Fourier Analysis Methods for PDE’s
R. Danchin
November 14, 2005
2
Contents
1 An introduction to Fourier analysis
1.1 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Littlewood-Paley decomposition . . . . . . . . . . . . . . . . . .
1.2.1 Bernstein inequalities . . . . . . . . . . . . . . . . . . . . . .
1.2.2 The nonhomogeneous Littlewood-Paley decomposition . . . .
1.2.3 About the periodic case . . . . . . . . . . . . . . . . . . . . .
1.3 Littlewood-Paley decomposition and functional spaces . . . . . . . .
1.3.1 Sobolev and Hölder spaces . . . . . . . . . . . . . . . . . . . .
1.3.2 Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 A few properties of Besov spaces . . . . . . . . . . . . . . . .
1.4 Paradifferential calculus . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Results of continuity for the paraproduct and the remainder .
1.4.3 Results of continuity for the product . . . . . . . . . . . . .
1.4.4 A result of compactness in Besov spaces . . . . . . . . . . . .
1.4.5 Results of continuity for the composition . . . . . . . . . . .
1.5 Calculus in homogeneous functional spaces . . . . . . . . . . . . . .
1.5.1 Homogeneous Littlewood-Paley decomposition . . . . . . . .
1.5.2 Homogeneous Besov spaces . . . . . . . . . . . . . . . . . . .
1.5.3 Paradifferential calculus in homogeneous spaces . . . . . . . .
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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periodic case
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transport equation
Framework and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A priori estimates in Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
Solving the transport equation in Besov spaces . . . . . . . . . . . . . . . . . . .
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2 The heat equation
2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A priori estimates in Besov spaces for the heat equation . .
2.2.1 Spectral localization . . . . . . . . . . . . . . . . . .
2.2.2 Estimates for the heat equation . . . . . . . . . . . .
2.2.3 A counterexample . . . . . . . . . . . . . . . . . . .
2.2.4 Estimates in nonhomogeneous Besov spaces, and the
2.3 Optimal well-posedness results for Navier-Stokes equations .
2.3.1 The model . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 About scaling and critical spaces . . . . . . . . . . .
2.3.3 Global well-posedness for small data . . . . . . . . .
2.3.4 Further results . . . . . . . . . . . . . . . . . . . . .
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The
3.1
3.2
3.3
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4
CONTENTS
3.4
3.5
On the Cauchy problem for a shallow water equation .
3.4.1 About Camassa-Holm equation . . . . . . . . .
3.4.2 A well-posedness result and a blow-up criterion
3.4.3 Uniqueness . . . . . . . . . . . . . . . . . . . .
3.4.4 The proof of existence . . . . . . . . . . . . . .
3.4.5 Blow-up criterion and energy conservation . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
4 A short insight into compressible fluid
4.1 About the model . . . . . . . . . . . .
4.1.1 Physical conservation laws . . .
4.1.2 The full model . . . . . . . . .
4.1.3 Simplifying assumptions . . . .
4.1.4 Barotropic fluids . . . . . . . .
4.2 Local well-posedness in critical spaces
4.2.1 The existence proof . . . . . .
4.3 Further results . . . . . . . . . . . . .
4.4 Exercises . . . . . . . . . . . . . . . .
Bibliographie
mechanics
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Introduction
Since the 80’s, Fourier analysis methods have known a growing interest in the study of
linear and nonlinear PDE’s. In particular, techniques based on Littlewood-Paley decomposition
and paradifferential calculus have proved to be very efficient. Littlewood-Paley decomposition
has been introduced more than fifty years ago in harmonic analysis but its systematic use in
the PDE’s framework is rather recent. Paradifferential calculus, as for it, has been introduced
in 1981 by J.-M. Bony for the study of the propagation of microlocal singularities in nonlinear
hyperbolic PDE’s (see [4]).
In the present notes, we aim at giving a survey of those techniques and a few examples of
how they may used to solve PDE’s. We focus on two linear models: the heat equation and
the transport equation. For each of them, an example of related nonlinear problem is given.
Although most of the results we present here belong to the mathematical folklore, we want
to point out that Fourier analysis methods are very efficient to tackle most of well-posedness
problems for evolutionary PDE’s in the whole space or in the torus.
The notes are structured as follows. The first chapter deals with Fourier analysis. We
introduce Littlewood-Paley decomposition and show how it may used to characterize functional
spaces. We also give a (non so) short insight into the theory of homogeneous spaces which
turn out to be well adapted to the study of many PDE’s and are omitted in most of textbooks
on functional analysis. Finally, we introduce some smatterings of paradifferential calculus and
prove estimates for the product of two temperate distributions, when it makes sense.
In the second chapter, we focus on the heat equation. We state a priori estimates in Besov
spaces with optimal gain of derivatives. As an application, we prove global well-posedness in
Besov spaces with critical regularity for the incompressible Navier-Stokes equations with small
data.
The third chapter is devoted to the study of transport equations associated to Lipschitz
vectorfields. We state a priori estimates in Besov spaces then apply our results to the study of
a shallow water equation.
In the last chapter, we give an example of coupling between heat equation and transport
equation, namely the compressible barotropic Navier-Stokes system. Well-posedness in Besov
spaces with critical regularity is stated.
Acknowledgments The author is grateful to J.-Y. Chemin for supplying most material for
the first chapter, to Zhouping Xin, Chiaojang Xu and Yinbin Deng for the invitation to deliver
this course at Wuhan Normal University, to Ping Zhang for his kind invitation at the Chinese
Academy of Sciences in Beijing and to Yuxin Ge for helping me to communicate between France
and China.
5
6
CONTENTS
Chapter 1
An introduction to Fourier analysis
1.1
Notations and definitions
• S stands for the Schwartz space of smooth functions over RN whose derivatives of all order
decay at infinity. The space S is endowed with the topology generated by the following
family of semi-norms:
kukM,S := sup (1 + |x|)M |∂ α u(x)| for all
u∈S
and M ∈ N.
x∈RN
|α|≤M
• The set S 0 of temperate distributions is the dual set of S for the usual pairing.
• For any u ∈ S, the Fourier transform of u denoted by u
b or Fu is defined by
Z
∀ξ ∈ RN , u
b(ξ) = Fu(ξ) :=
e−iξ·x u(x) dx.
RN
The Fourier transform maps S into and onto itself, and the inverse Fourier transform is
given by the formula F −1 = (2π)−N F.
• The Fourier transform is extended by duality to the whole S 0 by setting
<u
b, ϕ >:=< u, ϕ
b >S 0 ,S
whenever u ∈
S0
and ϕ ∈ S.
• Derivatives: for all multi-index α ∈ NN , we have
F(∂xα u) = (iξ)α Fu
and F(xα u) = (−i)|α| ∂ξα Fu.
• Algebraic properties: for (u, v) ∈ S × S 0 , we have u ∗ v ∈ S 0 and
F(u ∗ v) = Fu Fv.
The above formula also holds true for couples of distributions with compact supports.
• Multipliers: if A is a smooth function
with polynomial growth at infinity, and u ∈ S 0 (RN )
then we set A(D)u := F −1 A Fu .
• The open (resp. closed) ball with radius R centered at x0 ∈ RN is denoted by B(0, R)
(resp. B(0, R)).
• The shell {ξ ∈ RN | R1 ≤ |ξ| ≤ R2 } is denoted by C(0, R1 , R2 ).
• The notation A . B means that A ≤ CB for some “irrelevant” constant C (which may
change from line to line but whose meaning is clear from the context). Likewise, A ≈ B
means that A . B and B . A.
7
8
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
1.2
The Littlewood-Paley decomposition
1.2.1
Bernstein inequalities
As recalled in the previous section, in Fourier variables differentiating with respect to xj amounts
to multiplying by the function ξ 7→ iξj .
As far as one is concerned with estimates in Lebesgue spaces and whenever the distribution
we consider is well localized in Fourier variables, Bernstein lemma states that differentiating
amounts to multiplying by an appropriate constant
Lemma (Bernstein). Let k be in N. Let (R1 , R2 ) satisfy 0 < R1 < R2 . There exists a constant
C depending only on r1 , r2 , N, such that for all 1 ≤ a ≤ b ≤ ∞ and u ∈ La , we have
1
1
(1.1)
Supp u
b ⊂ B(0, R1 λ) ⇒ sup|α|=k k∂ α ukLb ≤ C k+1 λk+N ( a − b ) kukLa ,
(1.2)
Supp u
b ⊂ C(0, R1 λ, R2 λ) ⇒ C −k−1 λk kukLa ≤ sup|α|=k k∂ α ukLa ≤ C k+1 λk kukLa .
Proof: Arguing by rescaling, one can assume with no loss of generality that λ = 1.
Now, fix a smooth function φ compactly supported and such that φ ≡ 1 in a neighborhood
of the ball B(0, R1 ). We notice that u
b = φb
u. Hence, denoting g := F −1 φ, we get for all
multi-index α,
Z
∂ α u(x) =
∂ α g(y)u(x − y)dy.
Taking advantage of Young inequality, we thus get
k∂ α ukLb ≤ k∂ α gkLc kukLa
with
1
1 1
=1+ − ·
c
b a
Because
k∂ α gkLc ≤ k∂ α gkL∞ + k∂ α gkL1 ≤ C k+1 ,
the proof of the first inequality is complete.
For proving (1.2), we first notice that the inequality on the right is a particular case of
(1.1). Next, we introduce a smooth function ϕ
e with compact support in RN \ {0} and
such that ϕ ≡ 1 in a neighborhood of the shell C(0, R1 , R2 ).
P
As |α|=k (iξ)α (−iξ)α = |ξ|2k , we have
X
(1.3)
u=
gα ? ∂ α u with gbα (ξ) := (iξ)α |ξ|−2k ϕ(ξ).
e
|α|=k
Making use of Young inequality, one can now conclude to the left inequality in (1.2).
Remark. In other words, if u
b is supported in a ball of radius λ then differentiating once is not
worse that multiplying by λ. If u
b is supported in the shell {ξ ∈ RN | R1 λ ≤ |ξ| ≤ R2 λ} then,
up to an irrelevant constant, differentiating once amounts to multiplying by λ.
In most applications, the functions we deal with are not spectrally supported in a shell or
in a ball. Hence, if one wants to take advantage of the nice properties exhibited in Bernstein
lemma, one first has to split the function into pieces which are spectrally supported in a shell
or in a ball. This may be done by introducing a dyadic partition of unity in Fourier variables.
There are two main ways to proceed. Either the decomposition is made indistinctly over the
whole space RN (and we say that the decomposition is homogeneous), or the low frequencies
are treated separately (and the decomposition is said to be nonhomogeneous).
Both decompositions have advantages and drawbacks. The nonhomogeneous one is more
suitable for characterizing the usual functional spaces whereas the properties of invariance by
dilation of the homogeneous decomposition may be more adapted for studying certain PDE’s or
stating optimal functional inequalities having some scaling invariance.
1.2. THE LITTLEWOOD-PALEY DECOMPOSITION
1.2.2
9
The nonhomogeneous Littlewood-Paley decomposition
Let α > 1 and (ϕ, χ) be a couple of smooth functions valued in [0, 1], such that ϕ is supported
in the shell {ξ ∈ RN | α−1 ≤ |ξ| ≤ 2α}, χ is supported in the ball {ξ ∈ RN | |ξ| ≤ α} and
∀ξ ∈ RN , χ(ξ) +
X
ϕ(2−q ξ) = 1.
q∈N
For u ∈ S 0 , one can define nonhomogeneous dyadic blocks as follows. Let
∆q u := 0 if q ≤ −2,
∆−1 u := χ(D)u = e
h ? u with e
h := F −1 χ,
Z
∆q u := ϕ(2−q D)u = 2qN h(2q y)u(x − y)dy with h = F −1 ϕ,
if q ≥ 0.
One can prove that we have
u=
X
in S 0 (RN )
∆q u
q∈Z
for all temperate distribution u (see exercise 1.2). The right-hand side is called nonhomogeneous
Littlewood-Paley decomposition of u.
It is also convenient to introduce the following low frequency cut-off:
X
Sq u :=
∆p u.
p≤q−1
Of course, S0 u = ∆−1 u. Because ϕ(ξ) = χ(ξ/2) − χ(ξ) for all ξ ∈ RN , one can prove that,
more generally, we have
Z
−q
Sq u = χ(2 D)u = e
h(2q y)u(x − y)dy for all q ∈ N.
The Littlewood-Paley decomposition is “almost” orthogonal in L2 . Assuming for instance that
α = 4/3, we have the following result1 :
Proposition 1.2.1. For any u ∈ S 0 (RN ) and v ∈ S 0 (RN ), the following properties hold:
∆p ∆q u ≡ 0
if
∆q (Sp−1 u ∆p v) ≡ 0
|p − q| ≥ 2,
if
|p − q| ≥ 5.
Remark. At this point, one can wonder why it is so important to choose smooth cut-off functions
χ and ϕ for defining a Littlewood-Paley decomposition. Obviously, setting ∆0−1 u := 1|ξ|≤1 (D)u
and ∆0q u := 12q ≤|ξ|≤2q+1 (D)u would define a dyadic spectral decomposition which, in addition,
would be orthogonal in L2 .
In most applications however, it is crucial that we have
k∆q ukLp ≤ C kukLp
for some constant C independent of q.
Alas, unless p = 2, the above inequality fails to be true with ∆0q u instead of ∆q u (see
exercise 1.5).
1
Of course, similar properties may be proved for any α > 1.
10
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
1.2.3
About the periodic case
Throughout, a1 , · · · , aN denote N positive reals. We denote by TN
a the periodic box with
period 2πai in the i-th direction, and QN
:=
(0,
2πa
)
×
·
·
·
×
(0,
2πa
1
N ). We also introduce
a
N
N
e
Za := Z/a1 × · · · × Z/aN the dual lattice associated to Ta .
We claim that the analysis of the previous section for temperate distributions defined on the
whole space RN may be carried out to S 0 (TN
a ) with very few changes.
0
N
Indeed, decompose u ∈ S (Ta ) into Fourier series:
Z
X
1
u(x) =
u
bβ eiβ·x with u
bβ := N
e−iβ·y u(y) dy.
|Ta | TN
a
eN
β∈Z
a
Denoting
hq (x) =
X
ϕ(2−q β)eiβ·x ,
eN
β∈Z
a
one can now define the periodic dyadic blocks as follows:
Z
X
1
−q
iβ·x
hq (y)u(x − y) dy
∆per
u(x)
:=
ϕ(2
β)b
u
e
=
β
q
|TN
a | TN
a
for all
q∈Z
eN
β∈Z
a
and the low frequency cut-off:
Sqper u(x) := u
b0 +
X
∆per
p u(x) =
p≤q−1
X
χ(2−q β)b
uβ eiβ·x .
eN
β∈Z
a
It is obvious that ∆per
p u = 0 for negative enough p (depending on a) and that
X
0
N
u=u
b0 +
∆per
q u in S (Ta ).
q
Now, to any temperate distribution u over RN , one can associate the periodic distribution uper
defined by
X
uper (x) :=
u(x + α) where ZN
2πa := 2πa1 Z × · · · × 2πaN Z.
α∈ZN
2πa
For uper , both periodic and nonhomogeneous Littlewood-Paley decompositions are available. It
turns out that periodic blocks and nonhomogeneous blocks coincide in the following sense:
Proposition 1.2.2. For all temperate distribution u over RN , one has
per
per
∀q ∈ Z, ϕ(2−q D)u
= ∆per
.
q u
Proof: This is actually an easy consequence of the following Poisson formula for θ ∈ S 0 (RN ):
X
X
1
b
θ(2πα),
θ(β)
=
(2π)N
N
N
β∈Z
α∈Z
which, after a change of variables yields
(1.4)
∀u ∈ S 0 (RN ), ∀x ∈ RN ,
X
1 X iβ·x
e
u
b
(β)
=
u(x + α).
|TN
a |
N
eN
β∈Z
a
α∈Z2πa
1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES
11
Now, on one hand, for all x ∈ RN , we have by definition of ϕ(2−q D),
per
R
P
(x) = 2qN α∈ZN RN h(2q y)u(x+α−y) dy,
ϕ(2−q D)u
Z
2πa
= 2qN
h(2q y) uper (x − y) dz.
RN
On the other hand, by taking advantage of (1.4), we get
X
1 X iβ·(x−y)
h 2q (x + α − y) ,
e
ϕ(2−q β) = 2qN
N
|Ta |
N
α∈Z2πa
eN
β∈Z
a
whence
per (x)
∆per
q u
Z
1 X
=
eiβ(x−y) ϕ(2−q β)uper (y) dy,
|
|TN
N
a
e N Ta
β∈Z
a
X Z
= 2qN
h(2q (x+α−y)) uper (y) dy,
N
QN
a
α∈Z2πa
Z
= 2qN
h(2q (x−z)) uper (z) dz.
RN
The proof is complete.
In what follows, we shall focus on distributions defined on RN . We want to point out that all
the properties described in the next sections remain true in the periodic setting provided the
dyadic blocks have been defined as indicated above.
1.3
Littlewood-Paley decomposition and functional spaces
Many functional spaces over RN (or TN
a ) such as Hölder or Sobolev spaces may be characterized
in terms of Littlewood-Paley decomposition.
1.3.1
Sobolev and Hölder spaces
Let us first recall how nonhomogeneous Sobolev spaces H s are defined.
Definition. Let s ∈ R. A temperate distribution u belongs to H s (RN ) if u
b ∈ L2loc (RN ) and
Z
kukH s :=
2 s
2
(1 + |ξ| ) |b
u(ξ)| dξ
1
2
< ∞.
RN
It is classical that H s endowed with the norm k · kH s is a Banach space2 . Now, from the
definition of (χ, ϕ), one easily infers that
X
1
(1.5)
∀ξ ∈ RN , ≤ χ2 (ξ) +
ϕ2 (2−q ξ) ≤ 1,
3
q∈N
whence the following result:
Proposition 1.3.1. There exists a constant C such that for all s ∈ R, we have
X
1
2
kuk
22qs k∆q uk2L2 ≤ C |s|+1 kuk2H s .
s ≤
H
C |s|+1
q
Hence Littlewood-Paley decomposition supplies a simple characterization of Sobolev spaces:
u belongs to H s if and only if the sequence (2qs k∆q ukL2 )q∈Z belongs to `2 (Z).
2
Actually it is even an Hilbert space.
12
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Let us now focus on Hölder spaces.
Definition. Let r ∈ (0, 1). We denote by C r the set of bounded functions u : RN → R such
that there exists C ≥ 0 with
∀(x, y) ∈ RN × RN , |u(x) − u(y)| ≤ C|x − y|r .
(1.6)
More generally, if r > 0 is not an integer, we denote by C r the set of [r] times3 differentiable
functions u such that ∂ α u ∈ C r−[r] for all |α| ≤ r.
One can easily prove that the set C r endowed with the norm
kukC r :=
X
k∂ α ukL∞
|α|≤[r]
|∂ α u(x) − ∂ α u(y)|
+ sup
|x − y|r−[r]
x6=y
!
is a Banach space.
In the case r ∈ R+ \ N, the Littlewood-Paley decomposition supplies a very simple characterization of C r :
Proposition 1.3.2. There exists a constant C such that for all r ∈ R+ \ N and u ∈ C r we
have
C r+1
sup 2qr k∆q ukL∞ ≤
kukC r .
[r]!
q
Conversely, if the sequence (2qr k∆q ukL∞ )q≥−1 is bounded then
kukC r ≤ C
r+1
1
1
+
r − [r] [r] + 1 − r
sup 2qr k∆q ukL∞ .
q
Proof: Let us just sketch the proof (for more details see [10]).
R
Let us first notice that, owing to xα h(x)dx = 0 for all multi-index α, one can write
∀q ∈ N, ∆q u(x) = 2qN
(1.7)
Z
[r]
X
1 k
h(2q (x − y)) u(y) −
D u(x)(y − x)(k) dy.
k!
k=1
Applying the [r]-th order Taylor formula for bounding the right-hand side of (1.7) and
using the fact that k∆−1 ukL∞ ≤ C kukL∞ yields the first inequality.
For
the converse inequality, we notice that since for all multi-index we have ∂ α u =
P proving
α
q ∂ ∆q u, Bernstein lemma insures that
k∂ α ukL∞ ≤
C 1+r
sup 2qr k∆q ukL∞
r − [r] q
whenever |α| ≤ [r].
Next, for all multi-index α of size [r], all (x, y) ∈ RN × RN such that |x − y| ≤ 1 and all
Q ∈ N, we have
α
α
|∂ u(x) − ∂ u(y)| ≤
Q−1
X
α
α
|∂ ∆q u(x) − ∂ ∆q u(y)| +
q=−1
3
From now on, the notation [r] stands for the integer part of r.
+∞
X
q=Q
|∂ α ∆q u(x) − ∂ α ∆q u(y)|.
1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES
13
Making use of Bernstein lemma, we end up with
|∂ α u(x) − ∂ α u(y)| ≤ C r+1 sup 2qr k∆q ukL∞
Q
X
q≥0
2−q(r−[r]−1) |x − y| +
q=−1
X
2−q(r−[r]) .
q≥Q+1
Choosing Q = [− log2 |x − y|] + 1 yields the wanted inequality.
Remark. The above characterization of Hölder spaces is false if r is an integer (see exercise 1.9).
1.3.2
Besov spaces
From now on, we make the convention that for all non-negative sequence (aq )q∈Z , the notation
P
1
r r stands for sup a in the case r = ∞.
a
q q
q
q
The characterizations of Sobolev and Hölder spaces given in the previous part naturally lead
to the following definition:
Definition. Let 1 ≤ p, r ≤ ∞ and s ∈ R. For u ∈ S 0 (RN ), we set
kuk
s
Bp,r
:=
X
qs
2 k∆q ukLp
r
1
r
.
q
s is the set of temperate distributions u such that kuk s < ∞.
The Besov space Bp,r
Bp,r
Before going further into the study of Besov spaces, let us state two important lemmas. The
first one reads:
Lemma 1.3.3. Let (uq )q∈N be a sequence of bounded functions such that the Fourier transform
of uq is supported in dyadic shells. Let us assume that, for some M ≥ 0, we have
kuq kL∞ ≤ C2qM .
Then the series
P
q
uq is convergent in S 0 .
Proof: After rescaling, relation (1.3) rewrites
X
(1.8)
uq = 2−qk
2qN gα (2q ·) ? ∂ α uq .
|α|=k
Therefore, for any test function φ in S, we have
X
(1.9)
huq , φi = (−1)k 2−qk
huq , 2qN ǧα (2q ·) ? ∂ α φi with ǧ(z) = g(−z).
|α|=k
Hence
X
|huq , φi| ≤ C2−qk
2qM k∂ α φkL1 .
|α|=k
P
Let us choose k > M . Then
q huq , φi is a convergent series, the sum of which is less
than CkφkP,S for some large enough integer P. Thus the formula
X
hu, φi := lim
h∆q0 u, φi
q→∞
defines a temperate distribution.
q 0 ≤q
14
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
The second important lemma is the following one:
Lemma 1.3.4. Let s ∈ R and 1 ≤ p, r ≤ ∞. Let (uq )q≥−1 be a sequence of functions such that
X
qs
2 kuq kLp
r
1
r
< ∞.
q≥−1
(i) If Supp
b−1 ⊂ B(0, R2 ) and Supp u
bq ⊂ C(0, 2q R1 , 2q R2 ) for some 0 < R1 < R2 then
P u
s
u := q≥−1 uq belongs to Bp,r and there exists a universal constant C such that
s
kukBp,r
≤C
1+|s|
X
qs
2 kuq kLp
r
1
r
.
q≥−1
(ii) If s is positive and Supp u
bq ⊂ B(0, 2q R) for some R > 0 then u :=
s
Bp,r and there exists a universal constant C such that
s
kukBp,r
C 1+s
≤
s
X
qs
2 kuq kLp
r
P
q≥−1 uq
belongs to
1
r
.
q≥−1
Proof: Under the “hypothesis
of the first assertion and according to Bernstein lemma, we have
”
P
−s
q N
p
. Lemma 1.3.3 thus implies that q uq is a convergent series in S 0 .
kuq kL∞ ≤ C2
Next, we notice that there exists an integer N0 so that
|q 0 − q| ≥ N0 =⇒ ∆q0 uq = 0.
Therefore, with the convention that uq = 0 if q ≤ −2, we can write that
k∆q0 ukLp
X
= k
∆q0 uq kLp
|q−q 0 |<N0
X
≤ C
kuq kLp .
|q−q 0 |<N
0
So, we obtain that
0
2q s k∆q0 ukLp
≤ C
0
X
2q s kuq kLp
|q−q 0 |≤N0
X
≤ C 1+|s|
2qs kuq kLp ,
|q−q 0 |≤N0
and we deduce from convolution inequalities that
s
kukBp,r
≤C
1+|s|
X
rqs
2
kuq krLp
1
r
,
q∈N
which is exactly the first result.
For proving the second result, we first notice that for any q, we have
kuq kLp ≤ C2−qs .
1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES
15
P
As s is positive, this implies that q uq is a convergent series in Lp . Next, we notice that
there exists some N1 ∈ N such that
q 0 ≥ q + N1 =⇒ ∆q0 uq = 0.
Now, we write that
k∆q0 ukLp
X
=
q>q 0 −N
Lp
1
X
≤ C
,
∆q 0 u q
kuq kLp .
q>q 0 −N1
So, we get that
0
X
2q s k∆q0 ukLp ≤ C
0
2(q −q)s 2qs kuq kLp .
q≥q 0 −N1
In other words, we have
0
2q s k∆q0 ukLp ≤ C(ck ) ? (d` )
with ck = 1[−N1 ,+∞[ (k)2−ks
and d` = 2`s ku` kLp .
Applying convolution inequalities for series completes the proof of the second property.
s does not depend on the choice of the couple (χ, ϕ)
Corollary. The definition of the space Bp,r
defining the Littlewood-Paley decomposition.
Remark. The Besov spaces have been obtained by taking first the Lp norm on each dyadic
block, then taking a weighted `r norm. It turns out that taking first the weighted `r norm
and next the Lp norm over RN is also relevant. This yields new Banach spaces called Triebels . Using such spaces may be appropriate for studying certain
Lizorkin spaces and denoted by Fp,r
0 coincides with the Lebesgue space Lp
problems. In particular, if 1 < p < ∞, the space Fp,2
s
and, more generally, Fp,2 coincide with the potential space Hps of temperate distributions u
s
such that (I − ∆) 2 belongs to Lp .
The reader is referred to [36] or [38] for a more complete study of Triebel-Lizorkin spaces.
1.3.3
A few properties of Besov spaces
In the following proposition, we give a first list of important properties of Besov spaces.
Proposition 1.3.5. Let s ∈ R and 1 ≤ p, r ≤ ∞.
s is a Banach space which is continuously embedded in S 0 .
(i) Topological properties: Bp,r
s if and
(ii) Density: the space Cc∞ of smooth functions with compact support is dense in Bp,r
only if p and r are finite.
s
(iii) Duality: for all s ∈ R and 1 ≤ p, r < ∞, the space Bp−s
0 ,r 0 is the dual space of Bp,r .
s
s
If 1 ≤ p < ∞, the completion Bp,∞
of Cc∞ for the norm k · kBp,∞
is the predual of Bp−s
0 ,1 .
s is separable. The same holds for B s .
