
(1+|x|)
2
N ≥ 0, 2 < q < +∞ and 0 ≤ s < 1 − .
q
If ⃗v is a suitable weak Leray solution of the Navier–Stokes equations with the same
initial value ⃗u0 , then ⃗u = ⃗v on (0, T ).
506
The Navier–Stokes Problem in the 21st Century (2nd edition)
As in Barker’s proof [18], Theorem 14.11 will be a consequence of another weak-strong
uniquess theorem:
Theorem 14.12.
Let ⃗u0 be a divergence-free vector field with ⃗u0 ∈ L2 ∩ bmo−1
u be the mild solution
0 . Let ⃗
of the Navier–Stokes equations with initial value ⃗u0 such that ∥⃗u∥XT < 2C1 0 (with
T > 0 such that ∥et∆ ⃗u0 ∥XT < 4C1 0 ). Assume moreover that, for some γ < 1 and some
1
−γ
θ ∈ (0, 1), ⃗u0 belongs to [(L2 ( 1+|x|
2 dx))σ , (B∞,∞ )σ ]θ,∞ (where σ stands for divergencefree). If ⃗v is a suitable weak Leray solution of the Navier–Stokes equations with the
same initial value ⃗u0 , then ⃗u = ⃗v on (0, T ).
Proof. Step 1.
We first check that the mild solution ⃗u in XT of the Navier–Stokes equations with
t∆
u0 ∥XT < 4C1 0 ) is a suitable Leray
initial value ⃗u0 ∈ bmo−1
0 (with T > 0 such that ∥e ⃗
1
solution if moreover ⃗u0 ∈ L2 or a weighted Leray solution if ⃗u0 ∈ L2 ( 1+|x|
2 dx).
1
t∆
⃗u0 ∥XT ≤ δ < 4C1 0 . We
We write E for L2 or L2 ( 1+|x|
2 dx). Let δ such that ∥e
⃗ defined by U
⃗ 0 = et∆ ⃗u0 and U
⃗ n+1 = U
⃗ 0 − B(U
⃗ n, U
⃗ n ),
consider the Picard iterates U
R t (t−s)∆ n
⃗ n ∥X ≤ 2δ, that U
⃗n
where B(⃗v , w)
⃗ = 0e
P div(⃗v ⊗ w)
⃗ ds. We know that ∥U
T
1
n+1
⃗
⃗
δ.
converge to ⃗u and that ∥Un+1 − Un ∥XT ≤ 4 (4δC0 )
⃗ 0 (t, x)| ≤ M⃗u (x), so that ∥U
⃗ 0 (t, .)∥E ≤ CE ∥⃗u0 ∥E . Moreover,
We have |U
0
⃗n ⊗ U
⃗n − U
⃗ n−1 ⊗ U
⃗ n−1 )|
|e(t−s)∆ P div(U
1
⃗ n (s, .)−U
⃗ n−1 (s, .)∥∞ (M ⃗
≤ C√
∥U
⃗ n−1 (s,.) (x))
Un (s,.) (x) + (MU
t−s
From this, we get
∥⃗u∥L∞ ((0,T ),E) ≤ CE,δ ∥⃗u0 ∥E .
Moreover, we have
∥⃗u(t, .) − ⃗u0 ∥E ≤ ∥et∆ ⃗u0 − ⃗u0 ∥E + C sup ∥⃗u(s, .)∥E sup
0<s<t
√
s∥⃗u(s, .)∥∞ = o(1).
0<s<t
Finally, we remark that the mild solution ⃗u is smooth on (0, T ) × R3 , so that, for
0 < t0 ≤ t < T ,
⃗ ⊗ ⃗u|2 = ∆(|⃗u|2 ) − div((2p + |⃗u|2 )⃗u)
∂t (|⃗u|2 ) + 2|∇
and thus
Z
Z
=
ϕR (x)|⃗u(t, x)|2 dx + 2
ϕR (x)|⃗u(t0 , x)|2 dx +
Z tZ
Z
t0
tZ
⃗ ⊗ ⃗u(s, x)|2 dx ds
ϕR (x)|∇
∆(ϕR (x))|⃗u(t, x)|2 dx ds
t0
Z tZ
+
t0
⃗ R (x)) dx ds,
(2p + |⃗u|2 )⃗u · ∇(ϕ
A Theory of Uniformly Locally L2 Solutions
507
x
)w(x), θ is smooth and equal to 1 in a neighborhood of 0 and
where ϕR (x) = θ( R
1
2
w(x) = 1 or w(x) = 1+|x|
u ∈ L∞ (L2 (w dx)),
2 (so that E = L (w dx)). We have that ⃗
√
√
tui uj ∈ L∞ (L2 (w dx)), and thus t(2p + |⃗u|2 ) ∈ L∞ (L2 (w dx)) (as w ∈ A2 and
P
P3 ∂ ∂
p = − 1≤i≤3 j=1 i∆ j (ui uj )), so that
Z
ϕR (x)|⃗u(t, x)|2 dx + 2
Z tZ
⃗ ⊗ ⃗u(s, x)|2 dx ds
ϕR (x)|∇
t0
Z
≤C sup
0<s<T
Z TZ
+
0
T
Z
2
|⃗u(s, x)| w(x) dx + C
Z
|⃗u(s, x)|2 w(x) dx ds
0
√
ds
s 2p + |⃗u|2 |⃗u| w(x) dx √ < +∞.
s
We then let R go to +∞ and t0 go to 0.
t∆
u0 ∥XT ≤ δ <
A similar proof gives that, if ⃗u0 ∈ bmo−1
0 (with T > 0 such that ∥e ⃗
−γ
⃗ n satisfy
and if moreover ⃗u0 ∈ B
with 0 < γ < 1, ten thhe Picard iterates U
1
4C0 )
∞,∞
⃗n ⊗ U
⃗n − U
⃗ n−1 ⊗ U
⃗ n−1 )|
|e(t−s)∆ P div(U
1
⃗ n (s, .)−U
⃗ n−1 (s, .)∥∞ ∥U
⃗ n (s, .)∥∞ + ∥U
⃗ n−1 ∥∞ )
≤ C√
∥U
t−s
so that, for 0 < t < min(T, 1),
−γ .
∥⃗u(t, .)∥∞ ≤ Cδ,γ t−γ/2 ∥⃗u0 ∥B∞,∞
Step 2.
We now check that Theorem 14.11 is a corollary of Theorem 14.12.
The first step is to diminish the value of N . We know that the mild solution ⃗u is a weak
Leray solution as well. (As a matter of fact, the solutions of the Leray mollification will
converge in D′ to the mild solution and to a weak Leray solution [313]). In particular,
we have sup0<t<T ∥⃗u(t, .)∥2 < +∞, while sup0<t<T t1/2 ∥⃗u(t, .)∥∞ ≤ ∥⃗u∥XT < +∞.
Thus,
1
1
sup t 2 − q ∥⃗u∥q < +∞.
0<s<T
If 0 ≤ α ≤ 1, we find that
√
2
sup ( t)(1−α)(1− q )+αs ∥⃗u∥Lq (
0<t<T
1
(1+|x|)αN
dx)
< +∞.
For 0 < α < min(1, N4q ), we have 0 < sα = (1 − α)(1 − 2q ) + αs < 1 − 2q and αN < 4q .
1
As 4q < 3, we find that the weight (1+|x|)
αN belongs to the Muckenhoupt class Aq , so
2
1
that the fact that the mild solution ⃗u0 satisfies ⃗u ∈ L sα ,∞ ((0, T ), Lq ( (1+|x|)
αN dx) is
α
equivalent to the fact that ⃗u0 belongs to the Besov space BL−s
1
q(
(1+|x|)αN
√ s
1
sup0<t<T sup0<t<T ( t) α ∥et∆ ⃗u0 ∥Lq (
< +∞). For
αN dx)
(1+|x|)
2
sα < σ < 1 − ,
q
dx),∞
(i.e. that
508
The Navier–Stokes Problem in the 21st Century (2nd edition)
we have
α
BL−s
q(
1
(1+|x|)αN
dx),∞
⊂ HL−σ
q(
1
(1+|x|)αN
dx)
= (Id − ∆)σ (Lq (
1
dx)).
(1 + |x|)αN
We now recall the result proved in [324] on the complex interpolation of weighted
Sobolev spaces. Let θ ∈ (0, 1), s0 , s1 be real numbers, 1 < p0 , p1 < +∞ and s =
(1 − θ)s0 + θs1 and p1 = (1 − θ) p10 + θ p11 . Then, if w0 is a weight in the Muckenhoupt
class Ap0 and w1 is a weight in the Muckenhoupt class Ap1 ,
(Id − ∆)s Lp (w01−θ w1θ dx) = [(Id − ∆)s0 Lp0 (w0 dx), (Id − ∆)s1 Lp1 (w1 dx)]θ .
We are interested in s = σ, w01−θ w1θ =
θ ∈ (0, 1) such that
max(0,
1
(1+|x|)αN
, p0 = 2, s0 = 0 and w1 = 1. We pick
Nα 2
2
2
, − 2(1 − σ − )) < 1 − θ < .
2 q
q
q
We obtain, for
p = q, s = σ, w01−θ w1θ =
1
,
(1 + |x|)αN
and
p0 = 2, s0 = 0, w1 = 1,
the values