(iv) Separability: If 1 ≤ p, r < ∞ then the space Bp,r
p,∞
(v) Embeddings: we have
s ,→ B se whenever s
(a) Bp,r
e < s or se = s and re ≥ r,
p,e
r
s−N ( p1 − p1e )
s ,→ B
(b) Bp,r
pe,r
whenever pe ≥ p,
16
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
N
p
0
is continuously embedded in
(c) we have B∞,1
,→ C ∩ L∞ . If p < ∞ then the space Bp,1
the space C0 of continuous bounded functions which decay at infinity.
s
(vi) Fatou property: if (un )n∈N is a bounded sequence of Bp,r
which tends to u in S 0 then
s and
u ∈ Bp,r
s
s .
kukBp,r
≤ lim inf kun kBp,r
θs+(1−θ)e
s
s ∩ B se and θ ∈ [0, 1] then u ∈ B
(vii) Complex interpolation: if u ∈ Bp,r
p,r
p,r
and
1−θ
kukB θs+(1−θ)es ≤ kukθBp,r
.
s kuk s
Be
p,r
p,r
θs+(1−θ)e
s
s
se
(viii) Real interpolation: if u ∈ Bp,∞
∩ Bp,∞
and s < se then u belongs to Bp,1
θ ∈ (0, 1) and there exists a universal constant C such that
kukB θs+(1−θ)es ≤
p,1
for all
C
kuk1−θ
.
kukθBp,∞
s
s
e
Bp,∞
θ(1−θ)(e
s −s)
s is continuously embedded in S 0 . By definition, B s is a
Proof: Let us first prove that Bp,r
p,r
0
subspace of S . Thus it suffices to prove that there exist a constant C and an integer M
such that for any φ ∈ S we have
s kφkM,S .
|hu, φi| ≤ CkukBp,r
(1.10)
Taking advantage of (1.9) with ∆q u instead of uq , we see that for all k ∈ N, there exists
an integer Mk and a constant Ck such that
|h∆q u, φi| ≤ Ck 2−q 2q(1−k) k∆q ukL∞ kφkMk ,S .
According to Bernstein lemma, we have k∆q ukL∞ ≤ C2
chosen so large as to satisfy s − N/p ≥ 1 − k, we have
qN
p
k∆q ukLp . Hence, if k has been
s kφkM ,S
|h∆q u, φi| ≤ Ck 2−q kukBp,r
k
which, after summation on q, yields inequality(1.10).
s . Inequality (1.10) implies that for
Next, consider (u(n) )n∈N a Cauchy sequence in Bp,r
any test function φ in S , sequence (hu(n) , φi)n∈N is a Cauchy sequence in R. Thus the
formula
hu, φi := lim hu(n) , φi
n→∞
s , sequence (∆ u(n) )
defines a temperate distribution. By definition of the norm of Bp,r
q
n∈N
is a Cauchy sequence in Lp for any q . Thus an element uq of Lp exists such that
(∆q u(n) )n∈N converges to uq in Lp . As the sequence (u(n) )n∈N converges to u in S 0 we
actually have ∆q u = uq .
Fix a Q ∈ N and a positive ε. Since for all q ≥ −1, ∆q u(n) tends to ∆q u in Lp , we have
for all n ∈ N,
X
q≤Q
qs
2 k∆q (u
(n)
− u)kLp
r
1
r
= lim
X
m→∞
q≤Q
qs
2 k∆q (u
(n)
−u
(m)
)kLp
r
1
r
.
1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES
17
s
Because the argument of the limit in the right-hand side is bounded by ku(n) − u(m) kBp,r
(n)
s
and (u )n∈N is a Cauchy sequence in Bp,r , one can now conclude that there exists a n0
(independent of Q) such that for all n ≥ n0 , we have
X
qs
2 k∆q (u
(n)
− u)kLp
r
1
r
≤ ε.
q≤Q
s .
Letting Q go to infinity insures that (u(n) )n∈N tends to u in Bp,r
s
Let us tackle the proof of (ii). Consider first the case where r is finite. Let u be in Bp,r
and ε > 0. Since r is finite, there exists an integer q such that
ε
s
ku − Sq ukBp,r
< ·
2
(1.11)
Now let φ be in Cc∞ . For any q 0 ∈ N, Bernstein lemma insures that we have,
0
2q s k∆q0 (φSq u − Sq u)kLp
0
0
≤ 2−q 2q ([s]+2) k∆q0 (φSq u − Sq u)kLp
≤ Cs 2−q
0
k∂ α (φSq u − Sq u)kLp .
sup
|α|=[s]+2
From the above inequality, we get that
(1.12)
s
kφSq u − Sq ukBp,r
≤ Cs k(1 − φ)Sq ukLp +
sup
k∂ α ((1−φ)Sq u)kLp .
|α|=[s]+2
Let us consider a sequence (φn )n∈N such that all the derivatives of φn of order less than
or equal to [s] + 2 are uniformly bounded with respect to n and such that φn ≡ 1 in a
neighborhood of the ball B(0, n). If p is finite, combining Leibniz formula and Lebesgue
theorem, we discover that
lim k(1−φn )Sq ukLp +
n→∞
sup
k∂ α ((1−φn )Sq u)kLp = 0·
|α|=[s]+2
Thus a function φ in Cc∞ exists such that
Cs k(1 − φ)Sq ukLp +
ε
k∂ α ((1−φ)Sq u)kLp < · .
2
|α|=[s]+2
sup
Combining (1.11) and (1.12), we end up with
s
kφSq u − ukBp,r
< ε.
As Sq u is a smooth function, this completes the proof in the case p, r < ∞.
Now, it is obvious that the set Cb∞ of smooth functions with bounded derivatives at all
s . Therefore C ∞ cannot be a dense subset of B s .
orders is embedded in any space B∞,r
c
∞,r
s
Finally, the closure of Cc∞ for the Besov norm Bp,∞
is the space of temperate distributions
u such that
lim 2qs k∆q ukLp = 0,
q→∞
s . This completes the proof of (ii).
which is a strict subspace of Bp,∞
18
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
In order to prove the properties of duality, we use the fact that the map
u 7−→ (∆q u)q≥−1
s and4 `r (Lp ).
is a (bi)continuous isomorphism between Bp,r
s
Assume that 1 ≤ p < ∞. On one hand, if in addition 1 ≤ r < ∞, it is obvious that
0
0
0
`r−s (Lp ) is the dual of `rs (Lp ). On the other hand, `1−s (Lp ) is the dual space of the set of
Lp valued sequences (zq )q≥−1 such that
lim 2qs kzq kLp = 0.
q→+∞
s
Now, one can prove that Bp,∞
is the space of temperate distributions u such that
lim 2qs k∆q ukLp = 0,
q→+∞
whence the desired result.
The proof of (iv) also relies on the use of the map u 7−→ (∆q u)q≥−1 . The details are left
to the reader.
Let us now prove (v). Considering that `r (Z) ⊂ `re(Z) for r ≤ re, the first embedding is
straightforward. In order to prove the second embedding, we apply Bernstein lemma and
get
“
”
kS0 ukLpe ≤ CkS0 ukLp
qN
and k∆q ukLpe ≤ C2
1
− p1e
p
k∆q ukLp
if q ∈ N,
whence the result.
N
p
,→ Cb we use again Bernstein lemma and get that
For proving that Bp,1
k∆q ukL∞ ≤ 2
qN
p
k∆q ukLp .
P
This insures that the series
∆q u of continuous bounded functions converges uniformly on
N
R . Hence u is a bounded continuous function. Besides, it is obvious that the embedding
N
p
is continuous. If p is finite, one can use in addition that Cc∞ is dense in Bp,1
and conclude
that u decays at infinity.
s which
Let us now focus on the proof of (vi). Let (un )n∈N be a bounded sequence of Bp,r
0
tends to some u in S . This insures that for all q ∈ Z, sequence (∆q un )n∈N tends to ∆q u
in S 0 . Since (∆q un )n∈N is a bounded sequence in Lp ∩ Cb∞ , one can conclude that ∆q u
belongs to Lp ∩ Cb∞ and that
k∆q ukLp ≤ lim inf k∆q un kLp .
Now, for all Q ∈ N, we have
PQ
q=−1
2qs k∆
q ukLp
r
1
r
≤
PQ
q=−1
≤ lim inf
2qs lim inf
PQ
q=−1
2qs k∆q un kLp
s .
≤ lim inf kun kBp,r
Letting Q go to infinity completes the proof of (vi).
4
k∆q un kLp
Here `rs stands for the set of sequences (zq )q≥−1 such that k(2qs zq )k`r < ∞.
r
r
1
r
,
1
r
,
1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES
19
Property (vii) is a straightforward consequence of Hölder inequality.
For proving (viii), we write
X
X
kukB θs+(1−θ)es =
2q(θs+(1−θ)es) k∆q ukLp +
2q(θs+(1−θ)es) k∆q ukLp
p,1
q≤Q
q>Q
for some Q to be chosen hereafter.
Now, by definition of the Besov norms, we have
2q(θs+(1−θ)es) k∆q ukLp
s
≤ 2q(1−θ)(es−s) kukBp,∞
,
2q(θs+(1−θ)es) k∆q ukLp
≤ 2−qθ(es−s) kukBp,∞
.
s
e
Thus we infer that
s
kukB θs+(1−θ)es ≤ kukBp,∞
p,1
X
2q(1−θ)(es−s) + kukBp,∞
s
e
q≤Q
s
≤ kukBp,∞
X
2−qθ(es−s)
q>Q
2(Q+1)(1−θ)(es−s)
2(1−θ)(es−s) − 1
+ kukBp,∞
s
e
2−Qθ(es−s)
·
1 − 2−θ(es−s)
In order to complete the proof of (viii), it suffices to choose Q such that
kukBp,∞
s
e
s
kukBp,∞
≤ 2Q(es−s) < 2
kukBp,∞
s
e
s
kukBp,∞
.
s
s , or in other words, what happens if one lets θ tends
One can wonder how far is Bp,∞
from Bp,1
to 1 in proposition 1.3.5.(viii). Of course, one already knows that
s
s
s
Bp,1
,→ Bp,r
,→ Bp,∞
.
A more precise answer is given by the following logarithmic interpolation result:
Proposition 1.3.6. There exists a constant C such that for all s ∈ R, ε > 0 and 1 ≤ p ≤ ∞,
we have
kukBp,∞
s+ε
1+ε
s
s
kukBp,1
≤C
kukBp,∞
1 + log
.
s
ε
kukBp,∞
Proof: Let Q be a positive integer to be fixed hereafter. We have
X
X
s
kukBp,1
=
2qs k∆q ukLp +
2qs k∆q ukLp ,
q<Q
q≥Q
whence,
s
s
+
kukBp,1
≤ (Q + 1)kukBp,∞
Choosing for Q the closest positive integer to
2−Qε
kukBp,∞
s+ε .
1 − 2−ε
kukBp,∞
s+ε
1
log2
yields the result.
s
ε
kukBp,∞
We now want to study how multipliers operate on Besov spaces. Before stating our result, we
need to define the multipliers we are going to consider.
20
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Definition. A smooth function f : RN → R is said to be a S m -multiplier if for all multi-index
α, there exists a constant Cα such that
∀ξ ∈ RN , |∂ α f (ξ)| ≤ Cα (1 + |ξ|)m−|α| .
Proposition 1.3.7. Let m ∈ R and f be a S m -multiplier. Then for all s ∈ R and 1 ≤ p, r ≤ ∞
s to B s−m .
the operator f (D) is continuous from Bp,r
p,r
Proof: Let ϕ
e be a smooth function supported in a shell and such that ϕ
e ≡ 1 on Supp ϕ. It
is clear that we have
∆q f (D)u = ϕ(2
e −q D)f (D) ∆q u
for all
q ∈ N.
Hence, by virtue of convolution inequalities, we have
k∆q f (D)ukLp ≤ kkq kL1 k∆q ukLp
with
1
kq (x) :=
(2π)N
Z
eix·ξ f (ξ)ϕ(2
e −q ξ) dξ.
By performing an easy change of variables, we notice that kkq kL1 = k`q kL1 with
Z
1
`q (y) :=
eiy·η ϕ(η)f
e
(2q η) dη.
(2π)N
Now, for all M ∈ N, we have
(1 +
|y|2 )M `q (y)
Z
(1 − ∆η )M eiy·η ϕ(η)f
e
(2q η) dη,
Z
eiy·η (1 − ∆η )M ϕ(η)f
e
(2q η) dη,
Z
X
q|β|
cα,β 2
eiy·η ∂ α ϕ(η)
e ∂ β f (2q η) dη,
=
=
=
|α|+|β|≤2M
for some integers cα,β (whose exact value does not matter).
Hence, using the fact that integration may be restricted to Supp ϕ
e and that |∂ β f (2q η)| ≤
m−|β|
Cβ 1 + 2q |η|
, we get
(1 + |y|2 )M |`q (y)| ≤ CM 2qm .
Choosing M > N/2, we thus conclude that
kkq kL1 = k`q kL1 ≤ C2qm
whence
∀q ∈ N, 2q(s−m) kf (D)∆q ukLp ≤ C2qs k∆q ukLp .
Stating a similar inequality for q = −1 is left to the reader. This yields the proposition.
Proposition 1.3.8. A constant C exists which satisfies the following properties. Let s < 0,
s
if and only if (2qs kSq ukLp )q∈N ∈ `r (N).
(p, r) ∈ [1, ∞]2 and u ∈ S 0 . Then u belongs to Bp,r
Moreover, we have
1
s
kukBp,r
≤
2
X
q
qs
2 kSq ukLp
r
1
r
1
s .
≤C 1+
kukBp,r
|s|
1.4. PARADIFFERENTIAL CALCULUS
21
Proof: On one hand, we have
2qs k∆q ukLp
≤ 2qs (kSq+1 ukLp + kSq ukLp )
≤ 2−s 2(q+1)s kSq+1 ukLp + 2qs kSq ukLp ,
which proves the inequality on the left. On the other hand, we can write that
X
2qs kSq ukLp ≤ 2qs
k∆q0 ukLp
q 0 ≤q−1
X
≤
0
0
2(q−q )s 2q s k∆q0 ukLp .
q 0 ≤q−1
As s is negative, we get the result.
1.4
Paradifferential calculus
When dealing with nonlinear problems, one often has to study the functional properties of
products of two temperate distributions u and v.
Characterizing distributions such that the product uv makes sense is an intricate question
which is intimately related to the notion of wavefront (see e.g [1] for an elementary introduction).
In this section, we shall see that very simple arguments based on the use of Littlewood-Paley
decomposition yield sufficient conditions for uv to be defined, and continuity results for the map
(u, v) 7→ uv.
1.4.1
Definitions
For u and v two temperate distributions, we have the following formal decomposition:
X
uv =
∆p u∆q v.
p,q
The fundamental idea of paradifferential calculus is to split uv into three parts, both of them
being always defined. The first part, denoted by Tu v and called paraproduct of v by u corresponds to terms ∆p u ∆q v where p is small in comparison with q. The second term, Tv u is
the symmetric counterpart of Tv u (i.e. we keep only the terms corresponding to large frequencies of u multiplied by small frequencies of v ). The third and last term (the remainder term)
corresponds to the dyadic blocks of u and v with comparable frequencies.
This very simple splitting device goes back to the pioneering work by J.-M. Bony in [4]. In
what follows, we shall adopt the following definition for paraproduct and remainder:
Definition. Let u and v be two temperate distributions. We denote
X
X
Tu v =
∆p u ∆q v =
Sq−1 u ∆q v
p≤q−2
q
and
R(u, v) =
X
e qv
∆q u ∆
fq := ∆q−1 + ∆q + ∆q+1 .
with ∆
q
At least formally, we have the following Bony decomposition:
(1.13)
uv = Tu v + Tv u + R(u, v).
Of course, it may happen that the product uv is not defined. However, the reader may retain
the following principles:
22
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
• The paraproduct of two temperate distributions u and v is always defined. This is due to
the fact that the general term of the paraproduct is spectrally localized in dyadic shells.
Besides, the regularity of Tu v is mainly determined by the regularity of v. In particular,
Tu v cannot be more regular than v.
• The remainder may not be defined. Roughly, it is defined as soon as u and v belong to
functional spaces whose sum of regularity index is positive. In that case, the regularity
exponent of R(u, v) is the sum of the regularity exponents of u and v .
1.4.2
Results of continuity for the paraproduct and the remainder
The bilinear paraproduct and remainder operators benefit from continuity properties in most
usual functional spaces. In the present chapter, we focus on Besov spaces. The reader is referred
to [36] and [37] for a more complete study.
As regards the paraproduct, we have the following results:
Proposition 1.4.1. Let 1 ≤ p, r ≤ ∞ and s ∈ R.
s
s
(i) The paraproduct T is a bilinear continuous operator from L∞ × Bp,r
to Bp,r
and there
exists a constant C such that
|s|+1
s ;B s ) ≤ C
kT kL(L∞ ×Bp,r
.
p,r
(ii) If σ > 0 and 1 ≤ r, r1 , r2 ≤ ∞ are such that 1/r = 1/r1 + 1/r2 then T is bilinear
−σ × B s
s−σ
continuous from B∞,r
p,r2 to Bp,r and there exists a constant C such that
1
kT kL(B∞,r
−σ
s−σ
s
×Bp,r
;Bp,r
)
1
2
≤
C |s−σ|+1
·
σ
Proof: According to proposition 1.2.1, the sequence F(Sq−1 u∆q v) q∈Z is supported in dyadic
shells. Hence, in view of proposition 1.3.5, it suffices to prove that
X
1
r r
qs
s .
2 kSq−1 u∆q vkLp
. kukL∞ kvkBp,r
q
hkL1 kukL∞ , this is actually straightforward. This yields the first
Since kSq−1 ukL∞ ≤ ke
result.
For proving the second result, we use that, because σ > 0, we have
X σ(q0 −q) −q0 σ
2q(s−σ) kSq−1 u∆q vkLp ≤ 2qs k∆q vkLp
2
2
∆q 0 u
L∞
.
q 0 ≤q−2
Therefore, combining Hölder and convolution inequalities for series, we get
X
1 X
r r
q(s−σ)
≤
2−kσ kukB∞,r
,
2
kSq−1 u∆q vkLp
−σ kvkB s
p,r
q∈Z
k≥2
1
2
whence the desired inequality.
Remark. By combining the above results of continuity for the paraproduct with the embeddings
stated in proposition 1.3.5, one can get a score of other results of continuity. For instance,
N
p
−ε
−ε , we discover that T is continuous from
by using that for all ε > 0, we have Bp11,r1 ,→ B∞,r
1
N
p
−ε
s
s−ε for all 1 ≤ p, p , r, r , r ≤ ∞ such that 1/r = 1/r + 1/r .
Bp11,r1 × Bp,r
to Bp,r
1
1 2
1
2
2
1.4. PARADIFFERENTIAL CALCULUS
23
Proposition 1.4.2. Let (s1 , s2 ) ∈ R2 and 1 ≤ p, p1 , p2 , r, r1 , r2 ≤ ∞. Assume that
(1.14)
1
1
1
≤
+
≤ 1,
p
p1 p2
Bps11,r1
× Bps22,r2
s1 +s2 +N
1
1
1
p − p1 − p2
Then the remainder R maps
C such that
kR(u, v)k
1
1
1
≤
+
r
r1 r2
Bp,r
in
and s1 + s2 > 0.
s1 +s2 +N
Bp,r
1
− p1 − p1
p
1
2
and there exists a constant
|s1 +s2 |+1
≤C
kukBps1 ,r kvkBps2 ,r .
1 1
2 2
s1 + s2
Proof: It suffices to treat the case where 1/r = 1/r1 + 1/r2 and 1/p = 1/p1 + 1/p2 . The
general case then follows from the embeddings of proposition 1.3.5.(v).
Now, by definition of the remainder operator, we have
X
e q v.
R(u, v) =
∆q u ∆
q
e q , the support of F ∆q u∆
fq v is included in
On one hand, by definition of ∆q and ∆
B(0, 3 · 2q+3 ). On the other hand, by virtue of Hölder inequality for functions, we have
e q vkLp ≤ 2qs1 k∆q ukLp1 2qs2 k∆
e q vkLp2 .
2q(s1 +s2 ) k∆q u ∆
Applying Hölder inequality for series, we thus have
e q vkLp
2q(s1 +s2 ) k∆q u ∆
`r
≤ CkukBps1 ,r kvkBps2 ,r .
1
1
2
2
As s1 + s2 > 0 has been assumed, lemma 1.3.4 yields the desired result.
Remark. By combining proposition 1.3.5.(v) with the above proposition, one can get other
results of continuity for the remainder. Moreover, condition (1.14) may be somewhat relaxed
(see exercise 1.10).
1.4.3
Results of continuity for the product
We do not aim at giving an exhaustive list of the mapping properties of (u, v) 7→ uv in Besov
spaces. As a matter of fact, memorizing such a list would be quite useless: it is actually
far wiser to appeal to results of continuity for the paraproduct and remainder, and to Bony’s
decomposition.
For example, by combining propositions 1.4.1 and 1.4.2, and the continuous embeddings
stated in proposition 1.3.5, one gets the following important results:
s ∩ L∞ is an algebra and we have
Proposition 1.4.3. Let s > 0 and 1 ≤ p, r ≤ ∞. Then Bp,r
s
s
s .
kuvkBp,r
. kukL∞ kvkBp,r
+ kvkL∞ kukBp,r
Proof: According to propositions 1.4.2 and 1.4.1, we have
s
kR(u, v)kBp,r
s
kTu vkBp,r
s
kTv ukBp,r
s kvkB 0
. kukBp,r
,
∞,∞
s ,
. kukL∞ kvkBp,r
s .
. kvkL∞ kukBp,r
0
One can easily check that L∞ ,→ B∞,∞
, hence applying Bony’s decomposition yields the
proposition.
24
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Proposition 1.4.4. Let 1 ≤ p1 , p2 , p3 , p4 , r ≤ ∞ and (s1 , s2 , s3 , s4 ) ∈ R4 be such that
s1 + s2 −
N
N
= s3 + s4 − ,
p1
p4
1
1
+
≤ 1,
p1 p2
Then the product is continuous from Bps11,∞ ∩ Bps22,r
kuvk
N
s1 +s2 − p
1
Bp2 ,r
2
s1 + s2 > 0
and
s1 +s2 − pN
1
to Bp2 ,r
s1 , s3 <
N
.
p1
and we have
. kukBps1 ,∞ kvkBps2 ,r + kvkBps3 ,∞ kukBps4 ,r .
1
2
3
4
Proof: By virtue of propositions 1.4.1 and 1.4.2 and embeddings, we have
kTu vk
N
s1 +s2 − p
1
. kuk
Bp2 ,r
kR(u, v)k
N
s1 +s2 − p
1
Bp2 ,r
kTv uk
N
s3 +s4 − p
3
. kuk
Bp4 ,r
kvkBps2 ,r . kukBps1 ,∞ kvkBps2 ,r ,
N
s1 − p
2
B∞,∞ 1
1
2
. kukBps1 ,∞ kvkBps2 ,r ,
N
s3 − p
B∞,∞ 3
1
2
kvkBps4 ,r . kukBps3 ,∞ kvkBps4 ,r .
4
3
4
Taking advantage of Bony’s decomposition completes the proof.
Proposition 1.4.5. For all φ ∈ S, 1 ≤ p, r ≤ ∞ and s ∈ R, the map Mφ : u → φu is
s .
continuous in Bp,r
Proof: The proof relies on Bony decomposition. Indeed, according to proposition 1.4.1 and
Besov embeddings, we have
s
kTφ ukBp,r
s
kTu φkBp,r
s ,
. kφkL∞ kukBp,r

s kuk ∞
s kukB s
. kφkBp,r
 kφkBp,r
L
p,r
.
s
kφk
kuk
.
kφk
kuk
N
N
N

Bp,r
s− p −1
p +1
p +1
Bp,r
B∞,∞
if s >
if s ≤
Bp,r
N
p,
N
p,
and, by virtue of proposition 1.4.2,
(
s
kR(u, φ)kBp,r
.
s
kφkL∞ kukBp,r
if s > 0,
kφkB∞,∞
1−s kukB s
p,r
if s ≤ 0,
which completes the proof.
1.4.4
A result of compactness in Besov spaces
One can now state a result of compactness for Besov spaces which will prove to be very useful
for solving nonlinear PDE’s.
Proposition 1.4.6. Let 1 ≤ p, r ≤ ∞, s ∈ R and ε > 0. For all φ ∈ Cc∞ , the map u 7→ φu is
s+ε to B s .
compact from Bp,r
p,r
As a preliminary step, we need to state the following result :
Lemma 1.4.7. Let a1 , · · · , aN be positive and δ ∈ (0, (mini ai )/4). There exists a constant C
s (TN ) supported in a cube of size
such that for all 1 ≤ p, r ≤ ∞ and s ∈ R, and u ∈ Bp,r
a
2πa1 − 2δ × · · · × 2πaN − 2δ, we have
X
per
per
C −1 kukBp,r
kBp,r
with
u
:=
u( · + α).
s (RN ) ≤ ku
s (TN ) ≤ CkukB s (RN )
a
p,r
α∈ZN
2πa
1.4. PARADIFFERENTIAL CALCULUS
25
Proof: With no loss of generality, one can assume that
Supp u ⊂ QN
a,δ := [δ, 2πa1 − δ] × · · · × [δ, 2πaN − δ].
Let θ be a smooth function supported in QN
a,δ/2 and equals to one on a neighborhood of
N
qN
q
Qa,δ . Denoting hq := 2 h(2 ·), we have for all q ∈ N, M ∈ N and x ∈ RN ,
∆q u(x) = hu, θhq (x − ·)i,
= (Id−∆)−M u, (Id−∆)M θhq (x − ·) .
By virtue of Besov embeddings, one can choose M so large as to satisfy
s
Bp,r
(RN ) ,→ v ∈ S 0 (RN ) | (Id−∆)−M v ∈ L∞ (RN ) .
By taking advantage of Leibniz formula, we thus get for some large enough C,
Z
X
|∂α θ(y)||∂β hq (x − y)| dy.
|∆q u(x)| ≤ CkukBp,r
s (RN )
|α|+|β|≤2M
RN
5
If one assumes that x 6∈ QN
a then one has |x − y| ≥ |x − πa| − π|a| + δ/2 whenever y
belongs to Supp θ. Therefore one has for all K ∈ N,
X Z
C
|x − y|K |∂α θ(y)||∂β hq (x − y)| dy
|∆q u(x)| ≤
s (RN )
kukBp,r
δ K
N
|x−πa|−π|a|+ 2
|α|+|β|≤2M R
and it is easy to conclude that there exists a constant CK such that
(1.15) |∆q u(x)| ≤ CK |x−πa| − π|a| +
δ −K (2M−K)q
2
kukBp,r
for all x ∈ RN \ QN
s (RN )
a .
2
By choosing K = 2M + 1 with M large enough, one can now easily conclude that
(1.16)
k∆q ukLp (RN \QN
≤ C2−q kukBp,r
s (RN )
a )
for some constant C depending only on δ and on a.
Next, we notice that, by virtue of proposition 1.2.2, we have
X
per
∀x ∈ RN , ∆per
(x) − ∆q u(x) =
∆q u(x + α).
q u
α∈ZN
2πa \{0}
Therefore, taking M large enough and using (1.15) with K = 2M + 1, one gathers that
for all x ∈ QN
a , we have
per
|∆per
(x) − ∆q u(x)| ≤ C2−q kukBp,r
s (RN ) ,
q u
whence
(1.17)
per
k∆per
− ∆q ukLp (QN
≤ C2−q kukBp,r
s (RN ) .
q u
a )
Note that if one replaces the function hq by the function 2qN F −1 χ(2q ·) in the above
computations then one gets
per per
−q
(1.18)
max kSq ukLp (RN \QN
,
kS
u
−S
uk
p
N
s (RN ) .
q
L (Qa ) ≤ C2 kukBp,r
q
a )
5
We take the max norm in RN to simplify the computations.
26
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Let q0 ∈ N be such that 8C2−q0 ≤ 1. From (1.16), (1.17) and (1.18), we get
kSqper
uper krLp (QN )
0
a
+
X
2
qs
r
per
k∆per
kLp (QN
q u
a )
1
r
q≥q0
1
X
kukBp,r
s (RN )
r r
r
qs
≤
.
− kSq0 ukLp (RN ) +
2 k∆q ukLp (RN )
2
q≥q0
One can easily show that
kSqper
uper krLp (QN )
0
a
+
X
2
qs
r
per
k∆per
kLp (QN
q u
a )
1
r
≈ kuper kBp,r
s (TN )
a
q≥q0
and that
kSq0 ukrLp (RN )
+
X
qs
2 k∆q ukLp (RN )
r
1
r
≈ kukBp,r
s (RN )
q≥q0
whence the desired result.
One can now prove proposition 1.4.6. According to proposition 1.4.5, we already know that
s+ε (RN ) in B s (RN ). We shall prove the compactness by decomposing T
Tφ : u 7→ φu maps Bp,r
φ
p,r
into a product of continuous maps, one of them being compact.
With no loss of generality, one can assume that φ is supported in some cube QN
a,δ which
satisfies the assumptions of proposition 1.4.7. Now, one can use the decomposition
T φ = J ◦ I ◦ Π ◦ Mφ
where
• J stands for the extension map by 0 outside QN
a from the subset of those distributions
N
N
over Ta whose restriction to Qa is supported in QN
a,δ , to the set of temperate distributions
N
N
over R supported in Qa,δ ,
s+ε (TN ) to B s (TN ),
• I is the canonical embedding from Bp,r
a
p,r
a
• Π is the map u 7→ uper introduced in proposition 1.2.2,
s+ε (RN ) with values in B s+ε (RN ).
• Mφ is the map u 7→ φu over Bp,r
p,r
According to proposition 1.4.5, the map Mφ is continuous. Besides, according to lemma 1.4.7,
s (TN ) whose
the map J (resp. Π ) is continuous from the subspace of those distributions of Bp,r
a
N , to B s (RN ) (resp. from the subspace of distributions
restriction to QN
is
supported
in
Q
a
p,r
a,δ
s+ε (RN ) supported in QN , to B s+ε (TN )).
of Bp,r
p,r
a
a,δ
We claim that I may be approximated by a sequence of operators with finite rank. This
will yield compactness.
Indeed, introduce the finite rank operator I n defined by
X
I n (v) :=
∆per
q v.
q≤n
Using proposition 1.2.1, we discover that for all q ∈ N, we have
X
n
per
∆per
∆per
q (I − I )(v) =
q ∆ v.
j>n
|j−q|≤1
1.4. PARADIFFERENTIAL CALCULUS
27
We thus have for some constant C independent of q,
n
2qs k∆per
≤ C2−nε 2q(s+ε) k∆per
,
q (I − I )(v)kLp (TN
q vkLp (TN
a )
a )
whence
−nε
.
kI − In )(v)kBp,r
kvkBp,r
s+ε
s (TN ) ≤ C2
(TN
a
a )
s+ε (TN ); B s (TN ) . The proof of proposition 1.4.6 is
This insures that In tends to I in L Bp,r
a
p,r
a
complete.
1.4.5
Results of continuity for the composition
Let us state the main result of this section:
Proposition 1.4.8. Let I be an open interval of R. Let s > 0 and σ be the smallest integer
s
such that σ ≥ s. Let F : I → R satisfy F (0) = 0 and F 0 ∈ W σ,∞ (I; R). Assume that v ∈ Bp,r
s
has values in J ⊂⊂ I . Then F (v) ∈ Bp,r and there exists a constant C depending only on s,
I, J, and N, and such that
σ
s
s .
kF (v)kBp,r
≤ C 1+kvkL∞ kF 0 kWσ,∞ (I) kvkBp,r
Proof: The proof is based on Meyer’s first linearization method (see e.g. [1], chapter 2).
Of course, one can change F for a function Fe ∈ Wσ+1,∞ (R) compactly supported in I
and such that Fe ≡ F on a neighborhood of J . So let us assume that F belongs to
Wσ+1,∞ (R) and has compact support in I.
The starting point of the proof is the following formal decomposition6
X
(1.19)
F (v) =
F (Sq0 +1 v) − F (Sq0 v).
q 0 ≥−1
According to first order Taylor’s formula, we have for all q 0 ≥ −1,
Z
F (S
q 0 +1
v) − F (S v) = m ∆ v
q0
q0
q0
with m :=
q0
1
F 0 (Sq0 v + τ ∆q0 v) dτ.
0
One can easily prove that the mp ’s are Meyer multipliers, namely
(1.20)
∀k ∈ {0, · · · , σ}, Dk mq0
0
L∞
≤ Ck 2q k (1 + kvkL∞ )k kF 0 kWk,∞ .
In particular, inequality (1.20) with k = 0 implies that
F (Sq0 +1 v) − F (Sq0 v)
0
Lp
≤ C2−q s sup 2qs k∆q vkLp .
q
Since s is positive, we conclude that (1.19) holds true in Lp .
s . For all q ≥ −1, we have
We now have to prove that F (v) belongs to Bp,r
X
∆q F (v) =
X
∆ q mq 0 ∆ q 0 v .
∆ q mq 0 ∆ q 0 v +
q 0 ≥q
−1≤q 0 ≤q−1
|
6
Remind that S−1 v ≡ 0.
{z
∆1q
}
|
{z
∆2q
}
28
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Taking advantage of Bernstein lemma, we get for all (q 0 , q) such that −1 ≤ q 0 ≤ q − 1,
∆ q mq 0 ∆ q 0 v
Lp
. 2−qσ Dσ ∆q mq0 ∆q0 v
Lp
,
whence, combining Leibniz formula and (1.20),
2qs ∆q mq0 ∆q0 v
0
Lp
0
. kF 0 kWσ,∞ (I) (1 + kvkL∞ )σ 2(q −q)(σ−s) 2q σ ∆q0 v
Lp
.
Since σ > s, convolution inequalities enable us to conclude that
X
2
qs
r
∆1q Lp
1
r
s
. kF 0 kWσ,∞ (I) (1 + kvkL∞ )σ kvkBp,r
q
and proposition 1.3.4 insures that
P
s .
∆1q belongs to Bp,r
Bounding the term pertaining to ∆2q is easy. Indeed, we have according to (1.20),
2qs ∆q (mq0 ∆q0 v)
0
Lp
0
. kF 0 kL∞ (I) 2(q−q )s 2q s ∆q0 v
Lp
,
so that, since s > 0,
X
2
qs
r
∆2q Lp
1
r
s .
. kF 0 kL∞ (I) kvkBp,r
q
Applying once again proposition 1.3.4 completes the proof.
Finally, combining propositions 1.4.3 and 1.4.8 with the following equality:
Z
F (v) − F (u) = (v − u)
1
F 0 (u + τ (v − u)) dτ,
0
we readily get the following
Corollary 1.4.9. Let I be an open interval of R and F : I → R. Let s > 0 and σ be the
smallest integer such that σ ≥ s. Assume that F 0 (0) = 0 and that F 00 belongs to Wσ,∞ (I; R).
s have values in J ⊂⊂ I. There exists a constant C = C
Let u, v ∈ Bp,r
s,I,J,N such that
σ
s
kF (v) − F (u)kBp,r
≤ C 1+kvkL∞ kF 00 kWσ,∞ (I)
s
s
× kv − ukBp,r
sup ku + τ (v − u)kL∞ + kv − ukL∞ sup ku + τ (v − u)kBp,r
.
τ ∈[0,1]
1.5
τ ∈[0,1]
Calculus in homogeneous functional spaces
Nonhomogeneous functional spaces are not the most natural spaces for studying mathematical
problems which have a property of invariance by dilation. For instance, it is well known that
the Sobolev embedding H 1 (R3 ) ,→ L6 (R3 ) involves only the L2 norm of the gradient and not
the whole H 1 norm. We shall also see in the next chapters that many interesting PDE’s have
some properties of scaling invariance.
This is a good motivation for introducing a homogeneous Littlewood-Paley decomposition
where the low frequencies are treated exactly as the high frequencies, and to define homogeneous
functional spaces.
1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES
1.5.1
29
Homogeneous Littlewood-Paley decomposition
Let (χ, ϕ) be as in section 1.2.2. The homogeneous dyadic blocks are defined by
˙ q u := ϕ(2−q D)u
∆
for all
q ∈ Z.
We also introduce the following low frequency cut-off:
Ṡq u := χ(2−q D)
for all
q ∈ Z.
The above definition deserves two important preliminary remarks:
P
˙ q u = u modulo a polynomial only.
• For u ∈ S 0 , we have q∈Z ∆
• In contrast with the nonhomogeneous case, we do not have Ṡq u =
P
˙
p≤q−1 ∆p u.
In light of the first remark, working with distributions modulo polynomials seems to be the most
appropriate choice. As a matter of fact, this is the viewpoint of most authors of textbooks on
abstract functional analysis (see e.g. [38] or [36]).
Since in the next chapters, we aim at applying homogeneous Littlewood-Paley decomposition
for solving nonlinear PDE’s however, it is not suitable to work with distributions defined modulo
polynomials. This motivates the following definition (after J.-Y. Chemin in [12]):
Definition. We denote by Sh0 the space of temperate distributions u such that
lim Ṡj u = 0
j→−∞
in S 0 .
Remarks. (i) A polynomial u does not belong to Sh0 unless it is identically 0. Indeed, if u is
a polynomial then we have Ṡj u = u for all j in Z.
(ii) It is obvious that Sh0 is the space of temperate distributions u which satisfy
X
˙ j u in S 0 .
u=
∆
j
(iii) The space Sh0 is not a closed subspace of S 0 for the topology of weak convergence. Indeed,
consider a sequence (fn )n∈N with fn (x) = f (x/n) and f ∈ S such that f (0) = 1. Then
(fn )n∈N tends weakly to the constant function 1 which does not belong to Sh0 .
Examples. (i) If a temperate distribution u is such that its Fourier transform u
b is locally
0
integrable near 0, then u belongs to Sh .
(ii) If u is a temperate distribution such that for some function θ in Cc∞ (RN ) with value 1
near the origin, we have θ(D)u in Lp for some p ∈ [1, +∞[, then u belongs to Sh0 . In
s is included in S 0 .
particular, when p if finite, any nonhomogeneous Besov space Bp,r
h
1.5.2
Homogeneous Besov spaces
Definition 1.5.1. Let u be a temperate distribution, s a real number, and 1 ≤ p, r ≤ ∞. Then
we set
X
1
r
rqs ˙
r
2 k∆q ukLp
kukḂ s :=
p,r
q∈Z
with the usual change if r = ∞.
For the semi-norms we have defined, we can prove the following inequalities
30
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Theorem 1.5.2. A constant C exists such that for all s ∈ R,
r1 ≤ r2 ⇒ kukḂ s
≤ kukḂ s ,
p1 ≤ p2 ⇒ kuk
( p11 − p12 ) ≤ CkukḂps1 ,r .
p,r2
p,r1
s−N
Ḃp2 ,r
Moreover we have the following interpolation inequalities for any θ in ]0, 1[ and s < se:
kukḂ θs+(1−θ)es ≤ kukθḂ s kuk1−θ
,
Ḃ se
p,r
kukḂ θs+(1−θ)es
p,1
p,r
p,r
C
kukθḂ s kuk1−θ
.
≤
s
e
Ḃp,∞
p,∞
θ(1 − θ)(e
s − s)
The proof is the same as in the nonhomogeneous framework, and thus omitted. Finally, the
following logarithmic interpolation inequalities are available:
Proposition 1.5.3. There exists a constant C such that for all s ∈ R, ε > 0 and 1 ≤ p ≤ ∞,
we have
kuk s−ε + kuk s+ε 1+ε
Ḃp,∞
Ḃp,∞
≤ C
kukḂ s
(1.21)
1 + log
,
kukḂ s
p,∞
p,1
ε
kukḂ s
p,∞
(1.22)
kukḂ s
p,1
≤ C
1+ε ε
1 + kukḂ s
p,∞
log e + kukḂp,∞
s−ε + kuk s+ε
Ḃp,∞
.
Proof: The proof is almost the same as in the nonhomogeneous framework. We write for
some positive integer Q to be fixed hereafter:
X
X
X
2qs k∆q ukLp +
2qs k∆q ukLp +
2qs k∆q ukLp ,
kukḂ s =
p,1
q≤−Q
q≥Q
|q|<Q
whence, from elementary computations,
2−Qε
kuk
s−ε + kuk s+ε .
Ḃ
Ḃ
p,∞
p,∞
p,∞
p,1
1 − 2−ε
kuk s−ε + kuk s+ε 1
Ḃp,∞
Ḃp,∞
yields (1.21).
Choosing for Q the closest positive integer to log2
ε
kukḂ s
kukḂ s ≤ (2Q − 1)kukḂ s
+
p,∞
The second inequality may be easily deduced from (1.21).
Homogeneous Besov spaces have some invariance properties by dilation. More precisely, if for
any temperate distribution u and λ > 0, we introduce the temperate distribution uλ defined
for all x ∈ RN by uλ (x) := u(λx), then we have
Proposition 1.5.4. If kukḂ s
p,r
is finite, so is kuλ kḂ s
p,r
kuλ kḂ s ≈ λ
s− N
p
and we have
kukḂ s .
p,r
p,r
Besides, equality is true if λ = 2m for some m ∈ Z.
Proof: Let α be a positive real. We have
ϕ(α
−1
D)uλ (x) = α
N
Z
h(α(x − y))u(λy) dy.
1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES
31
By the change of variables z = λy , we get that
Z
ϕ(α−1 D)uλ (x) = αN λ−N h(αx − αλ−1 z)u(z) dz,
= ϕ(λα−1 D)u (αx).
For q ∈ Z, let us denote vq := ϕ(2[log2 λ]−log2 λ−q D)uλ . Taking α = 2q−[log2 λ] λ in the above
equality, we get
s− N
˙ q−[log λ] ukLp
2qs kvq kLp ≈ λ p 2(q−[log2 λ])s k∆
2
with an equality if log2 λ is an integer.
Thus we deduce that
X
qs
2 kvq kLp
1
r
≈λ
s− N
p
kukḂ s .
p,r
q
q
Since Supp vbq ⊂ C(0, 43 2q , 16
3 2 ), we have
˙ q uλ =
∆
X
˙ p uq .
∆
|p−q|≤2
Now, straightforward computations yield the desired result.
Next, we notice that k · kḂ s
is actually only a semi-norm in the sense that, if u is a polynomial
˙ q u = 0, (because the support of u
then, for any integer q we have ∆
b is the origin), and so
s
kukḂ s = 0. Therefore, if we define the homogeneous Besov space Ḃp,r as the set of temperate
p,r
distributions u such that kukḂ s is finite, we may run into troubles later when studying non
p,r
linear problems because we will not be able to tell a polynomial from a null function !
In the present lecture notes, we adopt the following definition:
p,r
s
Definition. Let s be a real number and (p, r) be in [1, ∞]2 . The space Ḃp,r
is the set of
0
distributions u in Sh such that kukḂ s is finite.
p,r
In the homogeneous framework, the results corresponding to proposition 1.3.4 read:
Proposition 1.5.5. Let s ∈ R and 1 ≤ p, r ≤ ∞ satisfy
(1.23)
s<
N
,
p
or
s=
N
p
and
r = 1.
(i) Let (uq )q∈Z be a sequence of functions such that
X
qs
2 kuq kLp
r
1
r
< ∞.
q
If Supp u
bq ⊂ C(0, 2q R1 , 2q R2 ) for some 0 < R1 < R2 then u :=
and there exists a constant C such that
X
1
r r
1+|s|
qs
(1.24)
kukḂ s ≤ C
2 kuq kLp
.
p,r
q
Let (uq )q∈Z be a sequence of functions such that
X
q
qs
2 kuq kLp
r
1
r
< ∞.
P
q∈Z uq
s
belongs to Ḃp,r
32
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
If Supp u
bq ⊂ B(0, 2q R) for some positive R and if in addition s is positive then u :=
P
s
q∈Z uq belongs to Ḃp,r and there exists a constant C such that
kukḂ s
(1.25)
p,r
1
r r
C 1+s X qs
2 kuq kLp
.
≤
s
q
Proof: Let us prove (i).
P
On one hand, using Bernstein lemma, it is easy to see that the series q≤0 uq is convergent
P
0
in L∞ and that u− :=
q≤0 uq belongs to Sh . On the other hand, since kuq kL∞ ≤
P
q( N −s)
0
, lemma 1.3.3 insures that the series
C2 p
q>0 uq is convergent in S . Besides, the
P
Fourier transform of u+ := q>0 uq is supported away from the origin, hence u+ belongs
to Sh0 . Since u = u− + u+ , one can conclude that u belongs to Sh0 . Now, the proof of
(1.24) may be done exactly as in lemma 1.3.4.
Proving (ii ) is similar. Indeed, we have for any q ∈ Z,
kuq kLp ≤ C2−qs
q( N
−s)
p
kuq kL∞ ≤ C2
and
.
P
P
∞ and belongs to S 0 whereas
p
Hence
q≤0 uq converges in L
q>0 uq converges in L
h
0
(and also belongs to Sh since its Fourier transform is bounded away from 0). Next, the
proof of (1.25) goes along the lines of lemma 1.3.4.
As an important corollary of proposition 1.5.5, we get that, under condition (1.23), the definition
s does not depend on ϕ.
of Ḃp,r
The following proposition, the proof of which is left to the reader, describes the relations
between homogeneous and nonhomogeneous spaces.
s ,→ B s .
Proposition 1.5.6. Let s be a negative number (or s = 0 and r = 1). Then Ḃp,r
p,r
s ,
Besides, if s < 0, a constant C (independent of s) exists so that, for any u belonging to Ḃp,r
we have
C
s
kukBp,r
≤
kukḂ s .
p,r
−s
s ,→ Ḃ s
Let s be a positive number (or s = 0 and r = ∞). Then Bp,r
p,r when p is finite, and
s
0
s
B∞,r ∩ Sh is a subset of Ḃ∞,r . If s > 0, a constant C exists (independent of s) so that, for any
s , we have
u belonging to Bp,r
C
s .
kukḂ s ≤ kukBp,r
p,r
s
Let us notice that there is no monotonicity property with respect to s for homogeneous
spaces. The reason why is that homogeneous Besov spaces carry on informations about both
low and high frequencies.
In homogeneous spaces, the counterpart of proposition 1.3.8 reads:
Proposition 1.5.7. There exists a constant C which satisfies the following properties. Let
s if and only if
s < 0, (p, r) ∈ [1, ∞]2 and u a distribution in Sh0 . Then u belongs to Ḃp,r
(2qs kṠq ukLp )q∈Z ∈ `r (Z).
Moreover, we have
1
kukḂ s ≤
p,r
2
X
q
qs
k(2 kṠq ukLp )q
r
1
r
1
≤C 1+
kukḂ s .
p,r
|s|
1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES
33
Proof: The proof goes along the lines of proposition 1.3.8.
Proposition 1.5.8. Let f be a smooth function on RN \ {0} which is homogeneous of degree
m. Let 1 ≤ p, r ≤ ∞. Assume that
s−m<
N
,
p
or
r=1
s−m≤
and
N
·
p
s to Ḃ s−m .
Then f (D) is continuous from Ḃp,r
p,r
Proof: Assume that r > 1. Introduce a smooth function ϕ
e supported in a shell and such
that ϕ
e ≡ 1 on Supp ϕ. As f is homogeneous of degree m, we have
˙ q f (D)u = 2qm [ϕf
˙ q u.
∆
e ](2−q D)∆
Since F −1 (ϕf
e ) belongs to S, we readily get
˙ q f (D)u
∆
Lp
˙ qu
≤ C2qm ∆
whence
X
˙ q f (D)ukLp
2q(s−m) k∆
r
Lp
,
1
r
. kukḂ s .
p,r
q
Now, we have
Ṡq f (D)u =
X
˙ q0 u,
Ṡq f (D)∆
q 0 ≤q
whence according to Bernstein lemma,
X q0 ( N +m−s) 0
˙ q0 u
Ṡq f (D)u ∞ .
2 p
2q s ∆
L
q 0 ≤q
Lp
.2
+m−s)
q( N
p
kukḂ s .
p,∞
Since N/p + m − s > 0, it is now clear that Ṡq f (D)u tends to 0 when q goes to −∞.
s−m . The proof in the case r = 1 is left to the reader.
Hence f (D)u belongs to Ḃp,r
s .
Let us now focus on the topological properties of the spaces Ḃp,r
s ,k · k
Proposition 1.5.9. For all s ∈ R and 1 ≤ p, r ≤ ∞, the couple (Ḃp,r
Ḃ s ) is a normed
p,r
s is densely embedded in Ḃ s .
space. If besides r is finite then Cc∞ ∩ Ḃp,r
p,r
Proof: It is obvious that k · kḂ s
p,r
is a semi-norm. Let us assume that kukḂ s = 0 for some
p,r
Sh0 .
u in
This implies that Supp u
b ⊂ {0} and thus that for any j ∈ Z we have Ṡj u = u.
0
As u belongs to Sh , we must have lim Ṡj u = 0 so that we can conclude that u = 0.
j→−∞
s , it is obvious that the sequence of general term
Now, if r is finite and u ∈ Ḃp,r
X
∆q u
|q|≤n
s and tends to u in Ḃ s . Arguing like in 1.3.5.(ii), it is then easy to
belongs to C ∞ ∩ Ḃp,r
p,r
∞
s
s .
exhibit a sequence of functions of Cc ∩ Ḃp,r which tends to u in Ḃp,r
Theorem 1.5.10. If s < Np or s =
is continuously embedded in S 0 .
N
p
s ,k·k
and r = 1 then (Ḃp,r
Ḃ s ) is a Banach space which
p,r
34
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
s
Proof: Let us first prove that Ḃp,r
is continuously embedded in S 0 . The case r = 1 and
P ˙
P ˙
s = N/p is easy because the series
∆j u is convergent in L∞ . As u = j ∆
j u, this
∞
implies that u belongs to L . Besides, we have
N
p
0
,→ Ḃ∞,1
,→ L∞ ,→ S 0 .
Ḃp,1
(1.26)
s− N
p
s ,→ Ḃ
Let us now assume that s < N/p. Using that Ḃp,r
∞,∞ and arguing like for proving
(1.10), one can find a large integer M such that for all nonnegative j, we have
˙ j u, φi| ≤ 2−j kuk
|h∆
s− N
p
Ḃ∞,∞
kφkM,S .
For negative j , one can write that for large enough M , we have
˙ j u, φi| . 2j
|h∆
“
j
“
. 2
N
p
N
p
−s
−s
”
kuk
s− N
p
Ḃ∞,∞
kφkL1
”
kukḂ s kφkM,S .
p,r
Because u belongs to Sh0 , we have hu, φi =
˙
P
j h∆j u, φi.
Therefore, for large enough M ,
|hu, φi| ≤ Cs kukḂ s kφkM,S
(1.27)
p,r
s ,→ S 0 .
and we can conclude that Ḃp,r
We still have to prove that for all triplet (s, p, r) satisfying the hypothesis of the theorem,
s
s .
the set Ḃp,r
is a Banach space. So let us consider a Cauchy sequence (un )n∈N in Ḃp,r
Using (1.26) or (1.27), this implies that a temperate distribution u exists such that the
sequence (un )n∈N converges to u in S 0 . We now have to state that u belongs to Sh0 . Let
us first assume that s < N/p. Since un belongs to Sh0 , we have, thanks to (1.27),
∀j ∈ Z , ∀n ∈ N , |hṠj un , φi| . 2
j
“
N
p
−s
”
kun kḂ s kφkM,S .
p,r
As the sequence (un )n∈N tends to u in S 0 , we have
j
∀j ∈ Z , |hṠj u, φi| ≤ Cs 2
“
N
p
−s
”
kφkM,S sup kun kḂ s .
n
p,r
Thus u belongs to Sh0 .
N
p
The case when u belongs to Ḃp,1
is a little bit different. Let ε > 0. As (un )n∈N is a
N
p
0 , there exists an integer n such that
Cauchy sequence in Ḃp,1
,→ Ḃ∞,1
0
∀j ∈ Z , ∀n ≥ n0 ,
X
k≤j
Let us choose j0 small enough so that
X
k≤j0
˙ k un kL∞ ≤
k∆
ε X ˙
+
k∆k un0 kL∞ .
2
k≤j
˙ k un kL∞ ≤ ε ·
k∆
0
2
As un belongs to Sh0 , we have
∀j ≤ j0 , ∀n ≥ n0 , kṠj un kL∞ ≤ ε.
1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES
35
As sequence (un )n∈N tends to u in L∞ , this implies that
∀j ≤ j0 , kṠj ukL∞ ≤ ε.
This proves that u belongs to Sh0 . Next, arguing like in the nonhomogeneous case completes the proof.
s is no longer a
Remark. It turns out that when s > N/p (or s = N/p and r > 1) the space Ḃp,r
Banach space. In the one dimensional case for instance, it is can be easily seen that the sequence
(fn )n∈N defined by
fbn (ξ) =
χ(ξ)
ξ log |ξ|
if |ξ| ≥ 2−n ,
and 0
1
elsewhere,
1
2
2
is a Cauchy sequence in Ḃ2,∞
but cannot have a limit in Ḃ2,∞
since the function ξ 7→ ξ χ(ξ)
log |ξ|
does not belong to S 0 !
This defect of convergence due to low frequencies is typical to homogeneous functional spaces
with high regularity index. It is sometimes called infrared divergence.
There is a way to modify the definition of homogeneous Besov spaces so as to get a Banach
space regardless of the regularity index. This is called realizing homogeneous Besov spaces. It
turns out that realizations coincide with definition 1.5.2 when s < N/p or s = N/p and r = 1.
In the other cases however, realizations are not functional spaces but spaces defined up to a
polynomial whose degree depends on s − N/p and on r (see e.g [5] or [32]). It goes without
saying that solving PDE’s in such spaces may be quite unpleasant.
1.5.3
Paradifferential calculus in homogeneous spaces
We designate homogeneous paraproduct of v by u and denote by Ṫu v the bilinear operator:
X
˙ q v.
Ṡq−1 u ∆
Ṫu v :=
q
We designate homogeneous remainder of u and v and denote by Ṙ(u, v) the bilinear operator:
X
˙ pu ∆
˙ q v.
Ṙ(u, v) =
∆
|p−q|≤1
It is clear that, formally, we have the following homogeneous Bony decomposition:
(1.28)
uv = Ṫu v + Ṫv u + Ṙ(u, v).
The properties of continuity of homogeneous paraproduct and remainder on homogeneous Besov
spaces are described in the following propositions.
Proposition 1.5.11. There exists a constant C such that for any couple of real numbers (s, σ)
with σ positive and for any (p, r, r1 , r2 ) in [1, +∞]4 with 1/r = 1/r1 + 1/r2 , we have
kṪ kL(L∞ ×Ḃ s
s
p,r ;Ḃp,r )
≤ C |s|+1
if condition (1.23) is satisfied, and
kṪ kL(Ḃ∞,r
−σ
s−σ
s
×Ḃp,r
;Ḃp,r
)
1
2
if s − σ < N/p, or s − σ = N/p and r = 1.
≤
C |s−σ|+1
·
σ
36
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
˙ q v. Arguing like in proposition 1.4.1, one can easily prove that
Proof: Let wq := Ṡq−1 u ∆
X
qs
2 kwq kLp
r
1
r
. kukL∞ kvkḂ s
p,r
and
X
2
q(s−σ)
kwq kLp
r
1
r
.
. kukḂ∞,r
−σ kvk s
Ḃ
1
q
q
p,r2
Since the sequence (Fwq )q∈Z is supported in dyadic shells, applying proposition 1.5.5
completes the proof.
Proposition 1.5.12. There exists a constant C which satisfies the following inequalities. For
any (s1 , s2 ), any 1 ≤ p1 , p2 , p ≤ ∞ and any 1 ≤ r, r1 , r2 ≤ ∞ such that
1
1
1
≤
+
≤1
p
p1 p2
s1 + s2 > 0,
and
1
1
1
≤
+
≤ 1,
r
r1 r2
we have
kṘkL(Ḃps1 ,r
1
s
σ12
×Ḃp22 ,r2 ;Ḃp,r
)
1
≤
C |s1 +s2 |+1
s1 + s2
with
σ12 := s1 + s2 − N
1
1
1
+
−
p1 p 2 p
,
provided that σ12 < N/p, or σ12 = N/p and r = 1.
fq v and arguing like in proposition 1.4.2, we easily get
Proof: Denoting wq := ∆q u ∆
X
2
qσ12
kwq kLp
r
1
r
. kukḂps1 ,r kvkḂps2 ,r .
1
q
1
2
2
Since the sequence the sequence (Fwq )q∈Z is supported in dyadic balls, applying proposition 1.5.5 completes the proof.