1
1 1 1−θ


) with q < p1 < +∞
= ( −


p
θ q
2

1


σ
s1 =
θ




αN
1


with
<2
 w0 =
αN
1
−θ
1−θ
(1 + |x|)
so that w0 ∈ Ap0 and w1 ∈ Ap1 . Moreover, L2 (
1
αN
(1+|x|) 1−θ
1
dx) ⊂ L2 ( 1+|x|
2 dx) and
−γ
1
with
⊂ B∞,∞
Hp−s
1
γ = s1 +
3
3
1−θ
1 1
2
1−θ
1
)=1+ ( +σ+ −1−
) < 1.
= (σ + − 3
p1
θ
q
2
θ q
q
2
Thus, under the assumptions of Theorem 14.11, we find that
1
1
⃗u0 ∈ [L2 (w0 dx), Hp−s
]θ ⊂ [L2 (w0 dx), Hp−s
]θ,∞ .
1
1
As ⃗u0 is divergence free and as the Leray projection operator P is bounded on
1
L2 (w0 dx) and on Hp−s
, we find that
1
1
⃗u0 ∈ [(L2 (w0 dx))σ , (Hp−s
)σ ]θ,∞ ⊂ [(L2 (
1
1
−γ
dx))σ , (B∞,∞
)σ ]θ,∞
1 + |x|2
so that ⃗u0 fulfills the assumptions of Theorem 14.12.
Step 3.
1
−γ
Now, we assume that ⃗u0 ∈ L2 ∩ bmo−1 ∩ [(L2 ( 1+|x|
2 dx))σ , (B∞,∞ )σ ]θ,∞ with γ < 1
and 0 < θ < 1, and we shall prove the following lemma of Barker:
A Theory of Uniformly Locally L2 Solutions
509
There exists a constant C1 (depending on ⃗u0 ) such that, for every t ∈ (0, T ), for every
suitable weak Leray solution ⃗v of the Navier–Stokes equations with initial value ⃗u0 ,
we have
∥⃗v (t, .) − ⃗u(t, .)∥L2 ( 1 2 dx) ≤ C1 tη
(14.55)
1+|x|
with η =
θ(1−γ)
2(1−θ) ,
where ∥⃗u∥XT ≤
1
2C0
and ⃗u is the mild solution on (0, T ).
Of course, we need to prove (14.55) only for t < T0 for some T0 depending on ⃗u0 , since
for t > T0 we can write
η
t
∥⃗v (t, .) − ⃗u(t, .)∥L2 ( 1 2 dx) ≤ ∥⃗v (t, .)∥2 + ∥⃗u(t, .)∥2 ≤ 2∥⃗u0 ∥2
.
1+|x|
T0
For every ϵ ∈ (0, 1) we can split ⃗u0 in
⃗u0 = ⃗v0,ϵ + w
⃗ 0,ϵ
with
−γ
div ⃗v0,ϵ = div w
⃗ 0,ϵ = 0, ∥⃗v0,ϵ ∥B∞,∞
≤ C2 ϵθ−1 and ∥w
⃗ 0,ϵ ∥L2 (
1
1+|x|2
dx)
≤ C 2 ϵθ ,
where C2 depends only on ⃗u0 . For 0 < t ≤ 1, ∥et∆⃗v0,ϵ ∥∞ ≤ C3 t−γ/2 ϵθ−1 . If 0 < T1 < 1,
we have
1−γ
√ t∆
sup
t∥e ⃗v0,ϵ ∥∞ ≤ C3 ϵθ−1 T1 2
0<t<T1
and
s
sup
1
t3/2
0<t<T1 ,x∈R3
Z tZ
0
1−γ
2
√
|et∆⃗v0,ϵ |2 dx ≤ C4 ϵθ−1 T1
B(x, t)
1−δ
2
so that ∥et∆⃗v0,ϵ ∥XT1 ≤ (C3 + C4 )ϵθ−1 T1
so that T1 < 1].
<
1
8C0
2
if T1 < C5 ϵ 1−γ (1−θ) ) [with C5 < 1
According to Step 1, we know that the Navier–Stokes equations with initial value ⃗v0,ϵ
will have a solution ⃗vϵ on (0, T1 ) such that ∥⃗vϵ (t, .)∥∞ ≤ C6 t−γ/2 ϵθ−1 . Moreover, by
1
Step 1, ⃗vϵ is a weighted Leray weak solution (since ⃗v0,ϵ = ⃗u0 − w
⃗ 0,ϵ ∈ L2 ( 1+|x|
2 dx)).
Now, if ⃗v is a suitable weak Leray solution of the Navier–Stokes equations with initial
value ⃗u0 , ⃗v is a weighted Leray weak solution as well and, by Lemma 14.7 (since
⃗vϵ ∈ L2 ((0, T1 ), L∞ )), we know that for every t0 ∈ (0, T1 ) we have (writing L2Φ for
1
L2 ( 1+|x|
2 dx))
∥⃗v (t0 , .) − ⃗vϵ (t0 , .)∥2L2
≤e
Φ
sup0<s<t (∥⃗
v (s,.)∥ 2 +∥⃗
vϵ (s,.)∥ 2 )4
0
L
L
Φ
Φ
C7 t0 (ν+
3
ν
) C7
e
R t0
0
∥⃗
v ϵ ∥2
∞
ν
(14.56)
dt
∥w
⃗ 0,ϵ ∥2L2 .
Φ
Recall that ∥w
⃗ 0,ϵ ∥L2 (
Z
1
1+|x|2
dx)
≤ C2 ϵθ , ∥⃗v ∥L2 (
t0
∥⃗vϵ ∥2∞ ds ≤ C62 ϵ2(θ−1)
0
1
1+|x|2
dx)
≤ ∥⃗v (s, .)∥2 ≤ ∥⃗u0 ∥2 ,
1 1−γ
1
t
≤ C62
C 1−γ ,
1−γ 0
1−γ 4
and, by Step 1,
sup ∥⃗vϵ ∥L2 (
0<s<T1
1
1+|x|2
dx)
≤ C8 ∥⃗v0,ϵ ∥L2 (
1
1+|x|2
dx)
≤ C8 (∥⃗u0 ∥2 + C2 ϵθ ).
510
The Navier–Stokes Problem in the 21st Century (2nd edition)
Thus, for a constant C9 depending only on ⃗u0 , ν and T , we get
∥⃗v (t0 , .) − ⃗u(t0 , .)∥2L2 ≤ ∥⃗v (t0 , .) −⃗vϵ (t0 , .)∥2L2 + ∥⃗u(t0 , .) −⃗vϵ (t0 , .)∥2L2 ≤ C9 ϵθ (14.57)
Φ
Φ
Φ
2
In particular, for t = 12 C5 ϵ 1−γ (1−θ) , we find
∥⃗v (t, .) − ⃗u(t, .)∥2L2 ≤ C10 tη .
(14.58)
Φ
This inequality has thus been proved for every t ∈ (0, T0 ) with T0 = min(T, C25 ).
Step 4.
1
We now prove weak-strong uniqueness when ⃗u0 ∈ L2 ∩ bmo−1 ∩ [(L2 ( 1+|x|
2 dx))σ ,
−γ
(B∞,∞ )σ ]θ,∞ with γ < 1.
Let ⃗u be the mild solution of the Navier–Stokes equations with initial value ⃗u0 such
that ∥⃗u∥XT < 2C1 0 and let ⃗v be a suitable weak Leray solution of the Navier–Stokes
equations with the same initial value ⃗u0 . As ⃗u ∈ L2 ((ϵ, T ), L∞ ) for every ϵ ∈ (0, T )
and as ⃗v is a suitable Leray solution on (t0 , T ) for almost every t0 ∈ (0, T ), we may
apply Lemma 14.8 and find for every t ∈ (t0 , T )
∥⃗u(t, .) − ⃗v (t. )∥22 + 2ν
Z
t
⃗ ⊗ (⃗u − ⃗v )∥22 ds
∥∇
t0
≤ ∥⃗u(t0 , .)
− ⃗v (t0 , .)∥22
ZZ
(14.59)
t
+2
⃗ u − ⃗v )) dx ds.
⃗u · ((⃗u − ⃗v ) · ∇(⃗
t0
We get
∥⃗u(t, .)
− ⃗v (t. )∥22
≤ ∥⃗u(t0 , .)
− ⃗v (t0 , .)∥22
1
+
ν
Z
t
∥⃗u∥2∞ ∥⃗u − ⃗v ∥22 ds.
(14.60)
t0
Letting t0 go to 0, we find
∥⃗u(t, .)
− ⃗v (t. )∥22
1
≤
ν
Z
t
∥⃗u∥2∞ ∥⃗u − ⃗v ∥22 ds.
(14.61)
0
We know by Step 3 that ∥⃗u(t, .) − ⃗v (t. )∥22 ≤ Ct2η , so that
√
1
1
( sup s∥⃗u(s, .)∥∞ )2
sup s−2η ∥⃗u(s, .) − ⃗v (s. )∥22
ν 0<s<t
2η 0<s<t
√
√
For t0 such that sup0<s<t0 s∥⃗u(s, .)∥∞ ≤ 2ην, we find
t−2η ∥⃗u(t, .) − ⃗v (t. )∥22 ≤
sup s−2δ ∥⃗u(s, .) − ⃗v (s. )∥22 = 0
0<s<t0
so that ⃗u = ⃗v on (0, t0 ); on the other hand, we get
∥⃗u(t, .) − ⃗v (t. )∥22 ≤
and thus ⃗u = ⃗v on (0, T ).
1
sup s∥⃗u(s, .)∥2∞
ν 0<s<T
Z
0
t
1
∥⃗u − ⃗v ∥22 ds
t0
A Theory of Uniformly Locally L2 Solutions
14.9
511
Further Results on Global Existence of Suitable Weak
Solutions
We have seen various cases of existence of global suitable weak solutions of the Cauchy
problem for the Navier–Stokes equations