Proposition 1.5.13. Let I be an open interval of R. Let s > 0 and σ be the smallest integer
s
such that σ ≥ s. Let F : I → R satisfy F (0) = 0 and F 0 ∈ Wσ,∞ (I; R). Assume that v ∈ Ḃp,r
s
has values in J ⊂⊂ I. Assume that condition (1.23) is satisfied. Then F (v) ∈ Ḃp,r
and there
exists a constant C = Cs,I,J,N such that
σ
kF (v)kḂ s ≤ C 1+kvkL∞ kF 0 kWσ,∞ (I) kvkḂ s .
p,r
p,r
Proof: Arguing like in the proof of proposition 1.4.8 and using Bernstein lemma, we get for
all j ∈ Z,
F (Ṡj+1 v) − F (Ṡj v) p ≤ C2−js 2js k∆j vkLp ,
L
j( N −s)
F (Ṡj+1 v) − F (Ṡj v) ∞ ≤ C2 p
2js k∆j vkLp .
L
Since s is positive, F (0) = 0 and condition (1.23) is satisfied, this insures that the series
X
F (Ṡj+1 v) − F (Ṡj v)
j∈Z
converges to F (v) in Lp + L∞ , thus in S 0 .
˙ q F (v) into
Next, for all q ∈ Z, we split ∆
X
X
˙ q F (v) =
˙ j v) +
˙ q (mj ∆
˙ j v).
∆
∆q (mj ∆
∆
j<q
j≥q
Following the lines of the proof of proposition 1.4.8 and applying proposition 1.5.5 leads
to the desired result.
1.6. EXERCISES
1.6
37
Exercises
Exercise 1.1. Prove that for any α > 1, there exists two smooth functions ϕ and χ such
that ϕ is supported in the shell {ξ ∈ RN | α−1 ≤ |ξ| ≤ 2α} and χ is supported in the ball
{ξ ∈ RN | |ξ| ≤ α}, and
X
∀ξ ∈ RN , χ(ξ) +
ϕ(2−q ξ) = 1.
q∈N
Exercise 1.2. Prove that for all temperate distribution u, the equality u =
true in S 0 (RN ).
P
q∈Z ∆q u
holds
Exercise 1.3. Prove proposition 1.2.1.
Exercise 1.4. Prove inequality (1.5).
Exercise 1.5. For q ∈ Z, denote ∆0q u := 12q ≤|ξ|≤2q+1 (D)u. Prove that the inequality
∆0q u
Lp
≤ C kukLp
for some constant C independent of q is false in the case p 6= 2.
Hint: Try with the function u = χ.
0 with the Lebesgue space Lp .
Exercise 1.6. Compare the Besov space Bp,r
s
s
s
Exercise 1.7. Let Bp,∞
be the completion of Cc∞ for the k · kBp,∞
norm. Prove that u ∈ Bp,∞
if and only if limq→+∞ 2qs k∆q ukLp = 0.
r−1 (RN ) is the set of temperate distributions u
Exercise 1.8. Let r ∈ (0, 1). Prove that B∞,∞
such that there exist N + 1 functions u0 , · · · , uN of C r verifying
u = u0 +
N
X
∂j uj .
j=1
Exercise 1.9. Let C?1 be the Zygmund space of bounded functions u such that
∀(x, y) ∈ RN × RN , |u(x + y) + u(x − y) − 2u(x)| ≤ C|y|
for some constant C.
1
Prove that C?1 = B∞,∞
.
Exercise 1.10. Let 1 ≤ p, p1 , p2 , r ≤ ∞ and s ∈ R. Let r0 := r/(r − 1). Assume that
1/p ≤ 1/p1 + 1/p2 .
−N
Prove that the remainder R maps Bps1 ,r × Bp−s
0 in Bp,∞
2 ,r
1
− p1 − p1
p
1
2
.
Exercise 1.11. Prove (1.20).
Exercise 1.12. Let I be an open interval of R and F : I → R. Assume that F (0) = 0 and
that F 0 is bounded.
0 with values in J ⊂⊂ I , the function F (v) belongs to B 0 and
Prove that for all v ∈ Bp,1
p,1
that
kF (v)kBp,1
. kF 0 kL∞ (I) kvkBp,1
0
0 .
Exercise 1.13. Prove that u belongs to Sh0 if and only if for any θ in Cc∞ (RN ) with value 1
near the origin, we have lim θ(λD)u = 0 in S 0 .
λ→∞
38
CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Exercise 1.14. Prove proposition 1.5.6.
Exercise 1.15. We say that a temperate distribution u tends weakly to 0 at infinity if u(λ ·)
tends to 0 in S 0 when λ goes to infinity.
Prove that u ∈ S 0 belongs to Sh0 if and only if u tends weakly to 0 at infinity.
Exercise 1.16. Let s be in ]0, N [. Prove that for any p in [1, ∞], we have
N
−s
1
p
∈
Ḃ
p,∞ .
s
|·|
s when s > N/p or s = N/p
Exercise 1.17. Find divergent Cauchy sequences for the space Ḃp,r
and r > 1.
Exercise 1.18. Let 1 ≤ p1 , r1 , p2 , r2 ≤ ∞ and (s1 , s2 ) ∈ R2 . Assume that
s1 <
N
p1
or s1 =
N
and r1 = 1.
p1
Prove that Lp1 ∩ Ḃps22,r2 and Ḃps11,r1 ∩ Ḃps22,r2 are Banach spaces.
Exercise 1.19. Let 1 ≤ p, r ≤ +∞, s ∈ R and ε > 0. Let φ ∈ Cc∞ (RN ).
s to Ḃ s−ε + Ḃ s .
Prove that the map u 7→ φu is compact from Ḃp,r
p,r
p,r
Exercise 1.20. Let s be a positive real number and (p, r) ∈ [1, ∞]2 . Prove the existence of a
constant C such that for all u in Sh0 , we have
C −1 kukḂp,r
kts et∆ ukLp
−2s ≤
Lr (R+ , dt
)
t
≤ CkukḂp,r
−2s .
Hint: use that for any positive s and c, we have
X
2j
sup
ts 22js e−ct2 < ∞.
t>0
j∈Z
Exercise 1.21. Let s be in ]0, 1[ and (p, r) ∈ [1, ∞]2 . Prove that there exists a constant C
such that for any u in Sh0 , we have
C −1 kukḂ s ≤
p,r
Exercise 1.22.
kτ−z u − ukLp
|z|s
Lr (RN ;
dz
|z|N
)
≤ CkukḂ s
p,r
1) Let (s1 , s2 ), (p1 , p2 , p) and (r1 , r2 ) be such that
1
1
1
1
1
≤
+
≤ 1 and
+
= 1.
p
p1 p2
r1 r2
:= s1 + s2 − N p11 + p12 − p1 satisfies σ12 < N/p, or σ12 = N/p and
s1 + s2 ≥ 0,
Assume that σ12
r = 1.
σ12 and that
Prove that the remainder is continuous from Ḃps11,r1 × Ḃps22,r2 to Ḃp,∞
kṘkL(Ḃps1 ,r
1
1
s
σ
12 )
×Ḃp22 ,r2 ;Ḃp,∞
≤ C |s1 +s2 |+1 .
2) Adapt exercise 1.12 to the homogeneous framework.
Chapter 2
The heat equation
In this chapter, we state estimates in Besov spaces for the heat equation. Such estimates are
fundamental for solving certain nonlinear PDE’s of parabolic type. As an example, we show
that incompressible Navier-Stokes equations are locally well-posed in Besov spaces with critical
index of regularity.
2.1
Generalities
The basic heat equation reads
(H)
∂t u − µ∆u = f,
u|t=0 = u0 .
Above, the external source term f = f (t, x) and the initial data u0 = u0 (x) are given. The
diffusion µ is a positive constant. We restrict ourselves to the evolution for positive times t,
and, for the sake of simplicity, we always assume that x belongs to the whole space RN . Similar
result would hold in the torus TN
a , though.
Giving an exhaustive list of properties of the heat equation is not our goal here. We however
have to recall a few important facts for (H) that will be needed for stating estimates in Besov
spaces.
Let u0 ∈ S(RN ) and f ∈ C(R+ ; S(RN )). Applying the partial Fourier transform with respect
to the space variable (still denoted by b ), we get the following linear ordinary differential
equation for all ξ ∈ RN :
(H)
∂t u
b(t, ξ) + µ|ξ|2 u
b(t, ξ) = fb(t, ξ),
u
b|t=0 (ξ) = u
b0 (ξ),
whence
(2.1)
−µ|ξ|2 t
u
b(t, ξ) = e
Z
u
b0 (ξ) +
t
e−µ|ξ|
2 (t−τ )
fb(τ, ξ) dτ.
0
Performing the inverse Fourier transform, we end up with the following well known representation
formula for all t ∈ R+ and x ∈ RN :
Z
Z
Z t
|x−y|2
|x−y|2
1
1
− 4µ(t−τ )
− 4µt
(2.2) u(t, x) =
e
u
(y)
dy
+
e
f
(τ,
y)
dy
dτ.
0
N
N
0 (4πµ(t−τ )) 2
RN
(4πµt) 2 RN
By arguing by duality, formulae (2.1) and (2.2) may be extended to temperate distributions.
39
40
CHAPTER 2. THE HEAT EQUATION
Introducing the heat semi-group es∆
rewritten in a more concise way:
µt∆
(2.3)
u(t) = e
s≥0
for the Laplacian operator, equality (2.2) may be
Z
t
u0 +
eµ(t−τ )∆ f (τ ) dτ.
0
Although formulae (2.1) and (2.2) are explicit, they are not so convenient for getting a priori
estimates in functional spaces. In fact, even for stating the basic energy equality, it is far more
efficient to multiply (H) by u, integrate over RN and perform an integration by parts in the
term with the Laplacian. At least formally, we end up with
Z
1d
kuk2L2 + µ kDuk2L2 = f u dx,
2 dt
thus, integrating over the time interval [0, T ], we get
Z T
Z
2
2
2
ku(T )kL2 + 2µ
kDu(t)kL2 dt = ku0 kL2 + 2
0
0
TZ
f (t, x)u(t, x) dt dx.
RN
We notice that starting from u0 ∈ L2 and f ∈ L2 (0, T ; Ḣ −1 ), the above equality provides an
estimate for Du in L2 (0, T × RN ). In other words, it gives a gain of one derivative for u with
respect to u0 and of two derivatives with respect to f. One can wonder if it is possible to gain
more than one derivative with respect to u0 by considering Lρ norms in time (ρ < 2) with
values in Sobolev spaces.
In the next section, we shall see that very simple arguments based on Littlewood-Paley
decomposition enable us to gain two derivatives when taking a L1 norm in time. Besides, the
method we are going to present apply indistinctly to the Lp framework for all p ∈ [1, +∞].
2.2
A priori estimates in Besov spaces for the heat equation
The fundamental idea is to localize the heat equation through a Littlewood-Paley decomposition.
˙ q u0
It is then easy to prove Lρ (0, T ; Lp ) estimates for each dyadic block in term of norms of ∆
˙ q f. If one assumes that u0 and f belong to some Besov spaces, performing a (weighted)
and ∆
r
` summation is the most natural next step. In doing so however, one does not obtain an
s ) since the time integration has been performed before
estimate in a space of type Lρ (0, T ; Ḃp,r
the summation.
This leads to the definition of the following spaces first introduced by J.-Y. Chemin and N.
Lerner in [13] then extended in [9].
Definition. For T > 0, s ∈ R, 1 ≤ r, ρ ≤ ∞, we set (with the usual convention if r = ∞):
X
1
r r
qs ˙
.
kukLeρ (Ḃ s ) :=
2 k∆q ukLρ (Lp )
T
p,r
T
q
s ) as the set of temperate distributions u over (0, T ) × RN
e ρ (Ḃp,r
We then define the space L
T
0
such that lim Ṡq u = 0 in S (0, T × RN ) and kukLeρ (Ḃ s ) < ∞.
q→−∞
T
p,r
In a similar way, we set
kukLeρ (Ḃ s
p,r )
:=
X
˙ q ukLρ (R+ ;Lp )
2 k∆
qs
r
1
r
,
q
s ) as the set of temperate distributions u over R+ × RN such that
e ρ (Ḃp,r
and define the space L
lim Ṡq u = 0 in S 0 (R+ × RN ) and kukLeρ (Ḃ s ) < ∞.
q→−∞
p,r
2.2. A PRIORI ESTIMATES IN BESOV SPACES FOR THE HEAT EQUATION
Remark 2.2.1.
(i) According to Minkowski inequality, we have
kukLeρ (Ḃ s
T
41
p,r )
≤ kukLρ (Ḃ s
p,r )
T
r ≥ ρ,
if
kukLeρ (Ḃ s
T
p,r )
≥ kukLρ (Ḃ s
T
if
p,r )
r ≤ ρ.
(ii) All the properties of continuity for the product, composition, remainder and paraproduct
s ). The general principle
e ρ (Ḃp,r
stated in chapter 1 may be easily generalized to the spaces L
T
is that the time exponent ρ behaves according to Hölder inequality.
For instance, we have the following tame estimate :
kuvkLeρ (Ḃ s
T
p,r )
. kukLρ1 (L∞ ) kvkLeρ2 (Ḃ s
T
p,r )
T
+ kvkLρ3 (L∞ ) kukLeρ4 (Ḃ s
T
T
p,r )
whenever s > 0, 1 ≤ p ≤ ∞, 1 ≤ ρ, ρ1 , ρ2 , ρ3 , ρ4 ≤ ∞ and
1
1
1
1
1
=
+
=
+ ·
ρ
ρ1 ρ2
ρ 3 ρ4
Remark. Of course similar definitions may be given in the nonhomogeneous framework, leading
s ). The details are left to the reader.
e ρ (Bp,r
to some functional spaces denoted by L
T
2.2.1
Spectral localization
In this section, we prove estimates for the semi-group of the heat equation restricted to functions
with compact supports away from the origin in Fourier variables. These estimates are based on
the following result.
Lemma 2.2.2. Let φ be a smooth function supported in the shell C(0, R1 , R2 ) with 0 < R1 < R2 .
There exist two positive constants κ and C depending only on φ and such that for all 1 ≤ p ≤ ∞,
τ ≥ 0 and λ > 0, we have
φ(λ−1 D)eτ ∆ u
Lp
≤ Ce−κτ λ
2
φ(λ−1 D)u
Lp
.
Proof: Performing a change of variable, one can assume with no loss of generality that λ = 1.
Now, let φe be a smooth function supported in C(0, R10 , R20 ) for some R10 < R1 and R20 > R2
and such that φe ≡ 1 in a neighborhood of C(0, R1 , R2 ). We have
−τ |ξ|2
e
F φ(D)eτ ∆ u (ξ) = φ(ξ)e
F(φ(D)u)(ξ).
Thus, φ(D)eτ ∆ u = kτ ? φ(D)u with
−N
Z
kτ (z) := (2π)
RN
2
e dξ.
e−τ |ξ| eiz·ξ φ(ξ)
According to convolution inequalities, we have
φ(D)eτ ∆ u
Lp
≤ kkτ kL1 kφ(D)ukLp .
Therefore it only remains to prove that there exist two positive constants κ and C such
that
(2.4)
∀τ ∈ R+ , kkτ kL1 ≤ Ce−κτ .
For that, we use the fact that for all m ∈ N, we have
Z
2
2
m
−N
e (Id − ∆ξ )m eiz·ξ dξ,
(1 + |z| ) kτ (z) = (2π)
e−τ |ξ| φ(ξ)
Z
2
−N
e
= (2π)
eiz·ξ (Id − ∆ξ )m e−τ |ξ| φ(ξ)
dξ.
42
CHAPTER 2. THE HEAT EQUATION
From the last equality and the fact that the integration may be restricted to the shell
C(0, R10 , R20 ), we easily conclude that there exists a constant Cm such that
∀z ∈ RN , (1 + |z|2 )m |kτ (z)| ≤ Cm e−κτ ,
whence inequality (2.4).
2.2.2
Estimates for the heat equation
Let us now state our main result for the heat equation.
s
Theorem 2.2.3. Let T > 0, s ∈ R and 1 ≤ ρ, p, r ≤ ∞. Assume that u0 ∈ Ḃp,r
and
2
2
s−2+ ρ
s+ ρ
s ) and there exists
e ρ (Ḃp,r
e ρ (Ḃp,r
e ∞ (Ḃp,r
f ∈L
). Then (H) has a unique solution u in L
)∩L
T
T
T
a constant C depending only on N and such that for all ρ1 ∈ [ρ, +∞], we have
1
1
−1
µ ρ1 kuk ρ s+ ρ2 ≤ C ku0 kḂ s + µ ρ kf k ρ s−2+ ρ2 .
e 1 (Ḃp,r
L
T
1
p,r
)
)
e (Ḃp,r
L
T
s ).
If in addition r is finite then u belongs to C([0, T ]; Ḃp,r
Proof: Since u0 and f are temperate distributions, equation (H) has a unique solution u in
S 0 (0, T × RN ), which satisfies
Z t
2
u
b(t, ξ) = e−µt|ξ| u
b0 (ξ) +
e−µ(t−τ ) fb(τ, ξ) dτ.
0
As F(Ṡq u0 ) (resp. F(Ṡq f )) tends to 0 in S 0 (RN ) (resp. S 0 (0, T × RN )) when q goes to
−∞, we easily gather that also Ṡq u goes to zero in S 0 (0, T × RN ) when q goes to −∞.
˙ q to (H) and using formula (2.3) yields
Next, we notice that, applying ∆
˙ q u(t) = eµt∆ ∆
˙ q u0 +
∆
t
Z
˙ q f (τ ) dτ.
eµ(t−τ ) ∆
0
Therefore,
˙ q u(t)
∆
Lp
˙ q u0
≤ eµt∆ ∆
Z
Lp
+
t
˙ q f (τ )
eµ(t−τ )∆ ∆
Lp
0
By virtue of lemma 2.2.2, we thus have for some κ > 0,
Z t
2q
−κµ22q t
˙
˙ q f (τ )
∆q u(t) p . e
k∆q u0 kLp +
e−κµ2 (t−τ ) ∆
L
0
dτ.
Lp
dτ.
Applying convolution inequalities, we get
˙ q uk ρ1 p .
k∆
L (L )
T
1 − e−κµT ρ1 2
κµρ1 22q
2q
1
ρ1
˙ q u0 kLp +
k∆
1 − e−κµT ρ2 2
κµρ2 22q
2q
1
ρ2
˙ qf k ρ p
k∆
L (L )
T
with 1/ρ2 = 1 + 1/ρ1 − 1/ρ.
Finally, taking the `r (Z) norm, we conclude that (with the usual convention if r = +∞)
(2.5)
kuk
s+ 2
e ρ1 (Ḃp,r ρ1
L
T
.
)
X
2q r
ρ1
1 − e−κµT ρ1 2
q
κµρ1 22q
˙ q u0 kLp
2 k∆
qs
r
1
r
X
1
2q 1
ρ2
r r
1 − e−κµT ρ2 2
q(s−2+ ρ2 ) ˙
ρ
+
2
k∆q f kL (Lp )
,
T
κµρ2 22q
q
2.2. A PRIORI ESTIMATES IN BESOV SPACES FOR THE HEAT EQUATION
43
2
s+ ρ
s ) and yields the desired inequality.
e ∞ (Ḃp,r
e ρ (Bp,r
)∩L
which insures that u ∈ L
T
T
s ) in the case where r is finite may be easily deduced from
That u belongs to C([0, T ]; Ḃp,r
s
s
the density of S ∩ Ḃp,r in Ḃp,r (see proposition 1.5.9).
s ) spaces enables us to gain two
e ρ (Ḃp,r
Remark. In the case u0 ≡ 0 (resp. f ≡ 0) the use of L
T
derivatives for u compare to the regularity of f (resp. u0 ). Remind that the classical result
of maximal regularity for the heat equation which states that D2 u ∈ Lp (0, T × RN ) whenever
2− 2
u0 ∈ Bp,p p and f ∈ Lp (0, T × RN ) breaks down for p = 1.
2.2.3
A counterexample
In this section, we aim at convincing the reader that the results stated in the previous section
s ) spaces is not a technical artifact.
e ρ (Bp,r
are optimal and that the appearance of L
T
Let us consider the simple case of the free heat equation ∂t u − ∆u = 0 with initial data u0
in L2 . From (2.2), it is easy to see that the function (t, x) 7→ tD2 u(t, x) belongs to L1 (R+ ; L2 ).
Can we expect u to be in L1loc (R+ ; H 2 ) ? The answer is negative. One can even state a more
accurate result:
Proposition 2.2.4. Let u0 be in Sh0 . The solution to (H) with f ≡ 0 belongs to L1 (R+ ; Ḣ 2 )
0 .
if and only if u0 is in Ḃ2,1
0 then theorem 2.2.3 states that u belongs to L
e 1 (Ḃ 2 ). As L
e 1 (Ḃ 2 ) =
Proof: If u0 is in Ḃ2,1
2,1
2,1
2 ) and Ḃ 2 ,→ Ḣ 2 , we thus have u ∈ L1 (R+ ; Ḣ 2 ) as expected.
L1 (R+ ; Ḃ2,1
2,1
Conversely, let us assume that u ∈ L1 (R+ ; Ḣ 2 ). According to Parseval formula and to
(2.1), we thus have
Z
+∞ Z
4 −2t|ξ|2
|ξ| e
I :=
2
1
2
dt < ∞.
|b
u0 (ξ)| dξ
RN
0
Z
2
1
2
|b
u0 (ξ)| dξ
Now, denoting cq :=
, easy computations yield
2q ≤|ξ|≤2q+1
I≥
XZ
p∈Z
41−p X
4−p
2(q−p) 2
24q e−2
cq
1
2
dt.
q∈Z
Keeping only the term q = p in the second sum, we end up with
X
I≥α
cp
p∈Z
for some positive constant α.
P
0 .
Hence
cq has to be finite. In other words, u0 has to belong to B2,1
2.2.4
Estimates in nonhomogeneous Besov spaces, and the periodic case
One can wonder if the estimates stated in theorem 2.2.3 remain true in nonhomogeneous Besov
spaces. On one hand, the block ∆−1 u corresponding to the low frequencies of u cannot be
handled by mean of lemma 2.2.2. On the other hand, by using the representation formula (2.2),
we easily get
Z
T
k∆−1 u(T )kLp ≤ k∆−1 u0 kLp +
0
k∆−1 f kLp dt,
44
CHAPTER 2. THE HEAT EQUATION
whence, if 1 ≤ ρ ≤ ρ1 ≤ ∞,
1
k∆−1 ukLρ1 (Lp ) ≤ T ρ1 k∆−1 u0 kLp + T
1+ ρ1 − ρ1
T
1
k∆−1 f kLρ (Lp ) .
T
Of course the other dyadic blocks may be treated as in the homogeneous case. We end up with
the following statement:
2
s−2+ ρ
s and f ∈ L
e ρ (Bp,r
Theorem 2.2.5. Let s ∈ R and 1 ≤ ρ, p, r ≤ ∞. Let T > 0, u0 ∈ Bp,r
).
T
2
s+ ρ
s ) and there exists a constant C
e ρ (Bp,r
e ∞ (Bp,r
Then (H) has a unique solution u in L
)∩L
T
T
depending only on N and such that for all ρ1 ∈ [ρ, +∞], we have
1
1
1
1+ 1 − 1
−1
s
+ (1 + T ρ1 ρ )µ ρ kf k ρ s−2+ ρ2 .
µ ρ1 kuk ρ s+ ρ2 ≤ C (1 + T ρ1 )ku0 kBp,r
e 1 (Bp,r
L
T
1
)
e (Bp,r
L
T
)
s ).
If in addition r is finite then u belongs to C([0, T ]; Bp,r
Remark. Compare to the homogeneous case, the constant appearing in the above inequality now
depends on T. When dealing with nonlinear parabolic PDE’s, this may preclude from proving
global in time existence results (even for small data). Avoiding the time dependence is thus a
good motivation for using homogeneous spaces when dealing with such PDE’s.
Let us finally make a short remark concerning the periodic case. In the case where u0 and
f are periodic and have zero average, it is clear that the unique solution u to (H) has also
zero average. The above analysis may be carried out and leads again to the estimates stated in
theorem 2.2.3.
In the general
not have zero average, the above estimates hold
R
R case where u0 and f need
−1
for u − |TN
|
u
dx
and
the
study
of
u
dx
has to be done separately.
a
TN
T
a
2.3
Optimal well-posedness results for Navier-Stokes equations
In this section, we give an example of application of theorem 2.2.3 for solving a nonlinear PDE’s
related to the heat equation. We focus on the incompressible Navier-Stokes equations which
have been extensively studied recently. It goes without saying that the same method works for
a great deal of semi-linear heat equations.
2.3.1
The model
The incompressible Navier-Stokes equations write

 ∂t u + u · ∇u − µ∆u + ∇Π = f,
div u = 0,
(N S)

u|t=0 = u0 .
Above u = u(t, x) stands for the unknown velocity field which is a time dependent vectorfield
and the scalar function Π = Π(t, x) stands for the pressure. From a mathematical viewpoint,
∇Π is the Lagrange multiplier associated to the divergence free constraint. The initial velocity
u0 and the external force f are given functions. It isPunderstood that the convective term
N
j
i
u · ∇u stands for the vector field whose i-th entry is
j=1 u ∂j u . Finally, the parameter µ
1
(the viscosity) has to be positive .
√
By performing the change of unknown function u(t, x) = v(t, x/ µ), one can restrict the study to the case
µ = 1, an assumption which is made in most of papers devoted to Navier-Stokes equations. Since we aim at
keeping track of the dependence with respect to µ , we will not perform this change of function.
1
2.3. OPTIMAL WELL-POSEDNESS RESULTS FOR NAVIER-STOKES EQUATIONS
45
Note that, strictly speaking, (NS) does not enter in the class of semi-linear parabolic equations. Since we will consider only the case where the space variable x belongs to the whole space
however, we shall see in a moment that (NS) may be reduced to a (system of) semilinear heat
equations with nonlocal nonlinearity.
We notice that for suitably smooth divergence-free vectorfields, the convective term rewrites
div u ⊗ u. This leads to the following definition of weak solution:
Definition. A distribution u ∈ S 0 (0, T × RN ; RN ) is called a weak solution of (N S) if div u = 0
in S 0 (0, T × RN ; R) and if 2
Z TZ
RN
0
Z TZ
u·∂t ϕ dxdt+
0
Z TZ
i j
u u ∂j ϕi dxdt−µ
RN
0
Z TZ
∇u·∇ϕ dxdt+
RN
0
Z
f ·ϕ dxdt =
RN
u0 ϕ(0) dx
RN
for all divergence free ϕ in Cc∞ [0, T ); S(RN ; RN ) .
If u is a smooth enough weak solution of (NS), taking ϕ = u in the above relation yields
the following energy equality:
(2.6)
ku(t)k2L2
Z
+ 2µ
0
t
kDu(τ )k2L2
dτ =
ku0 k2L2
Z tZ
f · u dx dt.
+2
0
Taking advantage of (2.6) and of compactness arguments, J. Leray in 1934 proved the existence
of global weak solutions with bounded energy in the case where f ≡ 0. Let us state his main
result (see [33] for more details):
Theorem (Leray). Let N ≥ 2. For all divergence free vectorfield u0 with coefficients in L2 ,
system (N S) with f ≡ 0 has a global weak solution u in L∞ (R+ ; L2 ) with Du ∈ L2 (R+ ; L2 ),
which satisfies
(2.7)
ku(t)k2L2
Z
+ 2µ
0
t
kDu(τ )k2L2 dτ ≤ ku0 k2L2 .
If N = 2, then (2.6) is true. Besides the solution u belongs to C(R+ ; L2 ) and is unique in the
set of divergence-free vector fields with coefficients in L∞ (R+ ; L2 ) and gradient in L2 (R+ ; L2 ).
Since the work by J. Leray in 1934, the problem of uniqueness in the energy space when
N ≥ 3 has remained unsolved. Considering smoother data and restricting the set of admissible
solutions is the usual way to get existence and uniqueness results. Again, this has been first
noticed by J. Leray in [33]:
Theorem (Leray). Let N = 3. There exists a positive constant c such that for all divergencefree vector field u0 with coefficients in H 1 which satisfies
ku0 kL2 k∇u0 kL2 ≤ cµ2
or
ku0 k2L2 ku0 kL∞ ≤ cµ3 ,
system (N S) with no external force has a unique global solution u ∈ C(R+ ; H 1 ) which also
satisfies D2 u ∈ L2 (R+ ; L2 ).
One can alternately consider large data in H 1 . One still obtain existence and uniqueness of
a solution but for small time only.
Whether global existence and uniqueness holds true is an outstanding open problem.
2
From now on, we adopt the summation convention over repeated indices.
46
2.3.2
CHAPTER 2. THE HEAT EQUATION
About scaling and critical spaces
In this section, we aim at finding functional spaces E as large as possible for which any vectorfield
u0 in E generates a unique solution on a small time interval. If one restricts to the Sobolev
spaces framework, the optimal result is due to H. Fujita and T. Kato in [28]. Their original
statement pertains to the three-dimensional case in bounded domains. The statement below in
dimension N = 2, 3 has been proved by J.-Y. Chemin in [9].
N
Theorem (Fujita-Kato). Let u0 be a solenoidal vector-field with coefficients in Ḣ 2 −1 . Let f
N
have coefficients in L2 (0, T ; Ḣ 2 −2 ). There exists a positive time T such that (N S) has a unique
N
N
solution in C([0, T ]; Ḣ 2 −1 ) with gradient in L2 (0, T ; Ḣ 2 −1 ).
Moreover, there exists a constant c depending only on N and such that (N S) has a unique
N
N
global solution in Cb (R+ ; Ḣ 2 −1 ) with gradient in L2 (R+ ; Ḣ 2 −1 ) whenever
ku0 k
(2.8)
1
Ḣ
N −1
2
+ µ 2 kf k
L2 (Ḣ
N −2
2
)
≤ cµ.
Compare to Leray’s results, there are two important advances in Fujita and Kato’s approach.
The first one is that suitably smooth solutions to (NS) may be interpreted as a fixed point of
some functional over Banach spaces.
Indeed, let P denote the orthogonal projector of L2 (RN ; RN ) over solenoidal vectorfields.
One can easily show that P is the 0 order multiplier defined by P := Id − ∇(−∆)−1 div . In
other words, in Fourier variables, we have for any u with coefficients in L2 ,
ξ·u
b(ξ)
c
ξ·
Pu(ξ)
=u
b(ξ) −
|ξ|2
Note that the above definition may be extended to distributions of Sh0 .
Now, we remark that Navier-Stokes equations rewrite
g
(N
S)
∂t u − P div(u ⊗ u) − µ∆u = Pf,
u|t=0 = u0 .
Hence, by virtue of (2.3), we have at least formally,
Z t
Z t
µt∆
µ(t−τ )∆
(2.9)
u(t) = e u0 +
e
Pf (τ ) dτ −
eµ(t−τ )∆ P div u(τ ) ⊗ u(τ ) dτ.
0
0
Any solution of (NS) which satisfies (2.9) is called (after the theory of analytic semi-groups) a
mild solution of Navier-Stokes equations. One can prove that any suitably smooth mild solution
is also a weak solution to (NS) (see [32] for a detailed study).
Thus solving (NS) amounts to finding a fixed point for the functional v 7→ Φ(v) defined by
Z t
µt∆
Φ(v)(t) := e u0 +
eµ(t−τ )∆ P (f − div(v ⊗ v))(τ ) dτ.
0
The second advance in Fujita-Kato’s approach has to do with the choice of the functional
framework for the solution u and for the data.
g
The idea is that when appealing to contracting mapping arguments for solving (N
S), it is
suitable that u belongs to a functional space X such that the linear term ∂t u − µ∆u and the
nonlinear term P div(u ⊗ u) have the same regularity.
g
This may be interpreted in terms of scaling invariance for (N
S). Indeed, we notice that if
g
u solves (N S) with data u0 and f, so does for any λ > 0 the vectorfield
uλ : (t, x) 7→ λu(λ2 t, λx)
2.3. OPTIMAL WELL-POSEDNESS RESULTS FOR NAVIER-STOKES EQUATIONS
47
with data
u0,λ : x 7→ λu(λx)
and
fλ : (t, x) 7→ λ3 f (λ2 t, λx).
It is also clear that a space where the heat and convective terms have the same regularity must
have a norm invariant for all λ by the change u 7→ uλ . This leads to the following definition:
Definition. A critical space for initial data is any Banach space E ⊂ S 0 (RN ) whose norm is
invariant for all λ by u0 7→ u0,λ .
A critical space for external forces is any Banach space F ⊂ S 0 (R+ × RN ) whose norm is
invariant for all λ by f 7→ fλ .
A critical space for solutions to (NS) is any Banach space X ⊂ S 0 (R+ × RN ) whose norm is
invariant for all λ by u 7→ uλ .
N
One can easily check that Fujita-Kato’s theorem enters in this framework. Indeed, Ḣ 2 −1
N
is invariant by u0 7→ u0,λ , the space L2 (R+ ; Ḣ 2 −2 ) is invariant by f 7→ fλ and the space of
N
N
divergence-free vector-fields with coefficients in C(R+ ; Ḣ 2 −1 ) and gradient in L2 (R+ ; Ḣ 2 −1 ) is
invariant by u 7→ uλ .
2.3.3
Global well-posedness for small data
In the present section, we investigate the problem of existence of global (mild) solutions to
Navier-Stokes equations for small data which belong to functional spaces which are invariant by
the scaling exhibited in the previous section.
Our main result is the following:
Theorem 2.3.1. Let 1 ≤ r ≤ ∞ and 1 ≤ p < ∞. There exists a constant c > 0 independent of
N
−1
p
µ such that for all divergence free vector-field u0 with coefficients in Ḃp,r
f with coefficients in
N
p
e 1 (Ḃp,r−1 )
L
such that
ku0 k
(2.10)
and external force
N −1
p
Ḃp,r
+ kPf k
N −1
p
e 1 (Ḃp,r
L
< cµ,
)
N
N
+1
−1
p
p
e ∞ (Ḃp,r
e 1 (Ḃp,r
system (N S) has a unique solution u in L
)∩L
) which satisfies
kuk
(2.11)
N −1
p
e ∞ (Ḃp,r
L
+ µkuk
)
N +1
p
e 1 (Ḃp,r
L
< 2cµ.
)
N
N
−1
+1
p
p
e 1 (Ḃp,r
Besides, if r is finite then u belongs to C(R+ ; Ḃp,r
) and uniqueness holds true in L
)∩
N
−1
p
e ∞ (Ḃp,r
L
) with no smallness condition.
N
−1
p
According to proposition 1.5.4, all the spaces Ḃp,r
N
2
−1
are scaling invariant for the Navier-
N
2
−1
Ḃ2,2 .
=
Hence theorem 2.3.1 is a natural generalStokes equations. Besides, we have Ḣ
ization of Fujita and Kato’s theorem.
Well-posedness in critical Besov spaces has been first proved by M. Cannone in [7] for 3 <
s ) spaces for
e ρ (Ḃp,r
p ≤ 6 and r = ∞, then extended in [8] and [35]. The idea of using L
T
solving Navier-Stokes equations is due to J.-Y. Chemin in [11]. Related results in more general
functional spaces have been proved in [31]. Critical spaces in which global well-posedness for
small data may be proved with the method below have been characterized in [2].
The proof of theorem 2.3.1 is based on the following lemma.
48
CHAPTER 2. THE HEAT EQUATION
Lemma 2.3.2. Let (X, k · kX ) be a Banach space and B : X × X → X be a bilinear continuous
operator with norm K. Then for all y ∈ X such that 4KkykX < 1, equation x = y + B(x, x)
1
). Besides x satisfies kxkX ≤ 2kykX .
has a unique solution x in the ball B(0, 2K
Proof: We rule out the case K = 0 which is obvious. Now, the result is a mere consequence
of the contracting mapping theorem.
R
) into
Indeed, let R := 4KkykX and F : x 7→ y + B(x, x). On one hand, F maps B(0, 2K
R 2
0
itself provided that R ≤ 1. On the other hand, for all (x, x ) ∈ B(0, 2K ) , we have
kF (x0 ) − F (x)kX ≤ Rkx0 − xkX .
R
whenever R < 1, which insures the
Hence F is a contracting mapping on B 0, 2K
existence of x.
1
Routine computations then lead to kxkX ≤ 2kykX and to uniqueness in B(0, 2K
).
Let us now prove theorem 2.3.1:
First step: Existence
The existence of a solution for (NS) with data u0 and f will be obtained by applying lemma
2.3.2 for convenient y, B and (X, k · kX ).
For X, we shall take the space of divergence free distributions over R+ ×RN with coefficients
N
N
+1
−1
p
p
e 1 (Ḃp,r
e ∞ (Ḃp,r
in L
)∩L
) endowed with the norm
kvkX := kvk
We then set y : t 7→ eµt∆ u0 +
formula
Rt
0
N −1
p
e ∞ (Ḃp,r
L
+ µkvk
)
N +1
p
e 1 (Ḃp,r
L
.
)
eµ(t−τ )∆ Pf dτ and define the bilinear functional B by the
Z
t
B(v, w)(t) = −
eµ(t−τ )∆ P div(v(τ ) ⊗ w(τ )) dτ.
0
We claim that y belongs to X, that B maps X × X in X and that there exists some constant
C such that
(2.12)
,
kykX ≤ C ku0 k Np −1 + kPf k
N −1
p
e 1 (Ḃp,r
L
Ḃp,r
)
∀(v, w) ∈ X 2 , kB(v, w)kX ≤ Cµ−1 kvkX kwkX .
(2.13)
N
N
−1
−1
p
p
e 1 (Ḃp,r
Indeed, as u0 is divergence free and belongs to Ḃp,r
, and as f is in L
), theorem 2.2.3
insures that y belongs to X and satisfies (2.12).
Next, using Bony’s decomposition and div v = div w = 0, one can write
div(v ⊗ w) = Ṫ∂j v wj + Ṫwj ∂j v + ∂j Ṙ(v, wj )
with the summation convention over repeated indices.
N
1
−
p 2
e ρ (Ḃp,r
Hence, combining propositions 1.4.1 and 1.4.2, remark 2.2.1 and the embedding L
) ,→
1
−2
e ρ (Ḃ∞,∞
L
) for ρ = 4/3 or ρ = 4, we get
kdiv(v ⊗ w)k
N −1
p
e 1 (Ḃp,r
L
≤ Ckvk
)
N +1
2
p
e 34 (Ḃp,r
L
kwk
)
N −1
2
p
e 4 (Ḃp,r
L
.
)
2.3. OPTIMAL WELL-POSEDNESS RESULTS FOR NAVIER-STOKES EQUATIONS
49
Putting forward complex interpolation, we have
kvk
N +1
p
2
e 43 (Ḃp,r
)
L
kwk
3
1
≤ kvk 4
kvk 4
≤ µ− 4 kvk,
≤ kwk
kwk
≤ µ− 4 kwk,
N +1
p
e 1 (Ḃp,r
L
)
1
4
N +1
p
e 1 (Ḃp,r
L
)
N −1
p
2
e 4 (Ḃp,r
)
L
3
N −1
p
e ∞ (Ḃp,r
L
)
3
4
N −1
p
e ∞ (Ḃp,r
L
)
1
whence
kdiv(v ⊗ w)k
N −1
p
e 1 (Ḃp,r
L
≤ Cµ−1 kvkX kwkX .
)
Finally, by using the fact that P is an homogeneous multiplier of degree 0, and by applying
theorem 2.2.3, be conclude that B(v, w) belongs to X and that (2.13) is satisfied.
Now, lemma 2.3.2 may be applied provided that 4CkykX < µ. According to (2.12) this
condition will be satisfied if
ku0 k
N −1
p
Ḃp,r
+ kPf k
N −1
p
e 1 (Ḃp,r
L
< cµ
)
for some small enough constant c.
This achieves the proof of existence of a global solution in X for (NS), and of uniqueness
under condition (2.11).
Second step: Uniqueness in the case where r is finite
Let u1 and u2 be two solutions of (NS) in X. Denoting δu := u2 − u1 , we have for all t ∈ R+ ,
Z
t
eµ(t−τ )∆ P div(δu ⊗ u2 + u1 ⊗ δu) dτ.
δu(t) = −
0
By going along the lines of the proof of (2.13), one can easily check that for all positive T and
N
N
4
+ 21
− 12
p
p
e 3 (Ḃp,r
e 4 (Ḃp,r
divergence free (v, w) in L
)×L
), we have
T
T
kP div(v ⊗ w)k
N −1
p
e 1 (Ḃp,r
L
T
≤ Ckvk
4
N +1
2
p
e 3 (Ḃp,r
L
T
)
kwk
N −1
2
p
e 4 (Ḃp,r
L
T
)
.
)
Hence, denoting XT the space of functions of X restricted to [0, T ]×RN , and applying theorem
2.2.3, we get with obvious notation,
kδukXT ≤ C kδuk 4 Np + 1 ku2 k
.
N − 1 + ku1 k 4
N + 1 kδuk
N −1
p
p
p
e 3 (Ḃp,r
L
T
2
)
e 4 (Ḃp,r
L
T
2
e 3 (Ḃp,r
L
T
)
2
)
e 4 (Ḃp,r
L
T
2
)
Using complex interpolation, we conclude that
(2.14)
with
kδukXT ≤ Z(T )kδukXT
3
Z(T ) := C µ− 4 ku2 k
1
N −1
p
2
e 4 (Ḃp,r
L
)
T
+ µ− 4 ku1 k
N +1
4
p
2
e 3 (Ḃp,r
L
)
T
.
Now, Lebesgue dominated convergence theorem insures that Z is a continuous nondecreasing
function which vanishes at zero. Hence δu ≡ 0 in XT for small enough T.
Finally, because the function t 7→ kδuk
is also continuous, a standard connectivity
N −1
p
e 1 (Ḃp,r
L
t
)
argument enable us to conclude that δu ≡ 0 on R+ × RN .
50
CHAPTER 2. THE HEAT EQUATION
Last step: Continuity
We still assume that r is finite. Let u ∈ X be a global solution to (NS). We have
∂t u − µ∆u = P f − div(u ⊗ u)
and u|t=0 = u0 .
Arguing like in the first step, one easily state that the right-hand side of the first equation
N
N
−1
−1
p
p
e 1 (Ḃp,r
belongs to L
). Since r is finite, theorem 2.2.3 insures that u ∈ C(R+ ; Bp,r
).
Remark. By going along the lines of the proof of uniqueness, it is easy to prove that, under
condition (2.10), the map (u0 , f ) 7→ u with u0 a divergence-free vectorfield with coefficients in
N
N
−1
+1 N
p
p
e 1 (Ḃp,r
Ḃp,r
and f ∈ L
) , is continuous with values in X. Hence, in the case where r is
N
−1
p
finite, Navier-Stokes equations with small data are globally well-posed in Ḃp,r
Hadamard.
2.3.4
in the sense of
Further results
N
N
−1
−1
p
p
of S ∩ Ḃp,∞
• In the case r = ∞, one can prove that if u0 belongs to the closure Ḃp,∞
N
−1
p
in Ḃp,∞
N
+1
p
e 1 (Ḃp,∞
and satisfies (2.10) then system (NS) has a unique solution in L
)∩
N
N
−1
−1
p
p
e ∞ (Ḃp,∞
L
) which is also continuous with values in Ḃp,∞
(see exercise 2.4).
N
N
−1
−1
p
p
e 1 (Ḃp,r
• For any (possibly large) u0 ∈ Ḃp,r
with div u0 = 0, and f ∈ L
), it is actually
possible to prove the existence and uniqueness of a solution on a small time interval.
The reader is referred to exercise 2.7 for more details.
−1
• The space Ḃ∞,∞
is the largest homogeneous Besov space invariant for the scaling of
Navier-Stokes equation. Hence one can wonder whether well-posedness holds true for data
−1 . This is unlikely to be true if no additional condition as the solution u is expected
in Ḃ∞,∞
−1 ) ∩ L
1
e 1 (Ḃ∞,∞
to be in L∞ (0, T ; Ḃ∞,∞
) only. With such a bad regularity, the meaning of
T
div(u ⊗ u) becomes unclear.
Let us mention however that well-posedness has been proved in [30] for data in the so-called
−1
−1 .
BM O−1 space. This space satisfies Ḃ∞,1
,→ BM O−1 ,→ Ḃ∞,∞
• From the previous section, one has to retain that our approach is not very sensitive to the
exact structure of the term div(u ⊗ u). As a matter of fact, the same method would lead
N
−1
p
to global results for small data (or local results for large data) in Ḃp,r
for any (possibly
not physical) Partial or Pseudo Differential Equation having the heat equation as a linear
part and a first order term which resembles div(u ⊗ u).
It has been noticed in [34] that the approach presented here works for the scalar equation
∂t u − ∆u + |D|(u2 ) = 0
but that one can find a class of (large) data for which finite time blow-up does occur.
Therefore, one cannot expect our method to give better results for (NS) as far as the very
structure of div(u ⊗ u) has not been used.
2.4. EXERCISES
2.4
51
Exercises
Exercise 2.1. Let 1 ≤ p, r, r1 , r2 , ρ, ρ1 , ρ2 ≤ ∞ with
and s ∈ R be such that
s−σ <
N
p
or s − σ =
N
p
1
r
=
1
r1
+
1
r2
and
1
ρ
=
1
ρ1
+
1
ρ2 .
Let σ > 0
and r = 1.
−σ ) × L
e ρ1 (Ḃ∞,r
e ρ2 (Ḃps ,r )
Prove that for all T ∈ (0, +∞], the homogeneous paraproduct Ṫ maps L
T
T
1
2 2
s−σ ) continuously.
e ρ (Ḃp,r
in L
T
Exercise 2.2. Let 1 ≤ p, p1 , p2 , r, r1 , r2 , ρ, ρ1 , ρ2 ≤ ∞ with
1
1
1
1
1
1
1
1
1
≤
+
≤ 1,
=
+
and
≤
+
≤ 1.
r
r1 r2
ρ
ρ 1 ρ2
p
p1 p2
Let s ∈ R and σ12 := s1 + s2 − N p11 + p1 − p1 . Assume that
2
σ12 <
N
p
or σ12 =
N
p
and r = 1.
e ρ1 (Ḃps1,r ) × L
e ρ2 (Ḃps2,r )
Prove that for all T ∈ (0, +∞], the homogeneous remainder Ṙ maps L
T
T
1 1
2 2
σ12 ) continuously.
e ρ (Ḃp,r
in L
T
Exercise 2.3. Let m ∈ R, 0 < R1 < R2 and 1 ≤ p ≤ ∞. Let φ be a smooth function supported
in the shell C(0, R1 , R2 ) and let A be a positive homogeneous function of degree m, smooth on
RN \ {0}.
1) Prove that there exist two positive constants κ and C depending only on φ and A, and
such that for all τ ≥ 0, λ > 0, we have
φ(λ−1 D)e−τ A(D) u
Lp
≤ Ce−κτ λ
m
φ(λ−1 D)u
Lp
.
2) Generalize theorem 2.2.3 to the solutions of
∂t u + A(D)u = f,
u|t=0 = u0 .
s
s for the norm of Ḃ s .
Exercise 2.4. Let Ḃp,∞
be the closure of S ∩ Ḃp,r
p,r
2
s−2+ ρ
s
e ρ (Ḃp,∞
Show that for all u0 ∈ Ḃp,∞
and f ∈ L
), equation (H) has a unique solution u in
T
2
s+ ρ
s ) and that u ∈ C([0, T ]; Ḃ s ).
e ρ (Ḃp,∞
e ∞ (Ḃp,∞
L
)∩L
p,∞
T
T
s ) for the gain of two derivatives in ordinary
Exercise 2.5. Find counterexamples in Lρ (0, T ; Ḃp,r
Besov spaces for the nonhomogeneous heat equation.
Hint: use exercise 1.20.
Exercise 2.6. Let (X, k · k) be a Banach space, B : X × X → X a bilinear continuous operator
with norm K and L : X → X a continuous linear operator with norm M < 1. Let y ∈ X
satisfy 4Kkyk < (1 − M )2 .
Prove that equation x = y + L(x) + B(x, x) has a unique solution in the ball B(0, 1−M
2K ).
N
−1
p
Exercise 2.7. Let u0 ∈ Ḃp,r
are finite.
N
−1
p
e 1 (Ḃp,r
with div u0 = 0, and f ∈ L
). We assume that p and r
52
CHAPTER 2. THE HEAT EQUATION
1) Let uL ∈ S 0 (R+ × RN ) satisfy
∂t uL − µ∆uL = Pf
Prove that lim kuL k
T →0
N −1+ 2
ρ
p
e ρ (Ḃp,r
L
T
and uL (0) = u0 .
= 0 whenever ρ is finite.
)
2) Check that u is a mild solution of (NS) on [0, T ] × RN with data u0 and f if and only if
u = uL + u with
Z t
eµ(t−τ )∆ P uL · ∇uL + u · ∇uL + uL · ∇u + u · ∇u .
∀t ∈ [0, T ], u(t) = −
0
3) Prove that there exists a positive T such that (NS) has a unique solution u on [0, T ] in
N
N
N
−1
+1
−1
p
p
p
e ∞ (Ḃp,r
e 1 (Ḃp,r
L
)
×
L
)
and
that
u
∈
C([0,
T
];
Ḃ
).
p,r
T
T
Hint: use the previous exercise.
Chapter 3
The transport equation
This chapter is devoted to the study of the following transport equation:
(T )
∂t f + v · ∇f = g,
f|t=0 = f0
where v : R × RN → RN stands for a given time dependent vectorfield, f0 : RN → RM and
g : R × RN → RM are known data.
Transport equations arise in many mathematical problems and, in particular, in most PDE’s
related to fluid mechanics. Although, most of the time, the velocity field v and the source term
g depend (nonlinearly) on f, having a good theory for the linear transport equation (T) is an
important first step for studying such PDE’s.
In this chapter, we aim at stating estimates in Besov spaces for (T). For the sake of conciseness, we shall always assume that x belongs to RN and restrict ourselves to the nonhomogeneous
setting. Our results, however, may be easily carried out to the periodic case and to homogeneous
spaces. We focus on the case where v has enough smoothness so that the initial regularity is conserved during the evolution. Roughly, it means that v must be at least lipschitz with respect to
the space variable (see [25] and references therein for the case where Dv is only quasi-lipschitz).
In the last section of the chapter, we use these estimates for proving local well-posedness for
a nonlinear shallow water equation commonly called Camassa-Holm equation.
3.1
Framework and basic properties
In this section, we briefly list some properties of smooth solutions to (T).
Let us first recall that for all x ∈ RN , the flow ψt,s (x) of v starting from x at time s is, by
definition, the solution of the following ordinary differential equation1 :
∂t ψt,s (x) = v(t, ψt,s (x)),
(ψt,s (x))|t=s = x.
x ∈ RN ,
Then we have
(3.1)
∀x ∈ RN ,
d
f (t, ψt,s (x)) = (∂t + v · ∇)f (t, ψt,s (x)).
dt
In other words, the operator ∂t + v · ∇ may be interpreted as the derivative along the flow. For
that reason, it is often called material derivative.
1
Of course, depending on the assumptions on v, the flow ψt,s may not be defined for all s, t.
53
54
CHAPTER 3. THE TRANSPORT EQUATION
As a consequence of (3.1), we gather that, at least formally, the solution of (T ) is given by
the formula
Z t
(3.2)
f (t, x) = f0 (ψ0,t (x)) +
g(τ, ψτ,t (x)) dτ.
0
To simplify the presentation, we restrict ourselves from now on to evolution for positive times 2
and assume that v and f are defined on [0, T ] × RN .
Then the previous calculations may be made rigorous for all (s, t) ∈ [0, T ]2 if v is smooth
enough, say, v is continuous and Dv ∈ L∞ ([0, T ] × RN ). Under this latter assumption, the flow
is uniquely defined for all (t, s, x) ∈ [0, T ] × [0, T ] × RN and the map (t, s, x) 7→ ψt,s (x) is C 1
with respect to s, t, and x. Besides, as a corollary of uniqueness, we have
∀(s, t, t0 ) ∈ [0, T ]3 , ψt,s ◦ ψs,t0 = ψt,t0 .
Hence denoting ψt := ψt,0 , we see that ψt is a diffeomorphism with inverse ψ0,t and that formula
(3.2) rewrites
Z t
−1
(3.3)
f (t, x) = f0 (ψt (x)) +
g τ, ψτ (ψt−1 (x)) dτ.
0
It is then easy to prove by direct calculation that if f0 ∈ Lp and g ∈ L1loc (0, T ; Lp ) then
f ∈ L∞ (0, T ; Lp ) and satisfies
Z t
R
R
1 t
kdiv v(τ )kL∞ dτ
− 1 τ kdiv v(τ 0 )kL∞ dτ 0
kg(τ )kLp dτ
kf0 kLp +
e p 0
(3.4)
kf (t)kLp ≤ e p 0
0
where it is understood that 1/p = 0 if p = ∞.
In the next section, we want to investigate estimates in the same spirit with Lp replaced by
s .
some Besov space Bp,r
3.2
A priori estimates in Besov spaces
In this section, we want to prove the following result3 :
Proposition 3.2.1. Let 1 ≤ p ≤ p1 ≤ ∞, 1 ≤ r ≤ ∞ and p0 := (1 − 1/p)−1 . Assume that
1 1
1 1
(3.5)
σ > −N min
,
or σ > −1 − N min
,
if div v = 0.
p1 p0
p1 p0
There exists a constant C depending only on N , p, p1 , r and σ, such that the following
estimates hold true:
R
Z t
Rτ
t
0 ) dτ 0
−C
Z(τ
0
σ +
σ dτ
(3.6)
kf kLe∞ (B σ ) ≤ kf0 kBp,r
e
kg(τ )kBp,r
eC 0 Z(τ ) dτ ,
t
with
p,r
0



 Z(t) = k∇v(t)k
N
p
Bp11,∞ ∩L∞


 Z(t) = k∇v(t)kBpσ−1
,r
1
if
if
σ <1+
N
σ > 1+
p1
N
,
p1
or
N
σ = 1+
p1
and r = 1 .
If f = v then for all σ > 0 (σ > −1 if div v = 0) estimates (3.6) and (3.12) hold with
Z(t) = k∇v(t)kL∞ .
2
3
It goes without saying that similar results may be proved for negative times.
From now on, we agree that, if div v = 0 then v · ∇f stands for div(f v)
3.2. A PRIORI ESTIMATES IN BESOV SPACES
55
Proof: Applying the operator ∆q to (T) yields
(∂t + v · ∇)∆q f = ∆q g + Rq ,
∆q f|t=0 = ∆q f0 ,
(Tq )
with Rq := v · ∇∆q f − ∆q (v · ∇f ).
Since Dv ∈ L1 (0, T ; L∞ ), one readily gets
Z
(3.7)
t
k∆q f (t)kLp ≤ k∆q f0 kLp +
k∆q g(τ )kLp dτ
Z0 t 1
+
kRq (τ )kLp + kdiv v(τ )kL∞ k∆q f (τ )kLp dτ.
p
0
Note that k∆q f (t)kLp may be replaced by supτ ∈[0,t] k∆q f (τ )kLp in the left-hand side.
Let us admit for a while the following lemma:
Lemma 3.2.2. Let σ ∈ R, 1 ≤ r ≤ ∞ and 1 ≤ p ≤ p1 ≤ ∞. Assume that (3.5) is
satisfied. There exists a constant C depending only on p, p1 , σ, r and N , and such that
the following inequalities are true:
2qσ kRq kLp
(3.8)
(3.9)
2qσ kRq kLp
`r
`r
≤ Ck∇vk
N
Bpp1 ,∞ ∩L∞
s
if σ < 1 +
kf kBp,r
σ
≤ Ck∇vkBpσ−1
kf kBp,r
if σ > 1 +
,r
1
N
,
p1
N
N
, or σ = 1 +
and r = 1.
p1
p1
If besides f = v then we also have
(3.10)
2qσ kRq kLp
`r
σ
≤ Ck∇vkL∞ kf kBp,r
if σ > 0, (or σ > −1 if div v = 0).