⃗
⃗

 ∂t ⃗u =ν∆⃗u − ⃗u · ∇⃗u − ∇p
(14.62)
div ⃗u =0


⃗u(0, .) =⃗u0
where ⃗u0 is a locally square integrable divergence free vector field:
• when ⃗u0 ∈ L2 , Leray’s mollification and Rellich’s theorem give a solution ⃗u ∈
L∞ ((0, +∞), L2 )∩L2 ((0, +∞), Ḣ 1 ) (Theorem 12.2) which is suitable (Theorem 13.6);
R
• when ⃗u0 ∈ L2 (R3 /2πZ3 ) with (−π,π)3 ⃗u0 dx = 0, Leray’s mollification and Rellich’s
⃗ ⊗ ⃗u ∈
theorem give a periodical solution ⃗u ∈ L∞ ((0, +∞), L2 (R3 /2πZ3 )) with ∇
2
2
3
3
L ((0, +∞), L (R /2πZ )) (see page 398); we can prove that this solution is suitable
in exactly the same way as for the case ⃗u0 ∈ L2 ;
R
• when ⃗u0 ∈ E 2 , i.e. when ⃗u0 ∈ L2uloc and limx0 →∞ B(x0 ,1) |⃗u0 |2 dx = 0, Theorem 14.8
provides a suitable weak solution ⃗u such that
\
2
3
2 1
3
⃗u ∈
(L∞
t Lx )uloc ((0, T ) × R ) ∩ (L Hx )uloc ((0, T ) × R );
0<T <+∞
1
• when ⃗u0 ∈ L2 (Φ dx) with Φ(x) = 1+|x|
2 , Theorem 14.9 provides a suitable weak
solution ⃗u such that
\
\
⃗ ⊗ ⃗u ∈
⃗u ∈
L∞ ((0, T ), L2 (Φ dx)) and ∇
L2 ((0, T ), L2 (Φ dx)).
0<T <+∞
0<T <+∞
2
3
We remark that the control in (L∞
u0 ∈ E 2 or in
t Lx )uloc ((0, T ) × R ) when ⃗
∞
2
2
L ((0, T ), L (Φ dx)) when ⃗u0 ∈ L (Φ dx) are not uniform with respect to T and may
be less and less precise when T goes to +∞. However, it is possible to recover uniform
controls in time for special classes of weak solutions:
Proposition 14.3.
For 0 < γ < 1, let Φγ (x) =
weak solution ⃗u such that
1
(1+|x|)γ .
When ⃗u0 ∈ L2 (Φγ dx), problem (14.62) has a suitable
⃗ ⊗ ⃗u ∈ L2 ((0, +∞), L2 (Φγ dx)).
⃗u ∈ L∞ ((0, +∞), L2 (Φγ dx)) and ∇
Proof. A proof similar to the proof of Theorem 14.9 provides, when ⃗u0 belongs to a weighted
Lebesgue space L2 (Φ dx), a suitable weak solution ⃗u such that
\
\
⃗ ⊗ ⃗u ∈
⃗u ∈
L∞ ((0, T ), L2 (Φ dx)) and ∇
L2 ((0, T ), L2 (Φ dx)),
0<T <+∞
under the following conditions on the weight Φ:
0<T <+∞
512
The Navier–Stokes Problem in the 21st Century (2nd edition)
ˆ (H0) Φ is a continuous Lipschitz function on R3
ˆ (H1) 0 < Φ ≤ 1.
3
⃗
ˆ (H2) There exists C1 > 0 such that |∇Φ|
≤ C1 Φ 2
ˆ (H3) Φ4/3 ∈ A4/3 (where A4/3 is the Muckenhoupt class of weights).
ˆ (H4) There exists C2 > 0 such that Φ(x) ≤ Φ( λx ) ≤ C2 λ2 Φ(x), for all λ ≥ 1.
(For details, see Fernández-Dalgo and Lemarié-Rieusset [173, 174]). In particular, we have
the inequality
Z
Z
√
d √
2
2
2
⃗
⃗
⃗
⃗ dx.
∥ Φ⃗u∥2 + 2ν∥ Φ∇ ⊗ ⃗u∥2 ≤ −ν ∇(|⃗u| ) · ∇Φ dx + (|⃗u|2 + 2p)⃗u · ∇Φ
dt
We have
so that
√
√
√
⃗ ⊗ ⃗u∥22 + 2C12 ∥ Φ⃗u∥22
⃗ ⊗ ( Φ⃗u)∥22 ≤ 2∥ Φ∇
∥∇
√
Φ⃗u ∈ Ḣ 1 , hence
√
√
⃗ ⊗ ( Φ⃗u)∥2
∥ Φ⃗u∥6 ≤ C∥∇
Writing (since Φ4/3 ∈ A4/3 )
√
√
√
5/4
3/4
∥Φ(|⃗u|2 + 2p)∥4/3 ≤ C∥ Φ⃗u∥28/3 ≤ C∥ Φ⃗u∥2 ∥ Φ⃗u∥6 ,
while
√
√
√
1/4
3/4
∥ Φ⃗u∥4 ≤ C∥ Φ⃗u∥2 ∥ Φ⃗u∥6 .
If we follow the proof of Theorem 14.9, we then write the inequality
√
d √
⃗ ⊗ ⃗u∥2
∥ Φ⃗u∥22 + 2ν∥ Φ∇
2
dt
√
√
⃗ ⊗ ⃗u∥2
≤2C1 ν∥ Φ⃗u∥2 ∥ Φ∇
√
√
√
3/2
⃗ ⊗ ⃗u∥3/2 + C 3/2 ∥ Φ⃗u∥3/2 )
+ CC1 ∥ Φ⃗u∥2 (∥ Φ∇
2
1
2
√
√
2
2
2
⃗
≤ν∥ Φ∇ ⊗ ⃗u∥2 + 2C1 ν∥ Φ⃗u∥2
3
√
√
2
5/2
3
4
∥ Φ⃗u∥62
+ CC1 ∥ Φ⃗u∥2 + (CC1 )
ν
which provides a local-in-time control of ⃗u: we have
Z t √
√
√
2
⃗ ⊗ ⃗u∥22 ds ≤ 2∥ Φ⃗u0 ∥22
∥ Φ⃗u∥2 + 2ν
∥ Φ∇
0
√
on (0, T ) with C12 νT (4 + CC12 ν −4 ∥ Φ⃗u0 ∥42 ) = 1/8).
1
In the case of Φ = Φγ = (1+|x|)
γ with 0 < γ < 1, , we can modify the proof in the
following way:
ˆ We have
|∂i ∂j Φγ | ≤ Cγ
as
∥
1
Φγ ;
|x|
p
1 p
⃗ ⊗ ( Φγ ⃗u)∥2 ,
Φγ ⃗u∥2 ≤ C∥∇
|x|
A Theory of Uniformly Locally L2 Solutions
we can write
Z
Z
⃗ u|2 ) · ∇Φ
⃗ γ dx =ν |⃗u|2 ∆Φγ dx
−ν ∇(|⃗
Z
γ(1 + γ)
1
=ν |⃗u|2
−
2γ
dx
(1 + |x|)γ+2
|x|(1 + |x|)γ+1
Z
(1 − γ)|x| + 2
= − νγ |⃗u|2
dx
|x|(1 + |x|)γ+2
Z
1
≤ − νγ(1 − γ) |⃗u|2
dx.
(1 + |x|)γ+1
3/2
ˆ We write (as Φγ ∈ A3/2 )
Z
⃗ γ dx ≤∥Φγ (|⃗u|2 + 2p)∥3/2 ∥ 1 ⃗u · ∇Φ∥
⃗ 3
(|⃗u|2 + 2p)⃗u · ∇Φ
Φγ
p
1
⃗ γ ∥1/2 ∥ 1 ⃗u · ∇Φ
⃗ γ ∥1/2
≤C∥ Φγ ⃗u∥23 ∥ ⃗u · ∇Φ
6
2
Φγ
Φγ
p
γ
1/2 p
3/2
1/2
≤CC1 ∥ Φγ ⃗u∥2 ∥ Φγ ⃗u∥6 ∥
⃗u∥
1 + |x| 2
1
≤νγ(1 − γ)∥
⃗u∥2
1 + |x| 2
1/3
γ
2/3 p
4/3 p
+ C′
C1 ∥ Φγ ⃗u∥2 ∥ Φγ ⃗u∥26 .
ν(1 − γ)
R 2 1
R 2
1
ˆ As |⃗u| (1+|x|)2 dx ≤ |⃗u| (1+|x|)
γ+1 dx, we get
p
p
p
d p
⃗ ⊗ ⃗u∥22 ≤ Cν,γ ∥ Φγ ⃗u∥4/3 ∥ Φγ ⃗u∥26 .
∥ Φγ ⃗u∥22 + 2ν∥ Φγ ∇
2
dt
We have
p
p
p
p
⃗ ⊗ ( Φγ ⃗u)∥2 ≤ C∥ Φγ ∇
⃗ ⊗ ⃗u∥2 + C∥(∇
⃗ Φγ ) ⊗ ⃗u∥2 .
∥ Φγ ⃗u∥6 ≤ C∥∇
p
⃗ ⊗ ⃗u|, we have
Writing v = Φγ |∇
⃗
|(∇
p
Φγ ) ⊗ ⃗u| =|
3
X
⃗
(∇
k=1
≤C
p
Φγ ) ⊗
∂k
∂k ⃗u|
∆
1 p
1
1
Φγ √
( p v).
1 + |x|
Φγ
−∆
We have
(1 + |y|)γ/2 ≤ C((1 + |x|)γ/2 + |x − y|γ/2 )
so that
⃗
|(∇
p
Φγ ) ⊗ ⃗u| ≤C
p
1
1
1
(√
v + Φγ √
γ v).
1 + |x| −∆
( −∆)1+ 2
Thus,
⃗
|∥(∇
1
1
∥L3,∞ ∥ √
v∥L6,2
1 + |x|
−∆
1 p
1
6 ,∞ ∥ √
6 ,2 .
+ C∥
Φγ ∥ 2+γ
γ v)∥
L
L 1−γ
1 + |x|
( −∆)1+ 2
p
⃗ ⊗ ⃗u∥2 .
≤C ′ ∥v∥2 = C ′ ∥ Φγ ∇
p
Φγ ) ⊗ ⃗u|∥2 ≤C∥
513
514
The Navier–Stokes Problem in the 21st Century (2nd edition)
Summing up those estimates, we find
p
p
p
d p
⃗ ⊗ ⃗u∥2 ≤ Cν,γ ∥ Φγ ⃗u∥4/3 ∥ Φγ ∇
⃗ ⊗ ⃗u∥2 .
(14.63)
∥ Φγ ⃗u∥22 + 2ν∥ Φγ ∇
2
2
2
dt
p
p
4/3
Thus, if Cν,γ ∥ Φγ ⃗u0 ∥2 < ν, we find that ∥ Φγ ⃗u∥22 is non-increasing and get a uniform
control on (0, +∞):
p
∥ Φγ ⃗u(t, .)∥22 + ν
t
Z
p
p
⃗ ⊗ ⃗u∥2 ds ≤ ∥ Φγ ⃗u0 (t, .)∥2 .
∥ Φγ ∇
2
2
0
We then finish the proof by noticing that, for ⃗u ∈ L2 (Φγ dx), we have
limλ→+∞ ∥λ⃗u(λ·)∥L2 (Φγ dx = 0; thus, for some λ0 > 1, we have a control on ⃗uλ0 (t, x) =
λ0 ⃗u(λ20 t, λ0 x):
p
∥ Φγ ⃗uλ0 (t, .)∥22 + ν
t
Z
p
p
⃗ ⊗ ⃗uλ ∥22 ds ≤ ∥ Φγ ⃗uλ ,0 ∥22 ,
∥ Φγ ∇
0
0
0
or equivalently:
r
r
Z t r
x
x ⃗
x
∥ Φγ ( )⃗u(t, .)∥22 + ν
⊗ ⃗u∥22 ds ≤ ∥ Φγ ( )⃗u0 ∥22 .
∥ Φγ ( )∇
λ0
λ0
λ0
0
Finally, since Φγ (x) ≤ Φγ ( λx0 ) ≤ λγ0 Φγ (x), we get
p
∥ Φγ ⃗u(t, .)∥22 + ν
Z
t
p
p
⃗ ⊗ ⃗u∥22 ds ≤ λγ ∥ Φγ ⃗u0 (t, .)∥22 .
∥ Φγ ∇
0
0
We may as well discuss the control in L2uloc of a weak solution associated to a large initial
value which does not vanish at infinity (so that Theorem 14.8 can not be used). A way to
get such a control is to assume that the uniform control in L2loc (uniform with respect to
1
spatial shifts of the argument) can be extended to a uniform control in L2 ( (1+|x|)
γ dx):
Proposition 14.4.
For 0 < γ < 2, let Φγ (x) =
1
(1+|x|)γ .
When ⃗u0 ∈ L2 (Φγ dx) is such that
sup ∥⃗u0 (x − x0 )∥L2 (Φγ dx) < +∞,
x0 ∈R3
problem (14.62) has a suitable weak solution ⃗u such that
\
2
3
2 1
3
⃗u ∈
(L∞
t Lx )uloc ((0, T ) × R ) ∩ (L Hx )uloc ((0, T ) × R ).
0<T <+∞
Proof. In Theorem 14.2, we saw that every local Leray solution could be controlled on a
small interval time whose size depends on the norm of ⃗u0 in L2uloc . Such a result is valid for
a weighted local Leray solution [173]: if ⃗u is a suitable weak solution to (14.62) such that,
for some γ ∈ (0, 2),
\
\
⃗ ⊗ ⃗u ∈
⃗u ∈
L∞ ((0, T ), L2 (Φγ dx)) and ∇
L2 ((0, T ), L2 (Φγ dx))
0<T <+∞
0<T <+∞
A Theory of Uniformly Locally L2 Solutions
515
(with limt→0 K |⃗u(t, x) − ⃗u0 (x)|2 dx = 0 for every compact subset K of R2 ), then for every
1
λ > 1 and Tλ ≈ Cγ,ν (1+∥λ⃗u0 (λx)∥
uλ (t, x) = λ2 ⃗u(λ2 t, λx),
)4 , we have a control for ⃗
2
R
L (Φγ dx)
which gives a control of ⃗u on (0, λ2 Tλ ):
sup
0<t<λ2 Tλ
∥⃗u(t, .)∥2L2 (Φγ dx) +
λ2 Tλ
Z
0
⃗ ⊗ u(t, .)∥2 2
∥∇
L (Φγ dx) dt
≤Cγ,ν λ∥λ⃗u0 (λx)∥2L2 (Φγ dx)
(14.64)
≤Cγ,ν λγ ∥⃗u0 (x)∥2L2 (Φγ dx) .
Assume now that supx0 ∈R3 ∥⃗u0 (x − x0 )∥L2 (Φγ dx) < +∞ for somme γ < 2. With no
loss of generality, we may asssume that 1 < γ < 2 (as Φγ2 ≤ Φγ1 for γ1 < γ2 ). Applying
the control (14.64) to ⃗u(x − x0 ) instead of ⃗u0 , we find that ∥λ⃗u0 (λx − x0 )∥L2 (Φγ dx) ≤
γ−1
λ 2 ∥⃗u0 (x − x0 )∥L2 (Φγ dx) which is uniformly small (with respect to x0 ) when λ is great, so
that, for T ≥ T0 , T0 large enough and independent from x0 , we have
sup ∥⃗u(t, x − x0 )∥2L2 (Φγ dx) +
0<t<T
T
Z
0
≤Cγ,ν
⃗ ⊗ u(t, .)∥2 2
∥∇
L (Φγ dx) dt
T
T0
γ/2
∥⃗u0 (x − x0 )∥2L2 (Φγ dx) .
Thus, ⃗u is controlled in L2uloc .
Our next example deals with a sum of plane waves with large amplitudes:
Theorem 14.13.
⃗1 . . . , A
⃗ N N vectors in R3 \ {0}, B
⃗1 =
Let N ≥ 1, ω
⃗ 1, . . . , ω
⃗ N N vectors in R3 \ {0}, A
N
N
N
⃗1 ∧ ω
⃗N = A
⃗N ∧ ω
A
⃗ 1, . . . , B
⃗ N . For θ = (θ1 , . . . , θN ) ∈ R /Z = T , define
⃗u0 (x, θ) =
N
X
⃗ k.
cos(⃗
ωk · x + 2πθk )B
k=1
Let 3 < γ ≤ 4 and let Φγ (x) =
1
(1+|x|)γ .
Then, for almost very θ ∈ TN , the problem
⃗ u − ∇p
⃗
∂t ⃗u =ν∆⃗u − ⃗u · ∇⃗
div ⃗u =0


⃗u(0, ·) =⃗u0 (·, θ)



(14.65)
⃗ ⊗ ⃗uθ ∈ L2 ((0, T ), L2 (Φγ dx)) for every T > 0.
has a global weak solution ⃗uθ with ⃗u, ∇
⃗ k are small enough, then
Proof. We have seen in Proposition 8.1 that, if the amplitudes B
we have a global mild solution, as proved by Dinaburg and Sinai [153] or as a consequence
of the Koch and Tataru theorem since ⃗u0,θ ∈ BM O−1 . Thus, the problem we study deals
with large amplitudes. We have ⃗u0 (·, θ) ∈ L2uloc , but it does not vanish at infinity1 , so we
cannot use Theorem 14.8. Similarly, we cannot use Theorem 14.9, as ⃗u0 (., θ) ∈ L2 (Φγ dx)
implies γ > 3, whereas we can construct weak solutions in L2 (Φγ dx) only for γ ≤ 2. Thus,
we need new ideas. Theorem 19.2 is a special case of the theory of homogeneous statistical
1 As a matter of fact, ⃗
u0 vanishes at infinity in the sense that limt→+∞ ∥et∆ ⃗
u0 ∥∞ = 0 but it does not
belong to E 2
516
The Navier–Stokes Problem in the 21st Century (2nd edition)
solutions developed by Višik and Fursikov (see Višik and Fursikov [189, 190], Foias and
Temam [180, 179] or Basson [23]).
The main property of the family of divergence-free vector fields (⃗u0 (·, θ))θ∈TN is its
stability under shifts of the argument: if τx0 f (x) = f (x − x0 ), we have
τx0 (⃗u0 (·, θ)) = ⃗u0 (·, τx∗0 θ)
with
1
1
ω
⃗ 1 · x0 , . . . , θN −
ω
⃗ N · x0 ).
2π
2π
The transform θ 7→ τx∗0 θ preserves the Lebesgue measure on TN .
We follow Basson’s ideas and approximate the (shift-invariant) family (⃗u0 (·, θ))θ∈TN by
another shift-invariant family (⃗u0,α (·, θ))θ∈TN (indexed by α > 0) defined by
τx∗0 θ = (θ1 −
⃗u0,α (x, θ) =
N
X
⃗ k,α
cos(⃗
ωk,α · x + 2πθk )B
k=1
⃗ k,α = A
⃗k ∧ ω
with ω
⃗ k,α ∈ Q3 \ {0}, |⃗
ωk − ω
⃗ k,α | < α and B
⃗ k,α . On any ball B(0, R), we have
|⃗u0,α (x, θ) − ⃗u0 (x, θ)| ≤ α
N
X
⃗ k |(R + |⃗
|A
ωk |).
k=1
An important property of the family (⃗u0,α (·, θ))θ∈TN is its periodicity: for some Lα > 0,
we have for every x ∈ R3 , θ ∈ TN and k ∈ Z3
⃗u0,α (x + kLα , θ) = ⃗u0,α (x, θ).
We then consider a mollified Cauchy problem
for the Navier–Stokes equations with initial
R
data ⃗u0,α (·, θ): we take φ ∈ D(R3 with φ dx = 1, we define, for ϵ > 0, φϵ (x) = ϵ13 φ( xϵ );
then we define ⃗vϵ,α,θ (t, x) the solution in L∞ ((0, +∞), L2 (R3 /Lα Z3 )) of

⃗
⃗

 ∂t⃗vϵ,α,θ =ν∆⃗vϵ,α,θ − (φϵ ∗ ⃗vϵ,α,θ ) · ∇⃗vϵ,α,θ − ∇pϵ,α,θ
(14.66)
div ⃗vϵ,α,θ =0