Now, inserting inequalities (3.8), (3.9) or (3.10) into (3.7) yields
Z
kf k
(3.11)
σ
L∞
t (Bp,r )
≤ kf0 k
σ
Bp,r
+C
0
t
t
Z
σ
Z(τ )kf kL∞
dτ +
τ (Bp,r )
0
σ dτ
kg(τ )kBp,r
with Z defined as in the statement of the proposition.
Applying Gronwall lemma completes the proof.
Remark. Note that, starting from (3.7) and using lemma 3.2.2, one can also get
Z
kf kLe∞ (B σ
t
p,r )
σ + C
≤ kf0 kBp,r
0
t
Z(τ )kf kLe∞ (B σ
τ
p,r )
dτ + kgkLe1 (B σ ) ,
t
p,r
and we thus have:
(3.12)
kf kLe∞ (B σ
t
p,r )
Rt
σ + kgk e 1
≤ kf0 kBp,r
eC 0 Z(τ ) dτ .
L (B σ )
t
p,r
For completeness, we now give the proof of lemma 3.2.2. In order to show that only the gradient
part of v is involved in the estimates, we have to split v into low and high frequencies : v =
∆−1 v + ve . Obviously, there exists a constant C such that
(3.13)
∀a ∈ [1, ∞], k∆−1 ∇vkLa ≤ C k∇vkLa
and
k∇e
v kLa ≤ C k∇vkLa .
56
CHAPTER 3. THE TRANSPORT EQUATION
Since there exists a R > 0 so that Supp F ve ∩ B(0, R) = ∅, Bernstein lemma yields
(3.14)
∀a ∈ [1, ∞], ∀q ≥ −1, k∆q ∇e
v kLa ≈ 2q k∆q vekLa .
Now, we have (with the summation convention over repeated indices):
Rq = v · ∇∆q f − ∆q (v · ∇f ),
= [e
v j , ∆q ]∂j f + [∆−1 v j , ∆q ]∂j f.
Hence, taking advantage of Bony’s decomposition, we end up with Rq =
P6
i
i=1 Rq
where
Rq1 = [Tvej , ∆q ]∂j f,
Rq2 = T∂j ∆q f vej ,
Rq3 = −∆q T∂j f vej ,
v j , ∆q f ) − ∂j ∆q R(e
v j , f ),
Rq4 = ∂j R(e
Rq5 = ∆q R(div ve, f ) − R(div ve, ∆q f ),
Rq6 = [∆−1 v j , ∆q ]∂j f.
In the following computations, the constant C depends only on σ, p, p1 , r and N and we
denote by (cq ) a sequence such that k(cq )k`r ≤ 1.
Bounds for 2qσ Rq1
Lp
:
By virtue of proposition 1.2.1, we have
X
Rq1 =
[Sq0 −1 vej , ∆q ]∂j ∆q0 f.
|q−q 0 |≤4
Using the definition of the operator ∆q leads to
[Sq0 −1 vej , ∆q ]∂j ∆q0 f (x) =
Z
h
i
h(y) Sq0 −1 vej (x) − Sq0 −1 vej (x − 2−q y) ∂j ∆q0 f (x − 2−q y) dy
so that applying first order Taylor’s formula, convolution inequalities and (3.13) yields
2qσ Rq1
Lp
≤ C k∇vkL∞
X
0
2q σ ∆ q 0 f
|q 0 −q|≤4
σ .
≤ Ccq k∇vkL∞ kf kBp,r
(3.15)
Bounds for 2qσ Rq2
Lp
:
By virtue of proposition 1.2.1, we have
Rq2 =
X
Sq0 −1 ∂j ∆q f ∆q0 vej .
q 0 ≥q−3
Hence, using inequalities (3.13) and (3.14) yields
(3.16)
2qσ Rq2
Lp
σ .
≤ Ccq k∇vkL∞ kf kBp,r
Lp
,
3.2. A PRIORI ESTIMATES IN BESOV SPACES
Bounds for 2qσ Rq3
57
:
Lp
One proceeds as follows :
X
Rq3 = −
(3.17)
∆q Sq0 −1 ∂j f ∆q0 vej ,
|q 0 −q|≤4
X
= −
(3.18)
∆q ∆q00 ∂j f ∆q0 vej .
|q 0 −q|≤4
q 00 ≤q 0 −2
Therefore, denoting 1/p2 = 1/p − 1/p1 and taking advantage of (3.13) and (3.14),
P
2qσ Rq3 Lp ≤ C |q0 −q|≤4 2qσ ∆q00 ∂j f Lp2 ∆q0 vej Lp1 ,
q 00 ≤q 0 −2
≤ C
(q−q 00 )(σ−1− pN ) q 00 σ
1 2
|q 0 −q|≤4 2
q 00 ≤q 0 −2
P
∆q00 f
Lp
2
q 0 pN
1
∆q0 ∇v
Lp1
.
Hence, if σ < 1 + N/p1 ,
2qσ Rq3
(3.19)
Lp
≤ Ccq k∇vk
N
p
Bp11,∞
σ .
kf kBp,r
Note that, starting from (3.17), one can alternately get
X
0
2qσ Rq3 Lp ≤ C
∇Sq0 −1 f Lp2 2q (σ−1) ∆q0 ∇v
Lp1
.
|q 0 −q|≤4
σ−1 ,→ Lp2 in the cases σ > 1 + N/p , or σ = 1 + N/p and r = 1,
As it may be proved that Bp,r
1
1
we eventually get
2qσ Rq3
(3.20)
Bounds for 2qσ Rq4
Lp
Lp
σ .
≤ Ccq k∇vkBpσ−1
kf kBp,r
,r
1
:
X
Rq4 =
e q0 f )
∂j ∆q (∆q0 vej ∆
q 0 ≥q−3
|q 0 −q|≤2
|
X
e q0 f ) −
∂j (∆q0 vej ∆q ∆
{z
|
}
Rq4,1
{z
}
Rq4,2
For the first term, we merely have (by virtue of (3.14)),
X
0
e q0 f
2qσ Rq4,1 Lp ≤ C k∇vkL∞
2q σ ∆
|q 0 −q|≤2
Lp
,
whence
(3.21)
2qσ Rq4,1
σ .
≤ Ccq k∇vkL∞ kf kBp,r
Lp
For Rq4,2 , we proceed differently according to the value of 1/p + 1/p1 . If 1/p + 1/p1 ≤ 1, we
define p2 by the relation 1/p2 := 1/p + 1/p1 . Then under condition σ > −1 − N/p1 , proposition
σ+ pN
σ insures that we have
1.4.2 combined with the embedding Bp2 ,r 1 ,→ Bp,r
(3.22)
2qσ Rq4,2
Lp
≤ Ccq ke
vk
N
p +1
Bp11,∞
σ .
kf kBp,r
Now, if 1/p + 1/p1 > 1, the above argument has to be applied with p0 instead of p2 and one
still ends up with (3.22) provided that σ > −1 − N/p0 .
Putting (3.21), (3.22) together and appealing to (3.13), one gets
(3.23)
2qσ Rq4
Lp
≤ Ccq k∇vk
N
p
Bp11,∞
σ .
kf kBp,r
58
CHAPTER 3. THE TRANSPORT EQUATION
Bounds for 2qσ Rq5
Lp
:
The arguments are the same as for Rq4 . Under condition σ > −N min(1/p1 , 1/p0 ), one gets
(3.24)
2qσ Rq5
Lp
≤ Ccq k∇vk
N
p
Bp11,∞
σ .
kf kBp,r
Bounds for 2qσ Rq6 Lp :
P
As Rq6 = |q0 −q|≤1 [∆q , ∆−1 v] · ∇∆q0 f, the first order Taylor formula yields
X
0
2qσ Rq6 Lp ≤ C
k∇∆−1 vkL∞ 2q σ ∆q0 f Lp ,
|q 0 −q|≤1
σ .
≤ Ccq k∇vkL∞ kf kBp,r
(3.25)
Combining inequalities (3.15), (3.16), (3.19) or (3.20), (3.23), (3.24) and (3.25) yields (3.8) and
(3.9). The proof of inequality (3.10) is left to the reader (see exercise (3.2)).
3.3
Solving the transport equation in Besov spaces
Let us now state our existence result for the transport equation with data in Besov spaces:
s
Theorem 3.3.1. Let p, p1 , r and σ be as in the statement of proposition 3.2.1. Let f0 ∈ Bp,r
s ). Let v be a time dependent vectorfield with coefficients in Lρ (0, T ; B −M )
and g ∈ L1 (0, T ; Bp,r
∞,∞
N
N
p
for some ρ > 1 and M > 0, and such that Dv ∈ L1 (0, T ; Bp11,∞ ∩ L∞ ) if σ < 1 + , and
p1
N
N
or σ = 1+
and r = 1.
) if σ > 1+
Dv ∈ L1 (0, T ; Bpσ−1
1 ,r
p1
p1
T
σ )
σ 0 ) and the
Then equation (T ) has a unique solution f ∈ L∞ (0, T ; Bp,r
∩σ0 <σ C([0, T ]; Bp,1
inequalities of proposition 3.2.1 hold true.
σ ).
If moreover r is finite then we have f ∈ C([0, T ]; Bp,r
Proof: Uniqueness readily stems from proposition 3.2.1 so let us tackle directly the proof of
the existence. For the sake of conciseness, we treat only the case σ < 1 + pN1 .
We first smooth out the data and the velocity field v by setting
f0n := Sn f0 ,
g n = ρ n ∗t S n g
and v n = ρn ∗t Sn v
where ρn := ρn (t) stands for a sequence of mollifiers with respect to the time variable4 .
∞ , g ∈ C([0, T ]; B ∞ ), v n ∈ C([0, T ]×RN ) and Dv n ∈ C([0, T ]; B ∞ )
We clearly haveTf0n ∈ Bp,r
p,r
p,r
∞
s
σ , g n is uniformly
with Bp,r := s∈R Bp,r . Moreover f0n is uniformly bounded in Bp,r
σ ), v n is uniformly bounded in Lρ (0, T ; B −M ) and Dv n is unibounded in L1 (0, T ; Bp,r
∞,∞
N
p
formly bounded in L1 (0, T ; Bp11,∞ ∩ L∞ ).
Let f n be the solution to
∂t f n + v n · ∇f n = g n ,
n
f|t=0
= f0n .
Of course f n is smooth and, according to proposition 3.2.1, we have
Z t
Rτ n 0
Rt n
0
n
C
Z
(τ
)
dτ
n
−C
Z
(τ
)
dτ
n
0
σ
σ +
σ dτ
(3.26)
kf (t)kBp,r
≤e 0
kf0 kBp,r
e
kf (τ )kBp,r
0
4
with no loss of generality, one can assume that v and g are defined on R × RN .
3.3. SOLVING THE TRANSPORT EQUATION IN BESOV SPACES
with Z n (t) := k∇v n (t)k
59
.
N
p
Bp11,∞ ∩L∞
Thus, according to the uniform bounds for f0n , g n and v n , one can conclude that sequence
σ ).
(f n )n∈N is uniformly bounded in C([0, T ]; Bp,r
There are several ways to prove the convergence of (f n )n∈N , all of them being quite
cumbersome if written into details. For instance, one can show that (f n )n∈N is a Cauchy
σ−1 ) (if σ is not too negative), or use a duality method or appeal
sequence in C([0, T ]; Bp,r
to compactness arguments.
In the present notes, we are going to use compactness. Let us first observe that (∂t f n −
−m ) for some α > 1, and some m > 0. This
g n )n∈N is uniformly bounded in Lα (0, T ; Bp,∞
may be proved by observing that
∂t f n − g n = ∆−1 v n · ∇f n + (Id − ∆−1 )v n · ∇f n .
(3.27)
s−1 ) and ∆ v n , uniformly
Indeed, because ∇f n is uniformly bounded in L∞ (0, T ; Bp,r
−1
bounded in Lρ (0, T ; Cb∞ ) (where Cb∞ stands for the set of smooth bounded functions with
bounded derivatives5 ), the first term in the right-hand side of (3.27) is uniformly bounded
s−1 ). Now, as for small enough ε the function (Id − ∆ )v n is uniformly
in Lρ (0, T ; Bp,r
−1
N
p
+1−ε
bounded in Lα (0, T ; Bp11+ε,r ) for some α > 1 (interpolate between the uniform bounds
N
p
+1
−M ) for (Id − ∆ )v n ), one can conclude by appealin L1 (0, T ; Bp11,∞ ) and in Lρ (0, T ; B∞,∞
−1
ing to propositions 1.4.1 and 1.4.2 that the last term in (3.27) is uniformly bounded in
−m ) for some large enough m > 0.
Lα (0, T ; Bp,∞
Rt
n
Integrating in time and denoting f (t) := f n − 0 g n (τ ) dτ, we thus gather that there exists
n
−m ),
some β > 0 such that the sequence (f )n∈N is uniformly bounded in C β ([0, T ]; Bp,∞
−m
hence uniformly equicontinuous with values in Bp,∞ .
Next, assuming that m is large enough, proposition 1.4.6 guarantees that for all ϕ ∈ Cc∞
s to B −m . Combining Ascoli theorem and Cantor
the map u 7→ ϕu is compact from Bp,r
p,∞
n
diagonal process thus insures that, up to a subsequence, sequence (f )n∈N converges in
−m ) for all ϕ ∈ C ∞ .
S 0 to some distribution f such that ϕf belongs to C([0, T ]; Bp,∞
c
σ ) and of the
Finally, by taking advantage once again of the uniform bounds in L∞ (0, T ; Bp,r
σ ) and, by interpolation, better
Fatou property for Besov spaces, we get f ∈ L∞ (0, T ; Bp,r
Rt
results of convergence so that it becomes easy to check that the function f := f + 0 g(τ ) dτ
is a solution to (T ) (that the data f0n , g n and v n converge to f, g and v may be easily
deduced from their definition).
σ ) in the case where r is finite. Just by
We still have to prove that f ∈ C([0, T ]; Bp,r
−M 0 ) for some large enough
looking at the equation (T), it is easy to get ∂t f ∈ L1 (0, T ; Bp,∞
0
−M ). Therefore ∆ f ∈ C([0, T ]; Lp ) for any q ≥ −1,
M 0 . Hence f belongs to C([0, T ]; Bp,∞
q
σ ) for all q ∈ N.
whence Sq f ∈ C([0, T ]; Bp,r
s valued functions (S f )
We claim that the sequence of continuous Bp,r
q q∈N converges uniformly on [0, T ]. Indeed, according to (1.1),
∆q0 (f − Sq f ) =
X
|q 00 −q 0 |≥1
q 00 ≥q
5
−M
Here comes the assumption that v ∈ Lρ (0, T ; B∞,∞
).
∆q0 ∆q00 f
60
CHAPTER 3. THE TRANSPORT EQUATION
whence
1

σ
kf − Sq f kBp,r
(3.28)
r
X ≤C
q0 σ
2
∆q 0 f
r
Lp
 .
q 0 ≥q−1
Using inequality (3.7) to bound the right-hand side of (3.28), we gather
σ
kf − Sq f kL∞
.
T (Bp,r )
X 2
q0 σ
∆q0 f0
r r1
Lp
Z
T
+
0
q 0 ≥q−1
X
σ
+kf kL∞
T (Bp,r )
P
q 0 ≥−1 cq 0 (t)
∆q0 g(t)
r
Lp
1
r
dt
q 0 ≥q−1
Z
with
2
q0 σ
0
T
X
r
cq0 (t)
1
!
r
Z(t) dt ,
q 0 ≥q−1
≤ 1 for all t ∈ [0, T ].
The first term clearly tends to zero when q tends to infinity. The terms in the integrals
also tend to zero for almost every t. Lebesgue dominated convergence theorem enables us
σ
to conclude that kf − Sq f kL∞
tends to zero when q tends to infinity. This achieves
T (Bp,r )
σ
to proving that u ∈ C([0, T ]; Bp,r ) in the case r < +∞.
0
σ
σ so that
When r = +∞, we use that for any σ 0 < σ we have the embedding Bp,∞
,→ Bp,1
0
σ . This yields f ∈ C([0, T ]; B σ 0 ).
the above argument may be repeated in the space Bp,1
p,1
3.4
On the Cauchy problem for a shallow water equation
Let us first point out that theorem 3.3.1 is a good starting point for the study of general (possibly
N -dimensional) equations of the type
∂t u + u · ∇u = f (u, ∇u, · · · ),
including the incompressible Euler equation, provided the nonlinearity is of first order at most.
In the present section, we show how it may be used to solve a nonlinear one-dimensional
shallow water equation which has been much studied recently: the so-called Camassa-Holm
equation.
3.4.1
About Camassa-Holm equation
The one-dimensional model we are interested in reads:
(3.29)
3
2
3
∂t v − ∂txx
v + 2κ∂x v + 3v∂x v = 2∂x v∂xx
v + v∂xxx
v.
Above, the scalar function u = u(t, x) stands for the fluid velocity at time t ≥ 0 in the x
direction and κ is a non-negative parameter. For simplicity, we assume that x belongs to R.
Similar results are true if x belongs to the circle, though.
Equation (3.29) has been derived independently by A. Fokas and B. Fuchssteiner in [27], and
by R. Camassa and D. Holm in [6]. Its systematic mathematical study has been initiated in a
series of papers by A. Constantin and J. Escher (see e.g [16]).
We shall concentrate on the case κ = 0 (which actually is not restrictive since the change of
variables u(t, x) = v(t, x − κt) + κ leads to (3.29) with κ = 0). It turns out that equation (3.29)
with κ = 0 has many remarkable structural properties. In particular, it has infinitely many
conservation laws, the most obvious ones being the conservation of the average over R and of
the H 1 norm for smooth solutions with sufficient decay at infinity. By taking advantage of this
3.4. ON THE CAUCHY PROBLEM FOR A SHALLOW WATER EQUATION
61
latter property, Z. Xin and P. Zhang proved that (3.29) has global weak solutions for any data
in H 1 (see [39]).
In the present chapter, we address the question of existence and uniqueness for the initial
value problem. For simplicity, we restrict ourselves to the evolution for positive times. Of
course, one would get similar results for negative times: this is just a matter of changing the
initial condition u0 into −u0 .
At this point, one can wonder which regularity assumptions are relevant for u0 so that the
initial value problem be well-posed in the sense of Hadamard (i.e. (CH) has a unique local
solution in a suitable functional setting, and continuity with respect to the initial data holds
true).
In order to go further into the quest for a good functional framework, let us observe that
one can get rid very easily of the third order terms. Indeed: applying the pseudo-differential
operator (1 − ∂x2 )−1 to (3.29), we discover that (CH) is equivalent to
∂t u + u∂x u = P (D) u2 + 21 (∂x u)2 ,
(CH)
u|t=0 = u0 ,
with P (D) = −∂x (1 − ∂x2 )−1 .
Hence Camassa-Holm equation is nothing but a generalized Burgers equation with a nonlocal nonlinearity of order 0. In light of proposition 3.3.1, we can thus expect that a necessary
condition for well-posedness for data in some functional space E is that E be embedded in the
space Lip of continuous bounded functions with bounded derivatives.
Moreover, as the solution u is expected to be only in C([0, T ]; E) (a gain of regularity cannot
be expected in a Burgers like equation), the application
G : u 7→ P (D) u2 + 12 (∂x u)2
must map E to E continuously.
3.4.2
A well-posedness result and a blow-up criterion
s , the condition
If we restrict ourselves to the framework of nonhomogeneous Besov space Bp,r
E ⊂ Lip is equivalent to s > 1 + 1/p (or s ≥ 1 + 1/p if r = 1), and no further restrictions
are needed for the continuity of the map G (up to the endpoint r = 1, s = 1, p = +∞ which
has to be avoided). We shall see however that for proving uniqueness, our method requires
that in addition we have s > max(1 + 1/p, 3/2). Let us mention in passing that in the periodic
framework, by using the Lagrangian formulation for Camassa-Holm equation, it is actually
possible to weaken the condition on s (see [26]).
For stating our local existence result, the following functional spaces are needed:
s (T )
Ep,r
s (T )
Ep,∞
s ) ∩ C 1 ([0, T ]; B s−1 )
:= C([0, T ]; Bp,r
p,r
:=
s )
L∞ (0, T ; Bp,∞
∩
if r < ∞,
s−1 )
Lip([0, T ]; Bp,∞
with T > 0, s ∈ R and 1 ≤ p, r ≤ ∞.
s . There exists a
Theorem 3.4.1. Let 1 ≤ p, r ≤ ∞ and s > max(3/2, 1 + 1/p). Let u0 ∈ Bp,r
s
time T > 0 such that (CH) has a unique solution u in Ep,r (T ).
We can also state a result of conservation of energy for smooth solutions:
s (T ) be a solution of (CH) on
Theorem 3.4.2. Let s, p, r be as in theorem 3.4.1. Let u ∈ Ep,r
s ∩ H 1 . Then the solution u to (CH) satisfies
[0, T ] × R with data u0 ∈ Bp,r
(3.30)
∀t ∈ [0, T ], ku(t)kH 1 = ku0 kH 1 .
62
CHAPTER 3. THE TRANSPORT EQUATION
Before stating blow-up criteria, let us give the definition of the lifespan of solutions with
s .
data in Bp,r
s . We define the lifespan T ? of the solutions of (CH) with initial
Definition. Let u0 ∈ Bp,r
u0
s (T ) on
data u0 as the supremum of positive times T such that (CH) has a solution u ∈ Ep,r
[0, T ] × R.
Our main blow-up criterion reads:
Theorem 3.4.3. Let u0 be as in theorem 3.4.1 and u be the corresponding solution. Then
Tu?0
3.4.3
Z
< ∞ =⇒
Tu?0
0
k∂x u(τ )kL∞ dτ = ∞.
Uniqueness
Let us start this section with a lemma:
Lemma 3.4.4. Let 1 ≤ p, r ≤ ∞, and (σ1 , σ2 ) ∈ R2 be such that
2
σ1 + σ2 > 2 + max 0, − 1 .
p
σ1 × B σ2 in B σ1 .
Then the function B : (f, g) 7→ P (D) f g + 12 ∂x f ∂x g maps Bp,r
p,r
p,r
σ2
Bp,r
,→ Lip,
σ1 ≤ σ 2
and
Proof: We notice that P (D) is a multiplier of degree −1. Hence, according to proposition
1.3.7, it suffices to prove that the functional
H : (f, g) 7→ f g + 21 ∂x f ∂x g
σ1 × B σ2 in B σ1 −1 .
maps Bp,r
p,r
p,r
The term f g is easy to handle so we focus on the study of ∂x f ∂x g. By virtue of Bony’s
decomposition, we have
∂x f ∂x g = T∂x f ∂x g + T∂x g ∂x f + R(∂x f, ∂x g).
σ1 ×
Proposition 1.4.1 insures that the application (f, g) 7→ T∂x f ∂x g is continuous from Bp,r
σ
2
Bp,r to
σ1 +σ2 −2− p1
• the space Bp,r
if σ1 < 1 + p1 ,
σ2 −1−ε for all ε > 0, if σ = 1 +
• the space Bp,r
1
σ2 −1 if σ = 1 +
• the space Bp,r
1
1
p
1
p
and r > 1,
and r = 1, or σ1 > 1 + p1 .
According to our assumptions on σ1 , σ2 , p and r, we thus can conclude that (f, g) 7→
σ1 × B σ2 in B σ1 −1 .
T∂x f ∂x g maps Bp,r
p,r
p,r
σ2 −1 ,→ L∞ , proposition 1.4.1 readily yields the continuity of (f, g) 7→ T
Since Bp,r
∂x g ∂x f
σ
σ
σ
−1
1
2
1
from Bp,r × Bp,r to Bp,r .
σ1 +σ2 −2− p1
σ1 ×B σ2 in B
Finally, the remainder term maps Bp,r
p,r
p,r
2
that σ1 + σ2 > 2 + max 0, p − 1 .
σ1 −1 ) provided
(and thus in Bp,r
Uniqueness in theorem 3.4.1 is a straightforward corollary of the following proposition.
3.4. ON THE CAUCHY PROBLEM FOR A SHALLOW WATER EQUATION
63
Proposition 3.4.5. Let 1 ≤ p, r ≤ +∞ and s > max(1 + 1/p, 3/2). Suppose that we are given
s ) ∩ C([0, T ]; B s−1 ) 2 two solutions of (CH) with initial data u , v ∈ B s .
(u, v) ∈ L∞ (0, T ; Bp,r
0 0
p,r
p,r
Then we have for every t ∈ [0, T ]:
R
s +kv(τ )kB s
C 0t ku(τ )kBp,r
dτ
p,r
ku(t) − v(t)kBp,r
.
s−1 ≤ ku0 − v0 k s−1 e
Bp,r
Proof: It is obvious that w := v − u solves the transport equation:
∂t w + u∂x w = −w∂x v + P (D) w(u+v) + 12 ∂x w ∂x (u+v) .
According to propositions 3.2.1 and 1.3.7, the following inequality holds true:
Z t Rt
R
C 0t k∂x uk s−1 dτ 0
C
k∂x uk s−1 dτ 0
Bp,r
Bp,r
(3.31)
kw(t)kBp,r
+
C
e τ
s−1 ≤ kw0 k s−1 e
Bp,r
0
× kw∂x vkBp,r
dτ,
s−1 + kB(w, u+v)k s−1
Bp,r
where B stands for the symmetric bilinear function introduced in lemma 3.4.4.
Since s > max( 23 , 1 + p1 ), we thus have
s
kB(w, u+v)kBp,r
+ kvkBp,r
.
s−1 ≤ Ckwk s−1 kukB s
Bp,r
p,r
Plugging this last inequality in (3.31) and applying Gronwall lemma completes the proof.
s−1 the difference between two
Remark. For proving uniqueness, we are led to estimate in Bp,r
s . Owing to the term (∂ u)2 , the additional condition
solutions although both solutions are in Bp,r
x
3
2
s > max( 23 , 1 + p1 ) is thus required. In fact, uniqueness is also in true in B2,1
, see [24]. It
has been discovered recently in [26] that if using the Lagrangian formulation, one can directly
prove uniqueness in a space with the regularity index s and forget about the restriction s >
max( 32 , 1 + p1 ).
3.4.4
The proof of existence
Let us first enumerate the main steps of the proof of theorem 3.4.1.
(i) Construction of approximate solutions of (CH) which are smooth solutions of some linear
transport equation.
s (T ).
(ii) Find a positive T for which these approximate solutions are uniformly bounded in Ep,r
(iii) Prove that the sequence of approximate solutions is a Cauchy sequence in a larger space.
(iv) Check that the limit has indeed the required regularity.
First step: approximate solution
We use a standard iterative process to build a solution. Starting from u0 := 0, we then define
by induction a sequence of smooth functions (un )n∈N by solving the following linear transport
equation:
(
(∂t + un ∂x )un+1 = P (D) (un )2 + 12 (∂x un )2 ,
(Tn )
un+1 |t=0 = un+1
:= Sn+1 ? u0 .
0
∞ , theorem 3.3.1 enables us to show by induction that for all
Since all the data belong to Bp,r
∞ ).
n ∈ N, the above equation has a global solution which belongs to C(R+ ; Bp,r
64
CHAPTER 3. THE TRANSPORT EQUATION
Second step: uniform bounds
According to theorem 3.3.1 and to lemma 3.4.4, we have the following inequality for all n ∈ N :
Z t
n (t)
n (τ )
n+1
CU
−CU
n
2
s
s
(3.32)
ku (t)kBp,r
≤ Ce
ku0 kBp,r
+
e
ku (τ )kBp,r
s dτ
0
with U n :=
Z
0
t
s dτ.
kun (τ )kBp,r
s T < 1 and suppose that
Let us fix a T > 0 such that 2C 2 ku0 kBp,r
s
∀t ∈ [0, T ], kun (t)kBp,r
≤
(3.33)
s
Cku0 kBp,r
s t
1 − 2C 2 ku0 kBp,r
·
Plugging (3.33) in (3.32) yields
!
s
kun+1 (t)kBp,r
≤
≤
1
q
s t
1−2C 2 ku0 kBp,r
s
Cku0 kBp,r
s t
1−2C 2 ku0 kBp,r
s
ku0 kBp,r
+ C 2 ku0 k2Bp,r
s
Rt
dτ
0 (1−2C 2 ku k s τ ) 32
0 Bp,r
,
·
s ). This clearly entails that un ∂ un
Therefore, (un )n∈N is uniformly bounded in C([0, T ]; Bp,r
x
s−1
is uniformly bounded in C([0, T ]; Bp,r ). As the right-hand side of (Tn ) has been shown to be
s ), one can conclude that the sequence (un )
uniformly bounded in C([0, T ]; Bp,r
n∈N is uniformly
s
bounded in Ep,r (T ).