⃗vϵ,α,θ (0, ·) =⃗u0,α (·, θ)
By usual arguments in the study of the mollified Cauchy problem, we see that we have
a unique global solution such that vϵ,α,θ ∈ L∞ ((0, +∞), L2 (R3 /Lα Z3 )), this solution is
smooth: for every T > 0, j ∈ N and k ∈ N3 ,
sup
sup sup |∂tj ∂xk⃗vϵ,α,θ (t, x)| + |∂tj ∂xk pϵ,α,θ (t, x)| < +∞.
0<t<T x0 ∈R3 θ∈TN
We have continuity in θ as well: for every T > 0, j ∈ N and k ∈ N3 , there exists a constant
⃗ 1, . . . , B
⃗ N , such that
Cα,T,j,k,ϵ which depends on α,T , j, k, ϵ, ω
⃗ 1, . . . , ω
⃗N, B
|∂tj ∂xk⃗vϵ,α,θ (t, x) − ∂tj ∂xk⃗vϵ,α,η (t, x)| + |∂tj ∂xk pϵ,α,θ (t, x) − ∂tj ∂xk pϵ,α,η (t, x)| ≤ Cα,T,j,k,ϵ |θ − η|.
Equations (14.66) preserve the stability of the family (⃗u0,α (·, θ))θ∈TN under the shifts
τx0 :
τx0 ⃗vϵ,α,θ (t, ·) = ⃗vϵ,α,τx∗ θ (t, ·).
0
We have the local energy balance
⃗ ⊗ ⃗vϵ,α,θ |2
∂t (|⃗vϵ,α,θ |2 ) =∆(|⃗vϵ,α,θ |2 ) − 2|∇
− 2 div(((⃗vϵ,α,θ · (φϵ ∗ ⃗vϵ,α,θ ))⃗vϵ,α,θ ) − 2 div(pϵ,α,θ ⃗vϵ,α,θ )
(14.67)
A Theory of Uniformly Locally L2 Solutions
517
We multiply by e−t Φγ (x) (with γ > 3 (so that Φγ ∈ L1 (R3 )) and integrate on ∆ =
(0, +∞) × R3 × TN and get
ZZZ
e−s |⃗vϵ,α,θ (s, x)|2 Φγ (x) dx dθ ds
∆
ZZZ
⃗ ⊗ ⃗vϵ,α,θ (s, x)|2 Φγ (x) dx dθ ds
+2
e−s |∇
∆
Z Z
ZZZ
=
|⃗u0,α (x, θ)|2 Φγ (x) dx dθ +
e−s ∆(|⃗vϵ,α,θ |2 )Φγ (x) dx dθ ds
(14.68)
TN
∆
ZZZ
−2
e−s div(((⃗vϵ,α,θ ·(φϵ ∗ ⃗vϵ,α,θ ))⃗vϵ,α,θ )Φγ (x) dx dθ ds
∆
ZZZ
−2
e−s div(pϵ,α,θ ⃗vϵ,α,θ )Φγ (x) dx dθ ds
∆
Now, we remark that if F (x, θ) is a bounded continuous function on R3 × TN such that
F (x − x0 , θ) = F (x, τ − x0 ∗ θ), we have
Z Z
Z Z
F (x, θ)Φγ (x) dx dθ =
F (x, τx∗0 θ)Φγ (x − x0 ) dx dθ
TN
TN
Z Z
Z Z
=
F (x, θ)Φγ (x − x0 ) dx dθ =
F (x + x0 , θ)Φγ (x) dx dθ.
TN
TN
If F is periodic (F (x + kLα , θ) = F (x, θ) for k ∈ Z3 ), we integrate this equality on x0 ∈
(0, Lα )3 and obtain
Z Z
Z
Z
1
F (x, θ)Φγ (x) dx dθ = ∥Φγ ∥1 3
F (x, θ) dx dθ.
Lα (0,Lα )3 TN
TN
We obtain
Z
TN
Z
+2
TN
∥e−t/2⃗vϵ,α,θ (t, x)∥2L2 ((0,+∞),L2 (Φγ (x) dx)) dθ
⃗ ⊗ ⃗vϵ,α,θ (t, x)∥2 2
∥e−t/2 ∇
L ((0,+∞),L2 (Φγ (x) dx)) dθ
Z Z
=
|⃗u0,α (x, θ)|2 Φγ (x) dx dθ
TN
≤ ∥Φγ ∥1 (
N
X
⃗ k | + α|A
⃗ k |)2 .
|B
k=1
We take (ϵn , αn ) →n→+∞ 0 (with αn ≤ 1) and we define
Mn (θ) =∥e−t/2⃗vϵn ,αn ,θ (t, x)∥2L2 ((0,+∞),L2 (Φγ (x) dx))
⃗ ⊗ ⃗vϵ ,α ,θ (t, x)∥2 2
+ 2∥e−t/2 ∇
n
n
L ((0,+∞),L2 (Φγ (x) dx)) .
and
Σ = {θ ∈ TN /
lim Mn (θ) = +∞}.
n→+∞
We may write
Σ=
\ [
\
j∈N k∈N n∈N,n≥k
{θ ∈ TN / Mn (θ) > 2j }.
(14.69)
518
The Navier–Stokes Problem in the 21st Century (2nd edition)
We have (noting |E| the Lebesgue measure of a subset E of TN )
\
{θ ∈ TN / Mn (θ) > 2j } ≤ {θ ∈ TN / Mk (θ) > 2j }
n∈N,n≥k
N
X
⃗ p | + |A
⃗ p |)2
≤2−j ∥Φγ ∥1 (
|B
p=1
and
|Σ| = lim
lim
j→+∞ k→+∞
\
{θ ∈ TN / Mn (θ) > 2j } = 0.
n∈N,n≥k
Now, we consider θ ∈
/ Σ. We know that there exists a sequence (ϵ(n) , α(n)
with (ϵ(n) , α(n) ) →n→+∞ 0 and supn∈N M(n) (θ) < +∞. In particular, the sequence
(⃗vϵ(n) ,α(n) ,θ )n∈N is locally (in time and space variables) bounded in L2 H 1 . In order to apply
the Aubin-Lions-Simon lemma (a generalization of the Rellich–Lions theorem (Theorem
12.1) in the case of a control of the time derivative in L1 H β with β < 0) [9, 337, 437] .
⃗ ϵ,α,θ can be computed as
By Proposition 6.3 and Lemma 6.4, we know that ∇p
⃗ R G) ∗ div(div((φϵ ∗ ⃗vϵ,α,θ ) ⊗ ⃗vϵ,α,θ ))
⃗ ϵ,α,θ = lim ∇(χ
∇p
R→+∞
where G is the Green function (fundamental solution of −∆), χR (x) = χ(x/R) and χ ∈ D
is equal to 1 on a neighborhood of 0. Thus, if ψ1 , ψ2 , ψ3 ∈ D(R3 ) with ψ2 = 1 on a
neighborhood of the support of ψ1 and ψ3 = 1 on a neighborhood of the support of ψ2 , and
∞
ψ ∈ CC
([0, +∞)), we have
⃗ ϵ,α,θ = ⃗q1 + ⃗q2
ψ(t)ψ1 (x)∇p
with
⃗ ∗ div(ψ3 (φϵ ∗ ⃗vϵ,α,θ ) · ∇(ψ
⃗ 2⃗vϵ,α,θ ))
⃗q1 = ψψ1 ∇G
and
⃗ ∗ div(div((1 − ψ2 )(φϵ ∗ ⃗vϵ,α,θ ) ⊗ ⃗vϵ,α,θ )).
⃗q2 = ψψ1 ∇G
We have (for ϵ ∈ (0, 1))
∥⃗q1 ∥3/2 ≤|ψ(t)∥ψ3 (φϵ ∗ ⃗vϵ,α,θ )(t, .)∥6 ∥ψ2⃗vϵ,α,θ ∥H 1
Z
⃗ ⊗ ⃗vϵ,α,θ |2 )
≤C|ψ(t)|Cψ1 ,ψ2 ,ψ3 (|⃗vϵ,α,θ (t, y)|2 + |∇
1
dy
(1 + |y|)4
(so that ⃗q1 ∈ L1 ([0, +∞), L3/2 ) ⊂ L1 H −1 ) and
Z
1
|⃗q2 | ≤|ψ(t)|Cψ1 ,ψ2 |(φϵ ∗ ⃗vϵ,α,θ )(t, y) ⊗ ⃗vϵ,α,θ (t, y)|
dy
(1 + |y|)4
Z
1
≤C|ψ(t)|Cψ1 ,ψ2 |⃗vϵ,α,θ (t, y)|2
dy
(1 + |y|)4
(so that ⃗q2 ∈ L1 L∞ ). Thus, we find
sup ∥ψ(t)ψ1 (x)∂t⃗vϵ(n) ,α(n) ,θ ∥L1 H −1 < +∞.
n∈N
We then apply the Aubin-Lions-Simon lemma and find a sequence (⃗vϵ[n] ,α[n] ,θ ) which converges strongly in (L2 L2 )loc to a limit ⃗uθ . We have the convergence (as distributions)
A Theory of Uniformly Locally L2 Solutions
519
⃗ vϵ ,α, θ ) to
of ∂t⃗vϵ[n] ,α[n] ,θ to ∂t ⃗uθ , of ∆⃗vϵ[n] ,α[n] ,θ to ∆⃗uθ , of div((φϵ[n] ∗ ⃗vϵ[n] ,α[n] ,θ ) ⊗ ∇⃗
[n]
[n]
⃗
⃗
div(⃗uθ ⊗ ⃗uθ ) and of ∇pϵ[n] ,α[n] ,θ to limR→+∞ ∇(χR G) ∗ div(div(⃗uθ ⊗ ⃗uθ )). Moreover, we find
Rt
Rt
that ⃗u0,α,θ + 0 ∂t⃗vϵ[n] ,α[n] ,θ ds converges to ⃗u0,θ + 0 ∂t ⃗uθ ds, so that ⃗uθ is a solution of the
Cauchy problem for the Navier-Stokes equations with initial value ⃗u0,θ .
Using probabilistic tools of the theory of homogeneous statistical solutions developed
by Višik and Fursikov [190], Basson [23] could prove a much stronger result: for 0 < ϵ < 1,
⃗ ⊗ ⃗uθ ∈ L2 (L2 (Φ4−ϵ )), pθ is locally L3/2 L3/2 and ⃗uθ is suitable in
⃗uθ ∈ L∞ (L2 (Φ4+ϵ )), ∇
the sense of Caffarelli, Kohn and Nirenberg.
Chapter 15
The L3 Theory of Suitable Solutions
In this chapter, we use the theory of local Leray solutions to get two major recent results: the
3
L∞
t Lx regularity result of Escauriaza, Seregin and Šverák [163] for suitable solutions of the
Navier–Stokes equations and the result of Jia and Šverák [244] on the (potential) existence
of a minimal-norm initial value for a blowing-up mild solution to the Navier–Stokes Cauchy
problem (first established by Rusin and Šverák [417] and by Gallagher, Koch and Planchon
[199]).
15.1
Local Leray Solutions with an Initial Value in L3
We first begin with a new construction of a local Leray solution associated to an initial
value in L3 that was initially studied by Calderón [77] (and further studied by Jia and
Šverák [244]):
Proposition 15.1.
Let M > 0. Let ⃗u0 ∈ L3 with div ⃗u0 = 0 and ∥⃗u0 ∥3 ≤ M . Let T0 > 0. Then there exists a
local Leray solution on (0, T0 ) × R3 to the Navier–Stokes problem
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u),
⃗u(0, .) = ⃗u0
that satisfies:
• ⃗u ∈ (L∞ L2 )uloc ∩ (L2 H 1 )uloc
−3/2
• ⃗u ∈ C([0, T0 ], Huloc )
• ∥⃗u(t, .) − Wνt ∗ ⃗u0 ∥L2uloc ≤ η(t), where the function η depends only on ν, T0 and M
and satisfies:
lim η(t) = 0.
t→0+
We begin the proof with an easy lemma:
Lemma 15.1.
Let ⃗v0 ∈ L2uloc and G ∈ (L2 L2 )uloc on (0, T ) × R3 with T < +∞. Then
Z
⃗v = Wνt ∗ ⃗v0 +
t
Wν(t−s) ∗ P div G ds
0
satisfies
2
2 1
⃗v ∈ (L∞
t Lx )uloc ∩ (L Hx )uloc
on (0, T ) × R3 .
DOI: 10.1201/9781003042594-15
520
The L3 Theory of Suitable Solutions
521
⃗ K,R + B
⃗ K,R and G = CK,R +
Proof. For a ball K = B(x0 , 1) and, for R > 2, write ⃗v0 = A
⃗
DK,R , where AK,R = 1B(x0 ,R)⃗v0 and CK,R = 1B(x0 ,R) G.
⃗ K,R ∈ L2 , we have Wνt ∗ A
⃗ K,R ∈ L∞ L2 ∩ L2 Ḣ 1 .
As A
We have, on K,
√
Z
√ 1
νt
⃗ K,R (x)| ≤ C
|Wνt ∗ B
|⃗v0 (y)| dy ≤ C νt ∥⃗v0 ∥L2uloc
4
R
|x−y|>R/2 |x − y|
and, for l = 1, . . . , 3,
⃗ K,R )(x)| ≤ C
|∂l (Wνt ∗ B
√
Z
|x−y|>R/2
√ 1
νt
|⃗v0 (y)| dy ≤ C νt 2 ∥⃗v0 ∥L2uloc .
5
|x − y|
R
⃗ K,R ) ∈ L∞ L2 ∩ L2 Ḣ 1 .
Thus, on (0, T ) with T < +∞, we have 1K (Wνt ∗ B
Rt
2
2
As CK,R ∈ L ((0, T ), L ), we have 0 Wν(t−s) ∗P div CK,R ds ∈ L∞ L2 ∩L2 Ḣ 1 on (0, T )×
3
R .
We have, on K,
Z t
Z tZ
dy ds
|
Wν(t−s) ∗ P div DK,R ds(x)| ≤C
|G(s, y)|
|x
− y|4
0
0
|x−y|>R/2
≤C
∥G∥(L2 L2 )uloc
R
and, for l = 1, . . . , 3,
Z t
Z tZ
|∂l (
Wν(t−s) ∗ P div DK,R ds)(x)| ≤C
0
0
|G(s, y)|
|x−y|>R/2
dy ds
|x − y|5
∥G∥(L2 L2 )uloc
≤C
.
R2
Thus, on (0, T ) with T < +∞, we have 1K (
Rt
0
Wν(t−s) ∗ P div DK,R ds) ∈ L∞ L2 ∩ L2 Ḣ 1 .
Proof of Proposition 15.1:
If ⃗u0 ∈ L3 with div ⃗u0 = 0, then, for any η > 0, we may split it as ⃗u0 = α
⃗ η + β⃗η , where
2
6
α
⃗ η ∈ L , div α
⃗ η = 0, β⃗η ∈ L , div β⃗η = 0 and
∥⃗
αη ∥2 ≤ C2 η∥⃗u0 ∥3 ,
∥β⃗η ∥6 ≤ C2 η −1 ∥⃗u0 ∥3
(just use the embedding L3 ⊂ [L2 , L6 ]1/2,∞ and the boundedness of the Leray projection
operator P on L2 and on L6 ).
Let θ be a mollifier, as we already used it for Leray mollifications, and θϵ = ϵ13 θ( xϵ ). Let
Z
B(⃗u, ⃗v ) =
0
t
3
X
Wν(t−s) ∗ P(
∂i (ui⃗v )) ds.
i=1
Let
ET = {(⃗u, ⃗v ) / ⃗u ∈ L4 ((0, T ), Ḣ 1/2 ), ⃗v ∈ L∞ ((0, T ), L6 )}
normed with
∥(⃗u, ⃗v )∥ET = ∥⃗u∥L4 Ḣ 1/2 + ∥⃗v ∥L∞ L6 .
522
The Navier–Stokes Problem in the 21st Century (2nd edition)
⃗ ∈ L∞ L2 of the mollified Navier–
For ⃗u0 ∈ L2 + L6 ⊂ L2uloc , we looked for a solution V
uloc
Stokes equations

⃗ = ν∆V
⃗ − P(P3 ∂i ((θϵ ∗ Vi )V
⃗ ))
∂t V
i=1
(15.1)

⃗
V (0, .) = ⃗u0
or equivalently of
⃗ = Wνt ∗ ⃗u0 − B(θϵ ∗ V
⃗ ,V
⃗ ).
V
⃗ with α
⃗ = ⃗γ + ⃗δ
Writing ⃗u0 = α
⃗ + β,
⃗ ∈ L2 and β⃗ ∈ L6 , we now look more precisely for V
⃗
with (⃗γ , δ) ∈ ET and

P3
⃗

∂
⃗
γ
=
ν∆⃗
γ
−
P(
∂
(θ
∗
γ
)⃗
γ
+
(θ
∗
γ
)
δ
+
(θ
∗
δ
)⃗
γ
)
i
ϵ
i
ϵ
i
ϵ
i
t

i=1








⃗γ (0, .) = α
⃗
(15.2)

P3

⃗
⃗
⃗
⃗

∂t δ = ν∆δ − P( i=1 (θϵ ∗ δi ).∇δ)







⃗δ(0, .) = β⃗
or equivalently of
⃗γ = Wνt ∗ α
⃗ − B(θϵ ∗ ⃗γ , ⃗γ ) − B(θϵ ∗ ⃗γ , ⃗δ) − B(θϵ ∗ ⃗δ, ⃗γ )
and
⃗δ = Wνt ∗ β⃗ − B(θϵ ∗ ⃗δ, ⃗δ).
Let Bϵ be the bilinear operator on ET × ET
Bϵ ((⃗γ1 , ⃗δ1 ), (⃗γ2 , ⃗δ2 )) = (B(θϵ ∗ ⃗γ1 , ⃗γ2 ) + B(θϵ ∗ ⃗γ1 , ⃗δ2 ) + B(θϵ ∗ ⃗δ1 , ⃗γ2 ), B(θϵ ∗ ⃗δ1 , ⃗δ2 )).
We have, for every T > 0,
∥B(θϵ ∗ ⃗γ1 , ⃗γ2 )∥L4 ((0,T ),Ḣ 1/2 ) ≤Cν T 1/8 ∥(θϵ ∗ ⃗γ1 ) ⊗ ⃗γ2 )∥L2 ((0,T ),Ḣ 1/4 )
≤Cν′ T 1/8 ∥⃗γ1 ∥L4 ((0,T ),
γ2 )∥L4 ((0,T ),Ḣ 1/2 )
Ḣ 7/4 ) ∥⃗
T 1/8
∥⃗γ1 ∥L4 ((0,T ),Ḣ 1/2 ) ∥⃗γ2 )∥L4 ((0,T ),Ḣ 1/2 )
ϵ5/4
≤Cν ∥(θϵ ∗ ⃗γ1 ) ⊗ ⃗δ2 )∥L2 ((0,T ),L2 )
≤C3
∥B(θϵ ∗ ⃗γ1 , ⃗δ2 )∥L4 ((0,T ),Ḣ 1/2 )
≤C3 T 1/4 ∥⃗γ1 ∥L4 ((0,T ),Ḣ 1/2 ) ∥⃗δ2 ∥L∞ ((0,T ),L6 )
∥B(θϵ ∗ ⃗δ1 , ⃗γ2 )∥L4 ((0,T ),Ḣ 1/2 ) ≤C3 T 1/4 ∥⃗γ2 ∥L4 ((0,T ),Ḣ 1/2 ) ∥⃗δ1 ∥L∞ ((0,T ),L6 )
∥B(θϵ ∗ ⃗δ1 , ⃗δ2 )∥L∞ ((0,T ),L6 ) ≤C3 T 1/4 ∥⃗δ1 ∥L∞ ((0,T ),L6 ) ∥⃗δ2 ∥L∞ ((0,T ),L6 )
so that
∥Bϵ ((⃗γ1 , ⃗δ1 ), (⃗γ2 , ⃗δ2 ))∥ET ≤ C4 min(T 1/4 ,
T 1/8
)∥(⃗γ1 , ⃗δ1 )∥ET ∥(⃗γ2 , ⃗δ2 ))∥ET
ϵ5/4
where C4 does not depend on ϵ nor T .
We have
⃗ E ≤ C5 (∥⃗
⃗ 6 ).
∥(Wνt ∗ α
⃗ , Wνt ∗ β)∥
α∥2 + ∥β∥
T
(15.3)
The L3 Theory of Suitable Solutions
523
1/8
⃗ 6 ) < A and if T is such that C4 min(T 1/4 , T5/4
If a number A is such that C5 (∥⃗
α∥2 + ∥β∥
)<
ϵ
1
3
⃗
γ , δ) on (0, T ) × R .
8A , then the Picard iterates will converge to a solution (⃗
Moreover, we have ⃗γ ∈ C([0, T ], L2 ) ∩ L2 ((0, T ), Ḣ 1 ) with
∥⃗γ ∥L∞ ((0,T ),L2 ) +∥⃗γ ∥L2 ((0,T ),Ḣ 1 )
≤ Cν (∥⃗
α∥2 +∥(θϵ ∗ ⃗γ )) ⊗ ⃗γ + (θϵ ∗ ⃗γ )) ⊗ ⃗δ + (θϵ ∗ ⃗δ)) ⊗ ⃗γ ∥L2 ((0,T ),L2 )
1/4
⃗
C ′ (∥⃗
α∥2 +ϵ−1/2 ∥⃗γ ∥2 4
∥⃗γ ∥ 4
1/2 ∥δ∥L∞ ((0,T ),L6 ) )
1/2 + T
ν
L ((0,T ),Ḣ
L ((0,T ),Ḣ
)
If moreover, we have ∥⃗γ (T, .)∥2 + ∥⃗δ(T, .)∥6 <
)
2A
C5 ,
then we can reiterate the construction on
(T, 2T ) × R , and so on, on (kT, (k + 1)T ) as long as ∥⃗γ (kT, .)∥2 + ∥⃗δ(kT, .)∥6 < 2A
C5 .
⃗
Remark that the equation on δ does not involve ⃗γ . From
3
1/4
∥B(θϵ ∗ ⃗δ1 , ⃗δ2 )∥L∞ ((0,T1 ),L6 ) ≤ C3 T1 ∥⃗δ1 ∥L∞ ((0,T1 ),L6 ) ∥⃗δ2 ∥L∞ ((0,T1 ),L6 )
and
⃗ L∞ L6 ≤ ∥β∥
⃗ 6,
∥Wνt ∗ β∥
we find that, if
1/4
⃗ 6 < 1,
T1 C3 ∥β∥
4
then we can define the solution ⃗δ on (0, T1 ) × R3 and
⃗ 6.
∥⃗δ∥L∞ ((0,T1 ),L6 ) ≤ 2∥β∥
Moreover, if we can define ⃗γ up to t = kT ≤ T1 , we have ⃗γ ∈ L2 ((0, kT ), Ḣ 1 ) and ∂t⃗γ ∈
L2 ((0, kT ), Ḣ −1 ) so that
∂t ∥⃗γ ∥22 =2⟨∂t⃗γ |⃗γ ⟩Ḣ −1 ,H 1
⃗ ⊗ ⃗γ ∥2 + 2
= − 2ν∥∇
2
Z
⃗δ.((θϵ ∗ ⃗γ ).∇⃗
⃗ γ ) dx
1/2
3/2
≤ − 2ν∥⃗γ ∥2Ḣ 1 + C∥⃗δ∥6 ∥⃗γ ∥2 ∥⃗γ ∥Ḣ 1
≤ − ν∥⃗γ ∥2Ḣ 1 + C6 ν −4 ∥⃗δ∥46 ∥⃗γ ∥22
and thus
∥⃗γ (kT, .)∥22 ≤ ∥⃗
α∥22 e16T1 C6 ν
−4
⃗ 4
∥β∥
6
Thus, ⃗γ will be defined up to t = T1 provided that
⃗ 4 < 2 ln 2,
16T1 C6 ν −4 ∥β∥
6
and we will have
∥⃗γ ∥L∞ ((0,T1 ),L2 ) ≤ 2∥⃗
α∥2
and
√
ν∥⃗γ ∥L2 ((0,T1 ),Ḣ 1 ) ≤ 2∥⃗
α ∥2 .
Finally, by Lemma 15.1, we know that we control ⃗δ in L∞ L2uloc ∩ (L2 H 1 )uloc independently
from ϵ.
⃗ 4 < min( ν 4 ln 2, 1 4 ), we will have solutions (⃗γϵ , ⃗δϵ ) of the mollified
Thus, if T1 ∥β∥
6
8C6
256C
3
equations on (0, T1 )×R3 with controls in L∞ L2uloc ∩(L2 L2 )uloc that are uniform with respect
524
The Navier–Stokes Problem in the 21st Century (2nd edition)
to ϵ. This will allow to use the Rellich–Lions theorem (Theorem 12.1) and to find a local
⃗ = ⃗γ + ⃗δ with
Leray solution V

⃗δ = ν∆⃗δ − P(P3 ∂i δi⃗δ )

∂
t

i=1




⃗δ(0, .) = β⃗






⃗ 2
∥⃗δ∥L∞ ((0,T1 ),L6 ) ≤ 2∥β∥
and

P3

∂t⃗γ = ν∆⃗γ − P( i=1 ∂i γi⃗γ + γi⃗δ + δi⃗γ )