Third step: convergence
s−1 ).
We are going to show that (un )n∈N is a Cauchy sequence in C([0, T ]; Bp,r
We remark that for all (m, n) ∈ N2 , we have
(∂t + un+m ∂x )(un+m+1 −un+1 ) = (un − un+m )∂x un+1 + B(un+m −un , un+m +un ).
s−1 is an algebra yields
Applying theorem 3.3.1 and lemma 3.4.4, and using that Bp,r
n+m+1
n+1
CU n+m (t)
∀t ∈ [0, T ], k(u
−u )(t)kBp,r
kun+m+1
− un+1
s−1 ≤ e
s−1
0
0 kBp,r
Z t
−CU n+m (τ )
n+m
n
n
n+1
n+m
s
s
s
+C
e
ku
− u kBp,r
+ ku kBp,r + ku
kBp,r dτ .
s−1 ku kB
p,r
0
s (T ) and
Since (un )n∈N is uniformly bounded in Ep,r
un+m+1
0
−
un+1
0
=
n+m+1
X
∆q u 0 ,
q=n+2
we finally get a constant CT independent of n, m and such that for all t ∈ [0, T ], we have
Z t
n+m+1
n+1
−n
n+m
n
k(u
− u )(t)kBp,r
2 +
k(u
− u )(τ )kBp,r
.
s−1 ≤ CT
s−1 dτ
0
Arguing by induction, one can easily prove that
ku
n+m+1
−u
n+1
kL∞ (Bp,r
s−1
)
T
n
k
X
(T CT )n+1 m
−(n−k) (T CT )
s ) + CT
≤
ku kL∞
2
.
(B
p,r
T
(n + 1)!
k!
k=0
3.4. ON THE CAUCHY PROBLEM FOR A SHALLOW WATER EQUATION
65
s
As kum kL∞
may be bounded independently of m, we can conclude to the existence of some
T (Bp,r )
new constant CT0 such that
0 −n
kun+m+1 − un+1 kL∞ (Bp,r
.
s−1 ≤ CT 2
)
T
(un )n∈N
s−1 ), whence it converges to some limit
Hence
is a Cauchy sequence in C([0, T ]; Bp,r
s−1 ).
function u ∈ C([0, T ]; Bp,r
last step: conclusion
s (T ) and satisfies (CH). Since (un )
We now have to check that u belongs to Ep,r
n∈N is uniformly
∞
s
bounded in L (0, T ; Bp,r ), Fatou property for Besov spaces guarantees that u also belongs to
s ).
L∞ (0, T ; Bp,r
s−1 ), an interpolation argument
On the other hand, as (un )n∈N converges to u in C([0, T ]; Bp,r
0
s ) for any s0 < s. It is then easy to
insures that convergence actually holds true in C([0, T ]; Bp,r
pass to the limit in (Tn ) and to conclude that u is indeed a solution to (CH).
s ), the right-hand side of the equation
Now, because u belongs to L∞ (0, T ; Bp,r
∂t u + u∂x u = P (D)(u2 + 12 (∂x u)2 )
s ). In the case r < ∞, theorem 3.3.1 enables us to conclude that
also belongs to L∞ (0, T ; Bp,r
s ). Finally, using again the equation, we see that ∂ u is in C([0, T ]; B s−1 ) if r
u ∈ C([0, T ]; Bp,r
t
p,r
s−1 ) otherwise. So finally, u belongs to E s (T ).
is finite, and in L∞ (0, T ; Bp,r
p,r
s , the arguments above give the existence
Remark. If v0 is in a small neighborhood of u0 in Bp,r
s (T ) to (CH) with initial datum v . Proposition 3.4.5 combined with
of a solution v ∈ Ep,r
0
s0 ) ∩
an obvious interpolation ensures continuity with respect to the initial data in C([0, T ]; Bp,r
s0 −1 ) for any s0 < s.
C 1 ([0, T ]; Bp,r
s ) ∩ C 1 ([0, T ]; B s−1 ) when r < +∞ is
The fact that continuity also holds in C([0, T ]; Bp,r
p,r
not obvious but belongs to the mathematical folklore. It may be proved through the use of a
s )∩
sequence of approximate solutions (u )>0 for (CH) which converges uniformly in C([0, T ]; Bp,r
1
s−1
C ([0, T ]; Bp,r ). A viscosity approximation gives the desired property of convergence.
3.4.5
Blow-up criterion and energy conservation
This section is devoted to the proof of theorems 3.4.3 and 3.4.2. Both theorems are based on
the following lemma:
s ) solving (CH) on [0, T )×R
Lemma 3.4.6. Let 1 ≤ p, r ≤ ∞ and s > 1. Let u ∈ L∞ (0, T ; Bp,r
s
with u0 ∈ Bp,r as an initial datum. There exist a constant C depending only on s and p, and
a universal constant C 0 such that for all t ∈ [0, T ), we have
(3.34)
(3.35)
s
ku(t)kBp,r
C
s e
≤ ku0 kBp,r
ku(t)kLip ≤ ku0 kLip eC
Rt
0
ku(τ )kLip dτ
R
0 t
0 k∂x u(τ )kL∞
,
dτ
.
Proof: Applying the last part of proposition 3.2.1 to (CH) and using the fact that P (D) is
a multiplier of order −1 yields
Z t
Rt
Rτ
0
2
s
s +C
e−C 0 k∂x ukL∞ dτ ku2 kBp,r
e−C 0 k∂x ukL∞ dτ ku(t)kBp,r
≤ ku0 kBp,r
s−1 +k(∂x u) k s−1 dτ.
Bp,r
0
As s − 1 > 0, we have, according to proposition 1.4.3,
2
ku2 kBp,r
s−1 + k(∂x u) k s−1 ≤ CkukLip kukB s .
Bp,r
p,r
66
CHAPTER 3. THE TRANSPORT EQUATION
Therefore
e−C
Rt
0 k∂x ukL∞
dτ
Z
s
s
ku(t)kBp,r
≤ ku0 kBp,r
+C
t
Rτ
e−C
0
k∂x ukL∞ dτ 0
0
s kukLip dτ.
kukBp,r
Applying Gronwall lemma completes the proof of (3.34).
By differentiating once equation (CH) with respect to x, and applying the L∞ estimate
for transport equations, we easily prove that
Z t R
R
τ
− 0t k∂x ukL∞ dτ
e
ku(t)kLip ≤ ku0 kLip +
e− 0 k∂x ukL∞ dτ kP (D) u2 + 21 (∂x u)2 kLip .
0
Now, exercise 3.3 guarantees that
kP (D) u2 + 21 (∂x u)2 kLip ≤ C 0 kukLip k∂x ukL∞
for some universal constant C 0 . Hence Gronwall lemma gives inequality (3.35).
R T?
T
s (T ) be such that
We can now prove theorem 3.4.3. Let u ∈ T <T ? Ep,r
0 k∂x u(τ )kL∞ dτ be
R T?
finite. According to inequality (3.35), 0 ku(τ )kLip dτ is also finite. Hence, (3.34) insures that
(3.36)
C
s
s e
∀t ∈ [0, T ? ), ku(t)kBp,r
≤ MT ? := ku0 kBp,r
R T?
0
ku(τ )kLip dτ
< ∞.
Let > 0 be such that 2C 2 MT ? < 1 where C stands for the constant used in the proof of
s () to (CH) with initial datum u(T ? − /2).
theorem 3.3.1. We then have a solution u
e ∈ Ep,r
For the sake of uniqueness, u
e(t) = u(t + T ? − /2) on [0, /2) so that u
e extends the solution u
?
beyond T . We conclude that T ? < Tu?0 and theorem 3.4.3 is proved.
s with s > max(3/2, 1+
Let us now prove theorem 3.4.2. Assume that u0 belongs to H 1 ∩Bp,r
s (T ). We
1/p), and that the corresponding solution u is defined on [0, T ] and belongs to Ep,r
want to prove that the H 1 norm of u is conserved. Clearly, this cannot be done by using the
approximation scheme of theorem 2.7 since it does not conserve the H 1 norm.
An alternative method is to mollify the initial datum u0 and to define un as the maximal
solution of (CH) corresponding to un0 then pass to the limit. This time, we set un0 := ρn ? u0
where (ρn )n∈N stands for a sequence of nonnegative mollifiers.
Owing to the nonnegativity of the mollifiers, one can easily check that
s
s
kun0 kBp,r
≤ ku0 kBp,r
and kun0 kH 1 ≤ ku0 kH 1 .
Following the second step of the proof of theorem 3.4.1, we discover that there exists a constant
C such that un is a solution of (CH) on [0, T̄ ] × R with
s
T̄ := C/kukL∞
T (Bp,r )
s (T̄ ) uniformly.
and un ∈ Ep,r
Now, we also have un0 ∈ H 4 so that, there exists some T n > 0 such that (CH) with data
n
u0 has a solution u
en ∈ C([0, T n ]; H 4 ). By virtue of uniqueness, we actually have u
en ≡ un on
[0, min(T̄ , T n )], and according to lemma (3.4.6) and to Besov embeddings,
C
∀t ∈ [0, min(T̄ , T n )], kun (t)kH 4 ≤ kun0 kH 4 e
Rt
0
s dτ
kun (τ )kBp,r
.
Note that the right-hand side may be bounded independently of n and of t ∈ [0, T̄ ]. Therefore,
arguing as in the proof of the first part of theorem 3.4.3, one can conclude that T n may be
chosen greater than T̄ .
3.4. ON THE CAUCHY PROBLEM FOR A SHALLOW WATER EQUATION
67
Now, the smoothness of un enables us to derive directly from (3.29) that
∀t ∈ [0, T̄ ], kun (t)kH 1 = kun0 kH 1 ≤ ku0 kH 1 .
(3.37)
Therefore, passing to the limit (use e.g. proposition 3.4.5) and using Fatou property for H 1 ,
one eventually gets ku(t)kH 1 ≤ ku0 kH 1 for t ∈ [0, T̄ ].
For proving the reverse inequality, one can solve the equation backward, starting from u(T̄ ).
Then, arguing as above and using uniqueness, one can assert that ku(T̄ − t)kH 1 ≤ ku(T̄ )kH 1 for
s ) . Repeating the argument several times, we finally get ku(t)kH 1 = ku0 kH 1 for
t ≤ C/kukL∞ (Bp,r
T̄
all t ∈ [0, T̄ ].
It is now easy to get equality on [0, T ]. Indeed, arguing as above yields equality on [T̄ , 2T̄ ],
[2T̄ , 3T̄ ], etc. until the whole interval [0, T ] is exhausted.
Let us end this chapter with the statement of a slightly weaker blow-up criterion in the spirit
of the celebrated Beale-Kato-Majda criterion for Euler equations (see for example [3] or [10]):
s with s > max(3/2, 1 + 1/p), we have:
Proposition 3.4.7. Under the assumption that u0 ∈ Bp,r
Z Tu?
0
(3.38)
Tu?0 < +∞ =⇒
ku(τ )kB∞,∞
dτ = +∞.
1
0
Proof: This merely stems from the following logarithmic interpolation inequality:
s
kukLip ≤ C 1 + kukB∞,∞
log e + kukBp,r
1
which holds true whenever s > 1 + 1/p and may be deduced from proposition 1.3.6 and
s−1− 1
s−1
1
the embeddings Ḃ∞,1
∩ L∞ ,→ Lip and Bp,1
,→ B∞,∞ p .
Plugging it into (3.36), we get
Ct C
s
s e
ku(t)kBp,r
≤ ku0 kBp,r
e
Rt
0
kukB 1
∞,∞
s ) dτ
log(e+kukBp,r
.
Therefore, easy calculations lead to
log e + ku(t)k
s
Bp,r
≤ log e + ku0 k
s
Bp,r
Z
+ Ct + C
0
t
s
kukB∞,∞
log e + kukBp,r
dτ.
1
Gronwall lemma thus yields
C R t kuk
dτ
1
s
s
log e + ku(t)kBp,r
≤ log e + ku0 kBp,r
+ Ct e 0 B∞,∞ .
RT
s ). Arguing as in
One can now conclude that if 0 kukB∞,∞
dt < ∞ then u ∈ L∞ (0, T ; Bp,r
1
theorem 3.4.3 completes the proof of the proposition.
Remark. The fact that k∂x u(t)kL∞ may be replaced with the weaker norm k∂x u(t)kB∞,∞
is
0
actually not very sensitive to the structure of the equation. As pointed out above, a similar
criterion may be stated (with the same method) for the incompressible Euler equations, and
also for a number of quasilinear first order equations.
We want to conclude this section with a more precise blow-up statement which is specific to the
Camassa-Holm equation:
s ∩ H 1 with s > max(3/2, 1 + 1/p), we
Proposition 3.4.8. Under the assumption that u0 ∈ Bp,r
have:
Z Tu?
0
?
(3.39)
Tu0 < +∞ =⇒
inf u(τ, x) dτ = −∞.
0
Proof: The reader is referred to [18].
x∈R
68
CHAPTER 3. THE TRANSPORT EQUATION
3.5
Exercises
Exercise 3.1. Let v : R×RN → RN be a smooth time-dependent vectorfield, globally Lipschitz
with respect to the space variable, and ψt,s , the flow of v.
1) Check that ψt,s is defined for all (t, s) ∈ R2 .
Z t
2) Prove that ∀(t, s) ∈ R , ∀x ∈ R , det Dψt,s (x) = exp
div v(τ, ψτ,s (x)) dτ .
2
N
s
3) Prove inequality (3.4) for smooth solutions which belong to L∞ (0, T ; Lp ).
4) Prove estimates in W 1,p spaces for the solutions to the transport equation (T).
Exercise 3.2. Prove inequality (3.10).
Exercise 3.3. Throughout the exercise, the function u is defined over R.
2 )−1 u =
1) Prove that (1 − ∂xx
1
2
e− |·| ∗ u.
2) From the first question, deduce that
2 )−1 u
(1−∂xx
L∞
≤ kukL∞ ,
2 )−1 ∂ (u2 )
(1−∂xx
x
L∞
≤ 2 kukL∞ k∂x ukL∞ ,
2 )−1 ∂ u
(1−∂xx
x
L∞
2 )−1 ∂ 2 (u2 )
(1−∂xx
xx
≤ kukL∞ ,
L∞
≤ 2 kukL∞ k∂x ukL∞ .
2 )−1 . Prove the existence of a universal constant C such that
3) Let P (D) := ∂x (1 − ∂xx
kP (D) u2 + 12 (∂x u)2 kLip ≤ CkukLip k∂x ukL∞ .
Chapter 4
A short insight into compressible
fluid mechanics
4.1
About the model
We first briefly explain how the system of equations for the flow of compressible fluids may be
derived from basic physics.
4.1.1
Physical conservation laws
We assume that the fluid under consideration fills in a time independent open domain D of RN
and may be characterized for every material point x ∈ D at time t ∈ R, by
• its velocity field u := u(t, x),
• its density ρ := ρ(t, x),
• its internal energy e := e(t, x),
• its entropy by unit mass s := s(t, x).
For each subdomain Ω of D, one can define the following physical quantities:
R
• the mass: M (Ω) := Ω ρ dx.
R
• the momentum: P (Ω) := Ω ρu dx.
R
• the energy: E(Ω) := Ω 21 ρ|u|2 + ρe dx.
R
• the entropy: S(Ω) := Ω ρs dx.
Denoting by ψt the flow1 of u and Ωt := ψt (Ω), and assuming that there is no production
or loss of mass, we have the following conservation law:
Z
d
d
M (Ωt ) =
ρ dx = 0.
(4.1)
dt
dt Ωt
For the momentum, we have
(4.2)
1
d
d
P (Ωt ) =
dt
dt
Z
Z
ρu dx =
Ωt
69
(σ · n) dΣ
ρf dx +
Ωt
See the definition in chapter 3.
Z
∂Ωt
70
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
where the first term on the right-hand side represents external volume forces with density f
(such as gravity for instance) and the second term, surface forces.
In the full generality, the strain about x ∈ ∂Ωt is some vector valued function T depending
on t, x and on the unit exterior normal vector n at x. Under the assumption of small strains,
this function may be linearized so that it reduces to T (t, x, n) = σ(t, x) · n for some strain
tensor σ. In the absence Rof mass couples, an assumption that we are going to do from now on,
the angular momentum Ωt x ∧ (ρu)(t, x) dx is also conserved and one can prove that σ is a
symmetric tensor.
As regards the energy conservation, we have
Z
Z
Z
Z
d
|u|2 d
(4.3)
E(Ωt ) =
dx =
ρf ·u dx +
(σ·n)·n dΣ −
q·n dΣ
ρ e+
dt
dt Ωt
2
Ωt
∂Ωt
∂Ωt
where q is the heat flux vector.
Finally, introducing the temperature T, the entropy balance writes:
Z
Z d
d
q · n
(4.4)
ρs dx ≥ −
dΣ.
S(Ωt ) =
dt
dt Ωt
T
∂Ωt
4.1.2
The full model
From the global conservation laws (4.1), (4.2), (4.3) and (4.4), one can obtain a system of PDE’s
involving ρ, u, e, and s. This is a mere consequence of the following (formal) lemma.
Lemma 4.1.1. Let Ω be an open subdomain of D, ψ be the flow of u and Ωt := ψt (Ω). The
following equalities hold true for all scalar function b:
Z
Z
d
b dx =
∂t b + div bu dx,
dt Ωt
ZΩt
Z
=
∂t b dx +
(b u · n) dΣ.
Ωt
∂Ωt
Combining lemma 4.1.1 with relations (4.1), (4.2), (4.3) and (4.4), we obtain the following
equations for the evolution of compressible flows:
(4.5)
∂t ρ + div(ρu) = 0,
(4.6)
∂t ρu + div(ρu ⊗ u) = ρf + div σ,
|u|2 |u|2 ∂t ρ e +
+ div ρ e +
u = ρf · u + div(σ · u) − div q,
2
2
q
∂t ρs + div(ρsu) ≥ − div
.
T
(4.7)
(4.8)
From a mathematical viewpoint, the above equations are too general to be handled. In the next
section, we shall restrict ourselves to particular fluids.
4.1.3
Simplifying assumptions
We shall first assume that the fluid is Newtonian, namely
• the tensor σ is a linear function of Du, invariant under rigid transforms,
• the fluid is isotropic (i.e the physical quantities depend only on (t, x)).
4.1. ABOUT THE MODEL
71
As a consequence, it may be shown that σ has the following form:
σ = (λ div u − p)Id + 2µD(u)
where the scalar function p = p(t, x) is the pressure, λ and µ are the viscosity coefficients and
D(u) := 12 (Du + tDu) is the deformation tensor. One can also introduce the viscous stress
tensor :
τ := λ div u Id + 2µD(u).
Finally, we shall assume that Fourier law is satisfied, namely q = −k∇T.
In full generality, the coefficients k, λ and µ may depend on ρ and T. For the sake of
simplicity however, we shall assume from now on that they are constant functions. We thus
obtain the following system of equations:
(4.9)
∂t ρ + div(ρu) = 0,
(4.10)
∂t ρu + div(ρu ⊗ u) − µ∆u − (λ + µ)∇ div u + ∇p = ρf,
2
2
+ div ρ e + |u|2 u + div pu − k∆T = ρf · u + div(τ · u) − div q,
∂t ρ e + |u|2
∂t ρs + div(ρsu) ≥ k div ∇T
.
T
(4.11)
(4.12)
In dimension N, the above system has N + 2 equations (and an inequation) for N + 5 unknown
scalar functions (namely ρ, u1 , · · · , uN , p, e, T and s). In order to close the system, three
additional equations are needed. As regards the entropy s, we assume that the so-called Gibbs
relation is satisfied:
1
T ds = de + p d
.
ρ
We still need two other equalities (often called state relations) involving p, ρ, e, s and T. One
can assume for example that p = P (ρ, T ) and e = ε(ρ, T ) for some given functions P and ε
which may be determined from further physical considerations depending on the nature of the
fluid. Combining Gibbs relation with the mass, momentum and energy equations, we get
∂t ρs + div(ρsu) = k div
∇T T
+
τ : D(u)
|∇T |2
+k
·
T
T2
Hence, according to the entropy inequality, we must have
τ : D(u) + k
|∇T |2
≥ 0.
T
This yields the following additional constraints on λ, µ and k :
(4.13)
k ≥ 0,
µ≥0
and 2µ + N λ ≥ 0.
Let us give some important examples:
• Monoatomic gases in dimension N = 3 satisfy 2µ + 3λ = 0. For more general fluids,
2µ + 3λ may be positive.
• Inviscid fluids are such that µ = λ = 0.
• By definition, a non conducting fluid is such that k = 0.
72
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
4.1.4
Barotropic fluids
From now, we shall consider a simplified model for compressible fluids, the so-called barotropic
Navier-Stokes equations:
∂t ρ + div ρ = 0,
(4.14)
∂t ρu + div(ρu ⊗ u) − µ∆u − (λ + µ)∇ div u + ∇p = ρf
with p = P (ρ) for some suitably smooth function P.
The above system may be derived from the general model (4.10), (4.11), (4.12) under the
assumption that s is a constant and that the fluid is non conducting. Note that in the viscous
case (that we are going to consider in the next sections) the assumption of constant entropy is
somewhat inconsistent with physics since the term τ : D(u) appearing in the entropy balance
may be positive. As a matter of fact, it turns out that, from a mathematical viewpoint, the
barotropic (or isentropic) model keeps most of the features of the full model.
4.2
Local well-posedness in critical spaces
From now on, we restrict ourselves to viscous fluids (i.e µ > 0) and we further assume (for the
sake of simplicity only) that λ+µ = 0 2 . We focus on flows whose density is a small perturbation
of a positive constant, say 1 to simplify the notation.
Denoting a = ρ − 1, the barotropic system for smooth enough solutions with positive density
thus reduces to
(
∂t a + u · ∇a + (1 + a) div u = 0,
(N SC)
µ
∆u + u · ∇u + ∇g = f,
∂t u − 1+a
where g = G(a) stands for the chemical potential expressed in term of a. The function G is
assumed to be conveniently smooth and, with no loss of generality, vanishes at 0.
The conducting thread for finding an appropriate framework is the same as in the incompressible case (see chapter 2): using functional spaces which have the same scaling invariance as
the system, if any.
For the velocity, one can expect the functional framework to be the same as the one used
in chapter 2. Now, one can notice that (NSC) is invariant for all λ > 0 by the rescaling
(a, u) 7→ (aλ , uλ ) with
aλ (t, x) = a(λ2 t, λx)
and
uλ (t, x) = λu(λ2 t, λx)
provided that the chemical potential has been changed into λ2 g.
This motivates us to solve system (NSC) in a functional space whose norm in invariant for all
λ (up to a an irrelevant constant) by the above transform. If we restrict to homogeneous Besov
N
p
N
p
−1
spaces, this suggests us to consider a0 ∈ Bp11,r1 and u0 ∈ Bp22,r2 for some 1 ≤ p1 , p2 , r1 , r2 ≤ ∞.
In order to avoid the appearance of vacuum however, a L∞ bound on the density is needed.
Hence, we shall assume that r1 = 1 which, in view of Besov embeddings, provides a L∞ control
for free. For technical reasons, we shall further assume that r2 = 1 and that p1 = p2 = p.
Finally, we thus expect to get existence in the following functional space:
n
o
N
N
eT (Ḃ p ) × C
eT (Ḃ p −1 ) N | kuk
ETp := (a, u) ∈ C
<
∞
N
p,1
p,1
p +1
L1T (Ḃp,1
)
eT (Ḃ s ) := C([0, T ]; Ḃ s ) ∩ L
e ∞ (Ḃ s ) for s ∈ R.
where C
p,1
p,1
p,1
T
2
Actually, the very same results may be proved under the weaker assumption that µ > 0 and λ + 2µ > 0 (see
[20])
4.2. LOCAL WELL-POSEDNESS IN CRITICAL SPACES
73
Let us now state our well-posedness result:
Theorem 4.2.1. Assume that the space dimension is N ≥ 2 and that 1 ≤ p < 2N. There exists
N
N
−1
−1
N
p
p
p
such
a positive constant ε such that for all u0 ∈ Ḃp,1
, f ∈ L1loc (R+ ; Ḃp,1
) and a0 ∈ Ḃp,1
that ka0 k Np ≤ ε then there exists a positive time T such that (N SC) has a solution (a, u) on
Ḃp,1
RN which
[0, T ] ×
belongs to ETp .
Besides there exists a positive constant η > 0 such that uniqueness holds true in ETp whenever
kak
≤ η if N ≥ 3 and p < N, or kak
≤ η if N = 2 or p = N.
N
N
p
p
L∞
T (Bp,1 )
4.2.1
e ∞ (B )
L
p,1
T
The existence proof
Existence in theorem 4.2.1 may be obtained by mean of the following form of Schauder-Tychonoff
fixed point theorem (stated by M. Hukuhara in [29]):
Theorem (Hukuhara). Let K be a convex subset of a locally convex topological linear space E,
and Φ be a continuous self-mapping of K. If besides Φ(K) is contained in a compact subset of
K then Φ has a fixed point in K.
Let us briefly enumerate the main steps of the proof.
In the first step, we show that our existence problem amounts to finding a fixed point for
some map Φ. In the next two steps, we state various a priori estimates for Φ which will help
us to find an appropriate functional framework for applying Hukuhara’s theorem. In the fourth
step, we show that Hukuhara’s theorem indeed applies. This yields existence in a space slightly
larger than ETp (time continuity is missing). In the fifth step, we show that the constructed
solution actually belongs to ETp . The last step is devoted to uniqueness (which actually relies
on independent arguments).
First step: construction of the functional Φ
Introducing e
a := a/(1 + a) and uL the solution to
∂t uL − µ∆uL = f,
uL |t=0 = u0 ,
we have to solve
∂t a + u · ∇a = −(1 + a) div u,
a∆u − ∇g
∂t u − µ∆u = −u · ∇u − µe
with initial data a|t=0 = a0 and u|t=0 = 0, and u := uL + u.
This motivates us to introduce the functional Φ : (b, v) 7→ (Φ1 (b, v), Φ2 (b, v)) defined by
v
Φ1 (b, v)(t) := a0 ◦ ψ0,t
−
t
Z
v
(1 + b) div v ◦ ψτ,t
dτ
with v := uL + v,
0
v
where ψs,s
0 stands for the flow of v, and
Z
Φ2 (b, v)(t) := −
0
with eb := b/(1 + b) and g := G(b).
t
eµ(t−τ )∆ v · ∇v + µeb∆v + ∇g dτ
74
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
Second step: a priori estimates
We shall prove that for suitably small T and a0 , the functional Φ has a fixed point in the
Banach space
N
N
N
−1
+1 N
p
p
p
∞
1
e
e p := L
e∞
.
)
×
L
(
Ḃ
)
∩
L
(
Ḃ
E
(
Ḃ
T
T
T
p,1
p,1
p,1 )
T
e p such that
Let (R, η) ∈ (0, 1)2 to be fixed hereafter. We denote by BTR,η the set of (b, v) in E
T
kbk
N
e ∞ (Ḃ p )
L
p,1
T
≤R
and kvk
N −1
e ∞ (Ḃ p
L
p,1
T
+ µkvk
N +1
p
L1T (Ḃp,1
)
≤ η.
)
In what follows, we assume that R has been chosen so small as to satisfy3
kbk
N
e ∞ (Ḃ p )
L
p,1
T
1
2
≤ R =⇒ |b(t, x)| ≤
We claim that if R0 := ka0 k
N
p
Ḃp,1
(t, x) ∈ [0, T ] × RN .
for all
, R, η and T are sufficiently small then the function Φ maps
BTR,η to BTR,η .