⃗γ (0, .) = α
⃗






∥⃗γ ∥L∞ ((0,T1 ),L2 ) ≤ 2∥⃗
α∥2
Now, the splitting of ⃗u0 into ⃗u0 = α
⃗ η + β⃗η obviously depends on η. For each η, we may
consider our splitting of the mollified Equation (15.1). By uniqueness of the solution of the
⃗η,ϵ , for fixed ϵ, will coincide as long as they are defined. By a
mollified equation, all the V
Cantor diagonal process, considering a decreasing sequence ηn → 0, we may ensure that
⃗η ,ϵ converges to V
⃗η for every n (the convergence occurs on (0, Tη ) × R3 ), and that
V
n k
n
n
−4
1
ν4
⃗
⃗
Vηn = Vηn+1 on (0, Tηn+ 1 ), where Tηn = O(min( 8C
ln
2,
)∥β
∥
)
= O(ηn4 ∥⃗u0 ∥−4
4
η
n 6
3 ).
256C
6
3
Of course, we begin with η0 large enough to ensure that T0 < Tη0 ≈ η04 ∥⃗u0 ∥−4
3 . We have
⃗η .
our local Leray solution ⃗u on (0, T0 ) × R3 , with ⃗u = V
0
−3/2
We check easily that ⃗u ∈ C([0, T0 ], Huloc ). Indeed, we have ⃗u ∈ (L∞ L2 )uloc ⊂
(L1 H −3/2 )uloc and ∂t ⃗u ⊂ (L1 H −3/2 )uloc . If g is a distribution on (0, T0 ) × R3 such that
g ∈ L1t H −3/2 and ∂t g ∈ L1 H −3/2 , then g ∈ C([0, T0 ], H −3/2 ) and, for 0 ≤ t ≤ τ ≤ 1,
Z τ
∥g(t, .) − g(τ, .)∥H −3/2 ≤
∥∂t g(s, .)∥H −3/2 ds.
t
To check it, it is enough to take ζ a smooth function on R which is equal to 1 on (−∞, 1/4)
and to 0 on (3/4, +∞) and to define ζT (s) = ζ( Ts ); we have
Z
t
∂t ((1 − ζT0 /3 )g) ds
g(t, .) =
if T0 /4 < t ≤ T0
0
and
Z
g(t, .) = −
T0
∂t (ζ3T0 g) ds
if 0 ≤ t < 3T0 /4.
t
It remains to estimate ∥⃗u(t, .) − Wνt ∗ ⃗u0 ∥L2uloc . We have, of course,
∥⃗u(t, .) − Wνt ∗ ⃗u0 ∥L2uloc ≤ 2∥⃗u∥(L∞ L2 )uloc ≤ 2(∥⃗γη0 ∥(L∞ L2 )uloc + ∥⃗δη0 ∥(L∞ L2 )uloc )
and thus
∥⃗u(t, .) − Wνt ∗ ⃗u0 ∥L2uloc ≤ C∥⃗u0 ∥3 max(η0 , η0−1 )
1/4
with η0 ≈ T0 ∥⃗u0 ∥3 .
The L3 Theory of Suitable Solutions
525
Now, we go back to the splitting ⃗u = ⃗γη + ⃗δη , valid on (0, Tη ) × R3 . We write, for
0 < t < Tη ,
⃗ η ∥L2uloc + ∥⃗δη (t, .) − Wνt ∗ β⃗η ∥L2uloc
∥⃗u(t, .) − Wνt ∗ ⃗u0 ∥L2uloc ≤ ∥⃗γη (t, .) − Wνt ∗ α
Z t
≤ ∥⃗γη (t, .)∥2 + ∥Wνt ∗ α
⃗ η ∥2 + ∥
Wν(t−s) ∗ P div(⃗δη ⊗ ⃗δη ) ds∥L2uloc
0
with
∥⃗γη (t, .)∥2 + ∥Wνt ∗ α
⃗ η ∥2 ≤ 3∥⃗
αη ∥2 ≤ 3C2 ∥⃗u0 ∥3 η
and (following Lemma 15.1)
Z t
∥
Wν(t−s) ∗ P div(⃗δη ⊗ ⃗δη ) ds∥L2uloc ≤C7 ∥⃗δη ⊗ ⃗δη ∥(L2 L2 )uloc ((0,t)×R3 )
0
≤C8 t1/4 ∥⃗δη ∥2L∞ L6
≤4C8 C22 ∥⃗u0 ∥23 η −2 t1/4 .
Thus, if η is small enough to get that 3C2 M η < ϵ/2 and 0 < T[ϵ] < Tη is small enough to
1/4
get that 4C8 C22 M 2 η −2 T[ϵ] < ϵ/2, we find that:
for 0 < t < T[ϵ] ,
∥⃗u(t, .) − Wνt ∗ ⃗u0 ∥L2uloc < ϵ.
As T[ϵ] depends only on ϵ, ν and ∥⃗u0 ∥3 , the proposition is proved.
15.2
Blow up in Finite Time
We apply the theory of local Leray solutions developed in Chapter 14 to get a first
criterion to check that a solution ⃗u ∈ C([0, T ), L3 ) blows up at time T ∗ = T , or that a local
Leray solution on (0, T ) × R3 with initial value in L3 blows up at time T ∗ ≤ T . Let us recall
that we defined the cylinder Qr (t, x) as Qr (t, x) = (t − r2 , t) × B(x, r).
Theorem 15.1.
Let ⃗u0 ∈ L3 be a divergence free vector field on R3 . Let ⃗u be the solution of the Navier–
Stokes equations equations
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u),
⃗u(0, .) = ⃗u0
in C([0, T ∗ ), L3x ), where T ∗ is the maximal existence time of the solution. Let 0 < T <
+∞ and let ⃗v be a local Leray solution on (0, T ) × R3 to the Navier–Stokes problem
∂t⃗v = ν∆⃗v − P div(⃗v ⊗ ⃗v ),
⃗u(0, .) = ⃗u0
that satisfies ⃗v ∈ (L∞ L2 )uloc ∩ (L2 H 1 )uloc . Then:
(A) ⃗u is a local Leray solution.
(B) ⃗v = ⃗u on (0, min(T, T ∗ )).
(C) T < T ∗ if and only if, for every (t, x) ∈ (0, T ] × R3 , there exists r > 0 such
that ⃗v is bounded on Qr (t, x).
526
The Navier–Stokes Problem in the 21st Century (2nd edition)
Proof. (A)√⃗u belongs to C([0, S], L3 ) for every 0 < S < T ∗ , hence to L∞ ([0, S], L3 ). More1
over, t⃗u is bounded on (0, S)×R3 . Hence, t 6 ⃗u ∈ L∞ ((0, S), L4 ). As ⃗u0 ∈ L3 ⊂ L2uloc ,
we have Wνt ∗ ⃗u0 ∈ (L∞ ((0, S), L2 ))uloc ∩ (L2 ((0, S), H 1 ))uloc . As ⃗u ∈ L4 ((0, S), L4 ),
we have
Z t
Wν(t−s) ∗ P div(⃗u ⊗ ⃗u) ds ∈ C([0, S], L2 ) ∩ L2 ((0, S), H 1 ).
0
Moreover, ⃗u is smooth on (0, T ∗ ) × R3 , hence suitable; Finally, for every compact
subset K of R3 , we have
Z
lim
|⃗u(t, .) − ⃗u0 |2 dx ≤ lim |K|1/3 ∥⃗u(t, .) − ⃗u0 ∥23 = 0.
t→0
t→0
K
(B) We have ⃗v = ⃗u on (0, min(T, T ∗ )) by the weak-strong uniqueness theorem (Theorem
14.7).
(C) If T < T ∗√
, we have that ⃗v = ⃗u on [0, T ], hence is continuous from [0, T ] to L3 .
3
3
Moreover, t⃗v is bounded on [0, T ] ×
√ R , thus, for 0 < t ≤ T and x ∈ R , ⃗v is
bounded on Qr (t, x) for every r ∈ (0, t].
Conversely, assume that T ≥ T ∗ , and T < +∞. Thus, T ∗ < +∞, and we know that
sup
∥⃗u(t, .)∥∞ = +∞.
T ∗ /2<t<T ∗
By Proposition 15.1, we know that ⃗u coincides on (0, T ∗ ) with a local Leray solution
defined on (0, 32 T ∗ ). From the proof of Theorem 14.5, we see that there exists R > 0
and M > 0 such that
sup
|w(t,
⃗ x)| ≤ M.
3 ∗
3 ∗
4 T <t< 2 T ,|x|>R
Let us assume that for every x ∈ R3 there exists rx > 0 such that w
⃗ is bounded on
Qrx (T ∗ , x). As B(0, R) is a compact set, we may find a finite covering
B(0, R) ⊂ ∪N
i=1 B(xi , rxi )
so that w
⃗ is bounded on (T ∗ −min1≤i≤N rx2i , T ∗ )×B(0, R), and finally on (T0 , T ∗ )×R3 ,
with T0 = max(T ∗ − min1≤i≤N rx2i , 34 T ∗ ). As ⃗u is bounded on (T ∗ /2, T0 ), we get a
contradiction.
For ⃗v a local Leray solution on (0, T ) of

⃗
⃗
⃗

 ∂t⃗v =ν∆⃗v − ⃗v · ∇⃗v − ∇p = ν∆⃗v − P(⃗v · ∇⃗v )
div ⃗v =0


⃗v (0, .) =⃗u0 ,
(15.4)
we say that a point (t, x) ∈ (0, T ] × R3 is regular if there exists r > 0 such that ⃗v is bounded
on Qr (t, x), and singular otherwise. Thus, Theorem 15.1 states that if T ∗ < +∞ there exists
at least one singular point (T ∗ , x) for ⃗u. Let ϵ0 be the constant in Theorem
14.4. We write
R
1
mr,x f for the average value of f on the ball B(x, r): mr,x f = |B(x,r)|
f
(y)
dy. We have
Bx,r
the following characterization of singular or regular points:
The L3 Theory of Suitable Solutions
527
Theorem 15.2.
Let ⃗u0 ∈ L2uloc be a divergence free vector field on R3 and let ⃗v a local Leray solution
on (0, T ) of equations (15.4) which belongs to (L∞ L2 )uloc ∩ (L2 H 1 )uloc . Let t ∈ (0, T )
and x ∈ R3 . Then:
(A) (t, x) is regular if and only if
ZZ
1
lim
|⃗v (s, y)|3 + |p(s, y) − mr,x p(s, .)|3/2 dy ds = 0.
r→0 r 2
Qr (t,x)
(B) (t, x) is singular if and only if
ZZ
1
inf√ 2
|⃗v (s, y)|3 + |p(s, y) − mr,x p(s, .)|3/2 dy ds ≥ ϵ30 .
0<r< t r
Qr (t,x)
Proof. First, we remark that, by√
the Caffarelli–Kohn–Nirenberg ϵ–regularity criterion Theorem 14.4, if there exists r ∈ (0, t) such that
ZZ
1
|⃗v (s, y)|3 + |p(s, y) − mr,x p(s, .)|3/2 dy ds < ϵ30
r2
Qr (t,x)
then (t, x) is regular.
RR
We now prove (A). If limr→0 r12 Qr (t,x) |⃗v (s, y)|3 + |p(s, y) − mr,x p(s, .)|3/2 dy ds = 0,
RR
then r12 Qr (t,x) |⃗v (s, y)|3 + |p(s, y) − mr,x p(s, .)|3/2 dy ds < ϵ0 for r small enough, and (t, x)
is regular. Conversely, let us assume that (t, x) is regular and let ρ > 0 such that ⃗v is
bounded on Qρ (t, x). On Qρ/2 (t, x), we have
p(s, y) =ϖ(s, x) +
3 X
3
X
Ri Rj (1B(x,ρ) vi vj )
i=1 j=1
+
3 X
3 Z
X
i=1 j=1
(∂i ∂j G(y − z) − ∂i ∂j G(x − z))vi (s, z)vj (s, z) dz
|x−z|>ρ
=p0,x (s) + p1,x (s, y) + p2,x (s, y).
As 1B(x,ρ) vi vj ∈ L∞ ((t − ρ2 , t), L1 ∩ L∞ ), we have p1,x ∈ L∞ ((t − ρ2 , t), L3 ), so that, for
r < ρ/2,
ZZ
1
3/2
|p1,x (s, y) − mr,x p1,x (s, .)|3/2 dy ds ≤ Cr3/2 ∥p1,x ∥L∞ L3 .
r2
Qr (t,x)
On the other hand, on Qρ/2 (t, x),
|p2,x (s, y)| ≤ C
1
sup ∥1B(z,ρ)⃗v (s, .)∥22
ρ4 z∈R3
so that p2,x is bounded on Qρ/2 (t, x), and, for r < ρ/2,
ZZ
1
3/2
|p2,x (s, y) − mr,x p2,x (s, .)|3/2 dy ds ≤ Cr3 ∥1Qρ/2 (t,x) p2,x ∥L∞ L∞ .
r2
Qr (t,x)
528
The Navier–Stokes Problem in the 21st Century (2nd edition)
Similarly, we have
1
r2
Thus, limr→0
15.3
1
r2
ZZ
|⃗v |3 dy ds ≤ Cr3 ∥1Qρ/2 (t,x)⃗v ∥3L∞ L∞ .
Qr (t,x)
RR
Qr (t,x)
|⃗v (s, y)|3 + |p(s, y) − mr,x p(s, .)|3/2 dy ds = 0.
Backward Uniqueness for Local Leray Solutions
Let us recall Escauriaza, Seregin and Šverák’s theorem [164] on backward uniqueness
for parabolic systems in a half-space R3+ = R2 × (0, +∞):
Backward uniqueness in a half-space
Theorem 15.3.
Let ω
⃗ be a vector field on Q+ = (−1, 0) × R3+ such that
• for every bounded subdomain Ω of Q+ , ω
⃗ and its weak derivatives ∂t ω
⃗ , ∂i ω
⃗
(1 ≤ i ≤ 3) and ∂i ∂j ω
⃗ (1 ≤ i ≤ 3, 1 ≤ j ≤ 3) are square-integrable on Ω
• for some positive constant C0 , we have
⃗ ⊗ω
|∂t ω
⃗ − ∆⃗
ω | ≤ C0 (|⃗
ω | + |∇
⃗ |) on Q+
• for some positive constants C1 and M, we have
2
|⃗
ω (t, x)| ≤ C1 eM |x| on Q+
• ω
⃗ (0, .) = 0 on R3+ .
Then ω
⃗ = 0 on Q+ .
The reader will find the proof of Theorem 15.3 in the papers of Escauriaza, Seregin and
Šverák [164, 163] or in Seregin’s book [429].
As we shall see later, Escauriaza, Seregin and Šverák applied their theorem to prove an
endpoint version of Serrin’s blow-up criterion [163] . In this section, we consider the case of
local Leray solutions for the Navier–Stokes problem with no force (f⃗ = 0) and initial value
in L3 .
We first see the consequences of Theorem 15.3 for local Leray solutions for the Navier–
Stokes problem. We consider a local Leray solution ⃗u on (T0 , T1 ) × R3 of
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u)
with ⃗u(T0 , .) ∈ L3 . We have seen in the preceding section that ∂t ⃗u ∈ (L1 H −3/2 )uloc and
−3/2
∂t ⃗u ∈ (L1 H −3/2 )uloc , so that the map t 7→ ⃗u(t, .) is continuous from [T0 , T1 ] to Huloc , and
in particular ⃗u(T1 , .) is well defined. We then have the following theorem:
The L3 Theory of Suitable Solutions
529
Backward uniqueness for local Leray solutions
Theorem 15.4.
Let ⃗u0 ∈ L3 (R3 ) with div ⃗u0 = 0. Let ⃗u be a local Leray solution on (T0 , T1 ) × R3 of
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u)
with ⃗u(T0 , .) = ⃗u0 . If ⃗u(T1 , .) = 0, then ⃗u = 0 on (T0 , T1 ) × R3 .
Proof. Step 1: behavior of ⃗u near t = T0 .
We know that we find a local-in-time solution of the Navier–Stokes problem in
C([T0 , T2 ], L3 ) for a small enough T2 > T0 . By the weak-strong uniqueness theorem
(Theorem 14.7), this solution coincides with ⃗u on (T0 , T2 ) × R3 .
In particular, for every T in (T0 , T2 ), ⃗u is bounded on [T, T2 ] × R3 . By Theorem 9.12,
⃗u is analytic in space and time variables on (T0 , T2 ) × R3 .
Step 2: behavior of ⃗u near x = ∞.
First, let us rematk that for almost every T3 ∈ (T0 , T1 ), ⃗u is a local Leray solution on
(T3 , T2 ). The only thing we have to check is the strong convergence of ⃗u(t, .) to ⃗u(T3 , .)
in L2loc when t → T3+ . But this is a consequence of the energy inequalities due to the
suitability of ⃗u: to get the convergence, it is enough to have that, for every N ∈ N, T3
is a Lebesgue point of t 7→ ∥1B(0,N ) ⃗u(t, .)∥2 .
Let T ∈ (T0 , T1 ). Let
1
S=
∥⃗
u∥2
C0 ν(1 +
(L∞ L2 )uloc
ν
)4
where C0 is the constant in Theorem 14.5. We pick up 0 < T3 < T such that T3 − T <
min(1, S) and ⃗u is a local Leray solution on (T3 , T )×R3 . Then (the proof of) Theorem
14.5 shows that, for |x| large enough (|x| > R, where R depends on ⃗u and T ), we have
3
|⃗u(t, x)| < C √T 1−T on ( T +T
2 , T ).
3
Let T4 ∈ (T0 , T1 ). We can reiterate the argument, descending from T1 to T4 /2 by steps
of size 12 min(1, S), and we find that there exists some R > 0 and M > 0 such that
|⃗u(t, x)| ≤ M on (T4 /2, T1 ) × (R3 \ B(0, R)).
Then we use the local regularity theory of Serrin [434] (see Theorem 13.1) and conclude
that there exists some constant M0 such that
for |α| ≤ 3,
|∂ α ⃗u(t, x)| ≤ M0 on (T4 , T1 ) × (R3 \ B(0, 2R)).
Step 3: Backward uniqueness for the vorticity.
⃗ ∧ ⃗u. Let Q+ = (T4 , T1 ) × R2 × (2R, +∞). For |α| ≤ 2, we have |∂ α ω
Let ω
⃗ =∇
⃗ (t, x)| ≤
2M0 on Q+ . Moreover, as
⃗ω+ω
⃗u
∂t ω
⃗ = ν∆⃗
ω − ⃗u.∇⃗
⃗ .∇⃗
we find that |∂t ω
⃗ (t, x)| ≤ 6M0 (ν + M0 ) on Q+ .
Similarly, we have
⃗ ω | + |⃗
⃗ u| ≤ M0 |∇
⃗ ⊗ω
|∂t ω
⃗ − ν∆⃗
ω | ≤ |⃗u.∇⃗
ω .∇⃗
⃗ | + 3M0 |⃗
ω | on Q+ .
530
The Navier–Stokes Problem in the 21st Century (2nd edition)
Finally, as ⃗u(T1 , .) = 0, we find that ω
⃗ (T1 , x) = 0 for x3 > 2R. Applying Theorem
15.3, we find that ω
⃗ = 0 on Q+ .
In particular, we have ω
⃗ = 0 on (T4 , T2 ) × R2 × (2R, +∞). As ω
⃗ is analytic in space
and time variables on (T0 , T2 ) × R3 , we find that ω
⃗ = 0 on (T0 , T2 ) × R3 .
Step 4: Backward uniqueness for the velocity.
⃗ ∧ω
As −∆⃗u = ∇
⃗ , we find that −∆⃗u = 0 on (T0 , T2 ) × R3 . But ⃗u ∈ C([T0 , T2 ], L3 ); in
3
L , −∆⃗u = 0 implies ⃗u = 0. Thus, we find that ⃗u = 0 on [T0 , T2 ] × R3 . In particular,
⃗u0 = 0. By the weak-strong uniqueness theorem (Theorem 14.7), we find that the local
Leray solution ⃗u must then coincide with the null solution of the Cauchy problem,
and thus ⃗u = 0 on (T0 , T1 ) × R3 .
15.4
Seregin’s Theorem
In 2003 Escauriaza, Seregin and Šverák [163] extended the celebrated Lp Lq criterion of
3
Serrin (Theorem 11.2) to the limit case L∞
t L : in the case of a null force, they showed that
a mild solution that remains bounded in the L3 norm cannot blow up. Seregin [428] then
gave a more precise statement: the L3 norm goes to +∞ near the blow-up time:
Seregin’s theorem
Theorem 15.5.
Let ⃗u0 ∈ L3 (R3 )) with div ⃗u0 = 0.
Let ⃗u be the solution of the Navier–Stokes equations equations
∂t ⃗u = ν∆⃗u − P div(⃗u ⊗ ⃗u),
⃗u(0, .) = ⃗u0
in C([0, T ∗ ), L3x ), where T ∗ is the maximal existence time of the solution. Then, if
T ∗ < +∞, we have
lim∗ ∥⃗u(t, .)∥3 = +∞.
t→T
Proof. We shall show that the assumption lim inf t→T ∗ ∥⃗u(t, .)∥3 < +∞ leads to a contradiction. Thus, we assume that there exists some M < +∞ and some sequence Tn ↑ T ∗ such
that ∥⃗u(Tn , .)∥3 ≤ M .
From Theorem 15.1, we know that there exists a point x0 ∈ R3 such that
√
for every r ∈ (0, T ∗ ),
sup
|⃗u(s, y)| = +∞,
(s,y)∈Qr (x0 )
where Qr (x0 ) = (T ∗ − r2 , T ∗ ) × B(x0 , r). Changing ⃗u into ⃗u(t, x + x0 ), we may assume with
√
√ ∗
no loss of generality that x0 = 0. Similarly, changing ⃗u into T ∗ ⃗u(T ∗ t, T x), we may
assume that T ∗ = 1.
Now, let us assume that there exists some M < +∞ and some sequence Tn → 1− such
that ∥⃗u(Tn , .)∥3 ≤ M . We define
p
p
⃗un (t, x) = 1 − Tn ⃗u(Tn + t(1 − Tn ), 1 − Tn x).
We have ⃗un ∈ C([0, 1), L3 ), ∥⃗un (0, .)∥3 ≤ M and ⃗un blows up at (1, 0).
The L3 Theory of Suitable Solutions
531
By Proposition 15.1 (and the weak-strong uniqueness theorem Theorem 14.7), we know
that ⃗un coincides on (0, 1) with a local Leray solution ⃗vn defined on (0, 2) with
sup ∥⃗vn (t, .)∥L2uloc ≤ Cν,M
0<t<2
and
∥⃗vn ∥(L2 H 1 )uloc ≤ Cν,M .
Thus, for every test function ϕ ∈ D′ ((0, T ∗ ) × R3 ), ϕ⃗vn remains bounded in L∞ L2 ∩ L2 Ḣ 1 ,
while ϕ∂t⃗vn remains bounded in L3/2 H −3/2 . We then use the Rellich–Lions theorem (Theorem 12.1): we may find a sequence nk → +∞ and a function ⃗v∞ ∈ (L∞ L2 )uloc ∩ (L2 Ḣ 1 )uloc
such that:
ˆ ⃗vnk is weak* convergent to ⃗v∞ in (L∞ L2 )uloc and in (L2 Ḣ 1 )uloc
ˆ ⃗vnk is strongly convergent to ⃗v∞ in L2loc ([0, 2] × R3 ).
The limit ⃗v∞ is a solution of the Navier–Stokes equations
⃗ ∞
∂t⃗v∞ = ν∆⃗v∞ − P div(⃗v∞ ⊗ ⃗v∞ ) = ν∆⃗v∞ − div(⃗v∞ ⊗ ⃗v∞ ) − ∇p
and satisfies the local energy inequality:
∂t (
2
|⃗v∞ |2
|⃗v∞ |2
⃗ ⊗ ⃗v∞ |2 − div((p∞ + |⃗v∞ | )⃗v∞ ).
) ≤ ν∆(
) − ν|∇
2
2
2
Moreover, ⃗vnk and ∂t⃗vnk are bounded in (L2 H −3/2 )uloc , so that, writing for 0 ≤ t ≤ 1
R2
⃗vnk (t, .) = − t ∂t (ζ⃗v∞ ) ds. , where ζ is a smooth function on R which is equal to 1 on
−3/2
(−∞, 5/4) and to 0 on (7/4, +∞), we find that ⃗v∞ ∈ C([0, 1], Huloc ) and that, for every
−3/2
t ∈ [0, 1], ⃗v∞ (t, .) is the weak* limit in Huloc of ⃗vnk (t, .).
As ∥⃗vnk (0, .)∥3 = ∥⃗u(Tnk , .)∥3 ≤ M , we find as well that ⃗v∞ (0, .) ∈ L3 . Moreover, by
Proposition 15.1, we know that, for 0 < t < 1,
∥⃗un (t, .) − Wνt ∗ ⃗u(Tn , .)∥L2uloc ≤ η(t)
where the function η depends only on ν and M and satisfies:
lim η(t) = 0.
t→0+
Thus, ∥⃗v∞ (t, .) − Wνt ∗ ⃗v∞ (0, .)∥L2uloc ≤ η(t), and since ⃗v∞ (0, .) ∈ L3 , limt→0+ ∥⃗v∞ (t, .) −
⃗v∞ (0, .)∥L2uloc = 0. Thus, ⃗v∞ is a local Leray solution.
Now, we shall prove that ⃗v∞ blows up at (1, 0): for every r ∈ (0, 1), we have
∥⃗v∞ ∥L∞ (Qr ) = +∞, where Qr = (1 − r2 , 1) × B(0, r). Indeed, let us assume that, for
some r0 , we have ∥⃗v∞ ∥L∞ (Qr0 ) < +∞. On Qr0 /2 , we have, for r0 < R,
p∞ (s, y) =ϖ∞ (s) +
3 X
3
X
Ri Rj (1B(0,r0 )) v∞,i v∞,j )
i=1 j=1
+
3 X
3 Z
X
i=1 j=1
+
3 X
3 Z
X
i=1 j=1
(∂i ∂j G(y − z) − ∂i ∂j G(−z))v∞,i (s, z)v∞,j (s, z) dz
r0 <|x−z|<R
(∂i ∂j G(y − z) − ∂i ∂j G(−z))v∞,i (s, z)v∞,j (s, z) dz
|z|>R
=ϖ∞ (s) + p∞,1 (s, y) + p∞,2,R (s, y) + p∞,3,R (s, y).
532
The Navier–Sto