Indeed, taking advantage of proposition 3.2.1, we get
Z T
CV (T )
kΦ1 (b, v)k
ka0 k Np +
≤e
e−CV (t) k(1 + b) div vk
N
p
e ∞ (Ḃ )
L
p,1
T
Rt
with V (t) := exp( 0 kDvk
Ḃp,1
N
p
Ḃp,1
0
dt
dτ ).
N
p
Ḃp,1
Hence, using proposition 1.4.3,
kΦ1 (b, v)k
N
e ∞ (Ḃ p )
L
p,1
T
CV (T )
≤e
Z
ka0 k
N
p
Ḃp,1
T
+
−CV (t)
e
kΦ1 (b, v)k
N
e ∞ (Ḃ p )
L
p,1
T
(1 + kbk
0
Using the definition of R0 and V, and the fact that kbk
(4.15)
N
p
L∞
T (Ḃp,1 )
N
p
Ḃp,1
)kdiv vk
dt .
N
p
Ḃp,1
≤ R, we end up with
≤ eCV (T ) R0 + eCV (T ) − 1 1 + R .
Next, according to theorem 2.2.3, we have the following inequality for Φ2 (b, v):
Z T
kΦ2 (b, v)k
kv·∇vk Np −1 +µkeb∆vk Np −1 +kG(b)k
N −1 +µkΦ2 (b, v)k
N +1 ≤ C
p
p
e ∞ (Ḃ
L
p,1 )
T
L1T (Ḃp,1 )
Ḃp,1
0
Ḃp,1
Appealing to propositions 1.5.11, 1.5.12 and 1.5.13, we thus get4
Z T
kΦ2 (b, v)k
kvk2 N + kbk
N −1 + µkΦ2 (b, v)k
N +1 ≤ C
p
p
e ∞ (Ḃ
L
p,1 )
T
e 1 (Ḃ
L
p,1 )
T
0
N
p
Ḃp,1
p
Ḃp,1
N
p
Ḃp,1
1 + µk∆vk
N −1
p
Ḃp,1
whence
(4.16)
kΦ2 (b, v)k
N −1
e ∞ (Ḃ p
L
p,1
T
)
+µkΦ2 (b, v)k
N +1
p
e1
LT (Ḃp,1 )
≤ C kuL k2
+ kvk2
N
p
L2T (Ḃp,1
)
N
N
p
p
∞
∞
N
e∞
Remind that L
T (Ḃp,1 ) ,→ L (0, T ; Ḃp,1 ) ,→ L (0, T × R ).
4
Here, N ≥ 2 and p < 2N are needed
3
N
p
L2T (Ḃp,1
)
+ R(T + µV (T )) .
dt.
dt,
4.2. LOCAL WELL-POSEDNESS IN CRITICAL SPACES
75
Let us assume that T is so small as to satisfy
T + µkuL k
N +1
p
L1T (Ḃp,1
)
1
≤η
and µ 2 kuL k
N
p
)
L2T (Ḃp,1
≤ η.
As (b, v) is in BTR,η , we also easily get by interpolation that
1
µ 2 kvk
N
p
L2T (Ḃp,1
)
≤ η.
Hence, plugging all these inequalities in (4.15) and (4.16), we end up (up to a change of C ) with
kΦ1 (b, v)k
kΦ2 (b, v)k
N
e ∞ (Ḃ p )
L
p,1
T
N −1
e ∞ (Ḃ p
)
L
p,1
T
≤e
Cη
µ
R0 + 2(e
+ µkΦ2 (b, v)k
Cη
µ
N +1
p
)
L1T (Ḃp,1
− 1),
≤ C(
η2
+ ηR).
µ
Now, if we fix a positive R ≤ min(1, (2C)−1 ), and assume that R0 ≤ R/4 and that η is so small
as to satisfy
Cη
Cη
1
≤
and 4(e µ − 1) ≤ R,
µ
2
then the function Φ maps BTR,η in BTR,η .
Third step: time derivatives
The compactness of the map Φ will be supplied by the following lemma:
Lemma 4.2.2. Let (b, v) be in BTR,η with T, R and η chosen according to the previous step.
Denote a := Φ1 (b, v) and u := Φ2 (b, v).
N
−1
N
2
−1
N
−2+α
p
p
p
Then ∂t a ∈ L2 (0, T ; Ḃp,1
) and ∂t u ∈ L 1+α (0, T ; Ḃp,1
+ Ḃp,1
) for any α ∈ [−1, 1] such
R,η
that α > max 2 1− Np , 2−N . Besides, there exists a constant CT,α depending only on R, η,
T, α, N, p, µ and G and such that
k∂t ak
N −1
p
L2T (Ḃp,1
)
+ k∂t uk
N −1
N −2+α
2
p
p
LT1+α (Ḃp,1
+Ḃp,1
)
R,η
≤ CT,α
.
Proof: Let us first focus on ∂t a. By definition of a, we have
(4.17)
∂t a = −v · ∇a − (1 + b) div v
N
with v := uL + v.
N
p
p
Because v belongs to L2 (0, T ; Ḃp,1
), a, b are in L∞ (0, T ; Ḃp,1
) and p < 2N, propositions
1.5.11 and 1.5.12 insure that
+
kbk
k∂t ak
kvk
N −1 ≤ C 1 + kak
N
N
N ,
p
p
p
p
L2T (Ḃp,1
L∞
T (Ḃp,1 )
)
L∞
T (Ḃp,1 )
whence, since (a, u) and (b, v) are in BTR,η ,
k∂t ak
1
N −1
p
L2T (Ḃp,1
)
≤ C(1 + 2η)ηµ− 2 .
Let us now consider ∂t u. We use that
(4.18)
∂t u = µ∆u − µeb∆v − v · ∇v − ∇g.
L2T (Ḃp,1 )
76
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
N
N
+1
−1
p
p
) ∩ L∞ (0, T ; Ḃp,1
), interpolation yields
As u and v are in L1 (0, T ; Ḃp,1
N
2
−2+α
p
∆u, ∆v ∈ L 1+α (0, T ; Ḃp,1
)
for any
α ∈ [−1, 1].
N
p
Because b and thus eb are in L∞ (0, T ; Ḃp,1
) (use proposition 1.5.13), we gather that the
N
2
−2+α
p
first two terms in the right-hand side of (4.18) are in L 1+α (0, T ; Ḃp,1
) (provided that
2N
α > max 2 − N, 2 − p for the second one) with a bound depending only on R, η, µ,
N, α and p.
N
2
N
−2+α
p
p
) and ∇v ∈ L α (0, T ; Ḃp,1
Next, since v ∈ L2 (0, T ; Ḃp,1
2
1+α
), propositions 1.5.11 and 1.5.12
N
p
−2+α
(0, T ; Ḃp,1
)
insure that v · ∇v belongs to L
under the same condition on α. Finally,
using proposition 1.5.13, it is easy to see that the last term in the right-hand side of (4.18)
N
−1
p
is in L∞ (0, T ; Ḃp,1
) and may be bounded in terms of R, µ, T, N, p, α and G.
Fourth step: the fixed point argument
Let us introduce the functional space
N
N
N
+1 N
p
p
e∞
e ∞ p −1
YTp := L
T (Ḃp,1 ) × LT (Ḃp,1 ) ∩ MT (Ḃp,1 )
N
N
+1
+1
p
p
) stands for the space of bounded measures on [0, T ] with values in Ḃp,1
.
where MT (Ḃp,1
p
The space YT endowed with the norm
Z T
+
kbk
+
dkv(t)k Np +1
k(b, v)kY p := kbk
N
N −1
p
p
T
e ∞ (Ḃ )
L
p,1
T
e ∞ (Ḃ
L
p,1
T
)
Ḃp,1
0
YTp
is a Banach space. Besides, we notice that
is the dual space of
N
N
−1 N
−N
p
p
e 1T (Ḃ −0 p ) × L
e 1T (Ḃ 1−
X p := L
)
)
+
C([0,
T
];
Ḃ
0
0
T
p ,∞
p ,∞
p ,∞
s ) stands for the completion of S([0, T ] × Rn ) for the norm of L
s ), and
e 1 (Ḃq,∞
e 1 (Ḃq,∞
where L
T
T
p
5
that XT is a separable Banach space (see exercise 4.2).
We claim that for R, T and η chosen according to the previous steps, Hukuhara’s theorem
applies with the functional space YTp endowed with the weak star topology and the function Φ
restricted to the set
R,η
R,η
BT,α
:= (b, v) ∈ BTR,η , k∂t bk
N −1 + k∂t vk
N −1
N −2+α ≤ C
2
T,α
p
p
p
L2T (Ḃp,1
)
LT1+α (Ḃp,1 +Ḃp,1
)
R,η
with max 2−N, 2(1− Np ), −1 < α < 1 and CT,α
defined in lemma 4.2.2.
R,η
(i) Properties of BT,α
and YTp .
On one hand, since YTp is the dual space of a Banach space, we gather that YTp endowed
with the weak star topology is a locally convex topological linear space.
R,η
On the other hand, BT,α
is obviously a convex subset of YTp .
R,η
R,η
Finally, according to lemma 4.2.2, we have Φ(BTR,η ) ⊂ BT,α
. Since BT,α
⊂ BTR,η , it is clear
R,η
.
that Φ is a self-map of BT,α
5
Having the property that the space YTp is the dual of some separable Banach space turns out to be crucial for
N
+1
N
+1
p
p
applying Hukuhara’s theorem, this is the reason why we take MT (Ḃp,1
) rather than L1 (0, T ; Ḃp,1
) above.
4.2. LOCAL WELL-POSEDNESS IN CRITICAL SPACES
77
(ii) Continuity of Φ.
R,η
is a bounded subset of YTp and YTp is the dual space of a separable Banach
Since BT,α
space, it suffices to state the sequential continuity of Φ for the weak star topology.
N
R,η
satisfies (bn , v n ) * (b, v) weak ∗ .
So let us assume that (bn , v n )n∈N ∈ BT,α
Let (an , un ) := Φ(bn , v n ). Since (an , un ) belongs to BTR,η for all n ∈ N, there exists a
distribution (a, u) verifying
kak
(4.19)
N
e ∞ (Ḃ p )
L
p,1
T
≤R
and kuk
N −1
e ∞ (Ḃ p
L
p,1
T
+ µkuk
N +1
p
MT (Ḃp,1
)
and such that, up to a subsequence, we have (an , un ) * (a, u)
≤η
)
weak ∗ .
Hence it is only a matter of showing that (a, u) = Φ(b, v).
The result will be obtained by “passing to the limit” in the equation satisfied by (an , un ).
To make this possible, we first have to prove that sequences (an , un )n∈N and (bn , v n )n∈N
converge in a stronger sense.
Let us first focus on (bn , v n )n∈N .
R,η
Because (bn , v n )n∈N belongs to BT,α
, Hölder inequality insures that sequence (bn )n∈N
1
N
−1
N
1−α
N
−1
p
p
p
(resp. (v n )n∈N ) is bounded in the space C 2 ([0, T ]; Ḃp,1
) (resp. C 2 ([0, T ]; Ḃp,1
+Ḃp,1
+
N N
N
−2+α
−1
p
p
p
)). The sequence (bn )n∈N of Ḃp,1
+ Ḃp,1
valued functions is thus uniformly
Ḃp,1
equicontinuous on [0, T ]. A slight generalization of proposition 1.4.6 to homogeneous spaces
N
N
N
−1 p
p
p
(see exercise 1.19) guarantees that the map z 7→ φz is compact from Ḃp,1
to Ḃp,1
+Ḃp,1
for all φ ∈ Cc∞ . Combining Ascoli’s theorem and Cantor diagonal process, we gather
N N
−1
p
p
+ Ḃp,1
: there exists a
that (bn )n∈N has a convergent subsequence in C [0, T ]; Ḃp,1
loc
subsequence of (bn )n∈N and some b0 ∈ Sh0 such that for all φ ∈ Cc∞ , (φbn )n∈N tends to φb0
N
−1
N
p
p
+ Ḃp,1
).
in C([0, T ]; Ḃp,1
A similar argument provides us with a subsequence of (v n )n∈N and some v 0 ∈ Sh0 such
N
N
−1
−2+α p
p
+ Ḃp,1
.
that for all φ ∈ Cc∞ , sequence (φv n )n∈N tends to φv 0 in C [0, T ]; Ḃp,1
Of course, since by assumption we have (bn , v n ) * (b, v) weak ∗, we actually have b0 = b
and v 0 = v. Arguing by interpolation, we conclude that for all φ ∈ Cc∞ and ε ∈ (0, 1), the
whole sequence (φbn , φv n )n∈N tends to (φb, φv) in
N
N
N
N
N
−ε
−1
−1−ε +1−ε N
p
p
p
p
p
C([0, T ]; Ḃp,1
∩ L1 (0, T ; Ḃp,1
+ Ḃp,1
) × C [0, T ]; Ḃp,1
+ Ḃp,1
) .
R,η
, the
Let us now focus on the convergence of (an , un )n∈N . Since Φ is a self-map of BT,α
n
n
sequence (a , u )n∈N is also uniformly bounded in the space
1−α
N
N
N
N
−1
−1
−2+α N
1
p
p
p
p
C 2 ([0, T ]; Ḃp,1
+ Ḃp,1
) × C 2 ([0, T ]; Ḃp,1
+ Ḃp,1
) .
Arguing as above, we thus gather that, up to extraction, we have (an , un ) → (a, u) in
N
N
N
N
N
−ε
−1
−1−ε +1−ε N
p
p
p
p
p
C([0, T ]; Ḃp,1
+Ḃp,1
)× C [0, T ]; Ḃp,1
+ Ḃp,1
∩L1 (0, T ; Ḃp,1
)
for all ε ∈ (0, 1).
Now, denoting un := un + uL and v n := v n + uL , we have for all n ∈ N,

n
n
n
n
n
 ∂t a + v · ∇a = −(1 + b ) div v n ,
b
n
n
n
n
∂t u − µ∆u = −(v · ∇v + µ 1+bn ∆v n + ∇ G(bn ) ,
(4.20)
 n
a|t=0 = a0 ,
un|t=0 = 0.
78
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
The properties of strong convergence which have been proved above enable us to pass to
the limit in the system. We get

 ∂t a + v · ∇a = −(1 + b) div v,
b
∆v + ∇ G(b) ,
∂t u − µ∆u = −(v · ∇v + µ 1+b
(4.21)

u|t=0 = 0.
a|t=0 = a0 ,
Since (b, v) belongs to BTR,η , uniqueness in theorems 3.3.1 and 2.2.3 ensures that (a, u) =
Φ(b, v) as wanted.
(iii) Compactness of Φ.
R,η
has a
It suffices to state that any sequence (an , un ) := Φ(bn , v n ) with (bn , v n ) ∈ BT,α
R,η
.
convergent subsequence in BT,α
Since in particular (an , un ) belongs to BTR,η for all n, there exists some distribution (a, u)
satisfying (4.19) and such that (an , un ) * (a, u) weak ∗. Proving that u actually belongs
N
+1
p
to L1 (0, T ; Ḃp,1
) is the main difficulty. For that, we use the fact that for all n ∈ N, we
have
∂t un − µ∆un = − v n · ∇v n + µebn ∆v n + ∇G bn .
Since (bn , v n ) ∈ BTR,η , it is easy to show that the right-hand side is uniformly bounded
N
−1
p
). Hence, according to theorem 2.2.3, there exists some constant C such
in L1 (0, T ; Ḃp,1
that for all n ∈ N, we have
k∂t un − µ∆un k
N −1
p
L1T (Ḃp,1
+ kun k
)
N −1
e ∞ (Ḃ p
L
p,1
T
≤ C.
)
N
−1
p
Taking advantage of inequality (2.5) and of the definition of the Besov space
P Ḃp,1 we
thus get, up to a change of C a nonnegative sequence (cq )q∈Z satisfying
cq ≤ 1 and
such that
X 1 − e−κµ(T2 −T1 )22q kun k
cq
N +1 ≤ C
p
κµ
)
L1 (T1 ,T2 ;Ḃp,1
q
whenever 0 ≤ T1 ≤ T2 ≤ T.
N
+1
p
In other words, the sequence (un )n∈N of Ḃp,1
valued functions is equiintegrable on
[0, T ]. Hence, according to Dunford-Pettis theorem, the limit function u belongs to
N
+1
p
L1 (0, T ; Ḃp,1
).
Finally, by making use of the uniform bounds in
for (∂t
an , ∂
t
un )
N
−1
p
L2 (0, T ; Ḃp,1
×
2
1+α
LT
N
−1
p
N
−2+α
p
(Ḃp,1 +Ḃp,1
N
)
R,η
and taking one more extraction if needed, we conclude that (a, u) ∈ BT,α
.
R,η
Now, Hukuhara’s theorem applies. Denoting by (a, u) a fixed point of Φ in BT,α
and setting
N
N
e ∞ (Ḃ p ) × L
e ∞ (Ḃ p −1 ) ∩
u := uL + u, we thus have proved the existence of a solution (a, u) in L
p,1
p,1
T
T
N
+1 N
p
1
for (NSC).
LT (Ḃp,1 )
Fifth step: time continuity
In order to complete the existence part in theorem 4.2.1, we still have to prove that a ∈
N
N
−1
p
p
C([0, T ]; Ḃp,1
) and that u ∈ C([0, T ]; Ḃp,1
).
4.2. LOCAL WELL-POSEDNESS IN CRITICAL SPACES
N
N
79
N
e ∞ (Ḃ p −1 )∩L1 (0, T ; Ḃ p +1 ), we readily check that the right-hand
e ∞ (Ḃ p ) and u ∈ L
As a ∈ L
p,1
p,1
p,1
T
T
N
p
). Therefore, theorem 3.3.1 insures
side of the density equation in (NSC) belongs to L1 (0, T ; Ḃp,1
N
p
).
that a ∈ C([0, T ]; Ḃp,1
N
−1
p
Similarly, the right-hand side of the velocity equation is in L1 (0, T ; Ḃp,1
), so theorem 2.2.3
N
−1
p
yields u ∈ C([0, T ]; Ḃp,1
).
Last step: uniqueness
Uniqueness in theorem 4.2.1 stems from the following proposition.
Proposition 4.2.3. Assume that N ≥ 2 and 1 ≤ p ≤ N. Let (a1 , u1 ) and (a2 , u2 ) be two
N
N
−1
p
p
× (Bp,1
solutions in ETp of (N SC) with the same initial data (a0 , u0 ) ∈ Ḃp,1
)N and external
N
−1
p
). There exists a constant η ∈ (0, 1) such that if
force f ∈ L1 (0, T ; Bp,1
(4.22)
ka1 k
N
≤η
in the case
N ≥3
and
(4.23)
ka1 k
N
≤η
in the case
N =2
or
p
L∞
T (Bp,1 )
e ∞ (B p )
L
p,1
T
p < N,
p = N,
then (a2 , u2 ) ≡ (a1 , u1 ) on [0, T ].
Proof: Throughout the proof, we denote e
ai := ai /(1+ai ).
Let us write a system of equations for (δa, δu) := (a2 −a1 , u2 −u1 ). We have
P
∂t δa + u2 · ∇δa = 3i=1 δFi ,
P
(4.24)
∂t δu − µ∆δu = 5i=1 δGi ,
δF1 := −δu · ∇a1 ,
with
δF2 := −δa div u2 ,
δF3 := −(1 + a1 ) div δu,
δG1 = −u2 · ∇δu,
δG2 = −δu · ∇u1 ,
δG3 = −µ e
a2 − e
a1 ∆u2 ,
δG4 = −µe
a1 ∆δu,
δG5 = −∇(G(a2 ) − G(a1 )).
Owing to the term δF1 in the first equation of (4.24), one can expect to be able to bound
N
N
−1
p
p
δa in L∞ (0, T ; Ḃp,1
) rather than in L∞ (0, T ; Ḃp,1
). This loss of one derivative will induce
also a loss of one derivative when bounding δu and then one will be in troubles for bounding
δG if p > N. This is the reason why we need a stronger condition on p for uniqueness
than for existence.
Let us first treat the case N ≥ 3 and 1 ≤ p < N which is easier6 . Uniqueness is going
to be proved in the following functional space (defined below for T̄ > 0, N ≥ 3 and
1 ≤ p < N ):
N
N
N −1
−2
N
p
p
p
∞
1
FT̄p := L∞
(
Ḃ
.
)
×
L
)
∩
L
(
Ḃ
(
Ḃ
p,1
p,1
p,1 )
T̄
T̄
T̄
Of course, we first have to check that (δa, δu) belongs to FTp . For δa, this is easy beN
−1
p
cause, according to lemma 4.2.2, we have ∂t ai ∈ L2 (0, T ; Ḃp,1
). Hence ai − a0 is in
N
−1
p
C([0, T ]; Ḃp,1
).
6
Of course, in view of Besov embeddings, we have EpT ,→ EpT0 for p0 ≥ p hence, if one makes the slightly
N
N
p
p
∞
e∞
stronger assumption that (4.22) holds with L
T (Ḃp,1 ) instead of LT (Ḃp,1 ) then the general case is a consequence
of the endpoint case p = N.
80
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
Dealing with δu is slightly more involved. Let ui := ui − uL with uL the solution to the
following linear heat equation:
∂t uL − µ∆uL = f − ∇(G(a0 )),
(4.25)
uL |t=0 = u0 .
We obviously have ūi (0) = 0 and
∂t ui − µ∆ui = −µe
ai ∆ui − ui · ∇ui − ∇ G(ai ) − G(a0 ) .
N
−1
p
Since p < N and N ≥ 3, and ai ∈ L∞ (0, T ; Ḃp,1
), it is now easy to check that the
N
N
N
−2
p
e ∞ (B p −2 ) ∩ L
e 1 (B p )
right-hand side belongs to L2 (0, T ; Ḃp,1
). Hence ui belongs to L
p,1
p,1
T
T
according to theorem 2.2.3, and one can now conclude that (δa, δu) belongs to FTp .
Next, applying proposition 3.2.1 to the first equation of (4.24), we get for T̄ ≤ T :
Cku2 k
kδak
N −1
p
L∞
)
(Ḃp,1
T̄
≤e
N +1
p
L1 (Ḃp,1
)
T̄
kδF k
N −1
p
L1T̄ (Ḃp,1
.
)
Propositions 1.5.11 and 1.5.12 yield
kδF1 k
N −1
p
L1T̄ (Ḃp,1
kδF2 k
N −1
p
L1T̄ (Ḃp,1
. k∇a1 k
)
. kδak
)
kδF3 k
1
.
N −1
p
L1T̄ (Ḃp,1
)
N
N −1
p
L∞
(Ḃp,1
)
T̄
N −1
p
L∞
(Ḃp,1
T̄
1
+ ka k
kδuk
N
p
L1T̄ (Ḃp,1
)
2
kdiv u k
,
N
p
L1T̄ (Ḃp,1
)
)
N
p
L∞
(Ḃp,1
)
T̄
kdiv δuk
,
N −1
p
L1T̄ (Ḃp,1
.
)
+1
p
Hence, as u2 ∈ L1 (0, T ; Ḃp,1
), there exists some constant CT independent of T̄ such
that
kδak
(4.26)
N −1
p
L∞
(Ḃp,1
T̄
with X(T̄ ) := CT ku2 k
N
p
L1T̄ (Ḃp,1
)
≤ X(T̄ )kδak
N −1
p
L∞
(Ḃp,1
T̄
)
+ CT kδuk
)
N
p
L1T̄ (Ḃp,1
)
,
.
Next, applying theorem 2.2.3 to the second equation of (4.24) yields
kδuk
N −2
e ∞ (Ḃ p
L
p,1
T̄
+ kδuk
N
p
L1T̄ (Ḃp,1
)
)
5
X
.
kδGi k
N −2
p
L1T̄ (Ḃp,1
i=1
.
)
N
p
We notice that, as Ḃp,1
(RN ) ,→ C(RN ), we have ai ∈ C([0, T ]×RN ). Hence if η has been
chosen small enough in (4.22) so that |a1 (t, x)| ≤ 14 on [0, T ] then we have, for small
enough T̄ ,
1
for i = 1, 2 and t ∈ [0, T̄ ].
(4.27)
ai (t) L∞ ≤
2
Therefore applying propositions 1.5.11, 1.5.12 and 1.5.13 yields (remind that p < N and
N ≥ 3 has been assumed)
kδG1 k
kδG2 k
kδG3 k
kδG4 k
kδG5 k
N −2
p
L1T̄ (Ḃp,1
)
N −2
p
L∞
(Ḃp,1
T̄
N −2
p
L∞
(Ḃp,1
T̄
N −2
p
L∞
(Ḃp,1
T̄
N −2
p
L∞
(Ḃp,1
T̄
)
)
)
)
. ku2 k
.
.
k∇δuk
N
kδuk
N
p
L∞
(Ḃp,1
)
T̄
N
p
L∞
(Ḃp,1
)
T̄
1
T (1+ka k
N −2
p
L2T̄ (Ḃp,1
p
L1T̄ (Ḃp,1
)
1
(1+ka k
. ka1 k
.
N
p
L2T̄ (Ḃp,1
)
1
k∇u k
N −2
p
L∞
(Ḃp,1
T̄
2
+ka k
N
,
)
N
p
L∞
(Ḃp,1
)
T̄
k∆δuk
p
L∞
(Ḃp,1
)
T̄
,
)
N −2
p
L1T̄ (Ḃp,1
)
2
+ka k
)k∆u2 k
N −1
kδak
N −1
.
p
L1T̄ (Ḃp,1
)
,
N
p
L∞
(Ḃp,1
)
T̄
)kδak
p
L∞
(Ḃp,1
)
T̄
N −1
p
L∞
(Ḃp,1
)
T̄
,
4.2. LOCAL WELL-POSEDNESS IN CRITICAL SPACES
81
Assuming that T has been chosen so small as to satisfy X(T ) ≤ 1/2, inequality (4.26)
implies that
kδak
N −1 ≤ 2CT kδuk
N .
p
p
L∞
(Ḃp,1
T̄
L1T̄ (Ḃp,1 )
)
Hence, we finally obtain
kδuk
(4.28)
N −2
e ∞ (Ḃ p
L
p,1
T̄
+ kδuk
)
N
e 1 (Ḃ p )
L
p,1
T̄
≤ Y (T̄ )kδuk
N
e 1 (Ḃ p )
L
p,1
T̄
with, up to a change of CT ,
Y (T̄ ) := Cka1 k
N
p
L∞
(
Ḃ
)
p,1
T̄
+ CT T̄ + ku1 k
N +1
p
L1T̄ (Ḃp,1
)
+ ku2 k
N +1
p
L1T̄ (Ḃp,1
)
+ ku2 k
N
p
L2T̄ (Ḃp,1
)
.
We stress that C depends only on N and p and that
lim sup Y (T̄ ) ≤ Cη.
T̄ →0+
Hence assuming Cη < 1 entails (δa, δu) ≡ 0 on a suitably small time interval [0, T̄ ].
Standard connectivity arguments then lead to T̄ = T.
Let us now treat the limit case7 where p = N. Obviously, the above proof fails because
some terms in the right-hand side of the equation of δu (such as δa∆u2 for instance)
N
N
−2
−1
p
p
2
cannot be estimated in L1 (0, T ; Ḃp,1
) in terms of norms of δa in L∞
T (Ḃp,1 ) and u
N
+1
p
in L1 (0, T ; Ḃp,1
). On the other hand, according to exercise 1.22, those terms may be
N
−2
p
).
bounded in the slightly larger space L1T (Ḃp,∞
−1
e 1 (Ḃ 1 ), but we
At this point, we could try to get bounds for δu in L∞ (0, T ; ḂN,∞
)∩L
T
N,∞
1 ,
then have to face the lack of control on δu in L1 (0, T ; L∞ ) (because, in contrast with ḂN,1
1
the space ḂN,∞
is not embedded in L∞ ) so that we run into troubles when estimating
δF1 . The key to that difficulty relies on the following logarithmic interpolation inequality
s ) spaces:
e ρ (Ḃp,r
which is a straightforward generalization of proposition 1.5.3 to L
T
(4.29)
kwkL1 (Ḃ 1
T
N,1 )
kwkLe1 (Ḃ 0 ) +kwkLe1 (Ḃ 2 ) N,∞
T
N,∞
T
log
e
+
.
N,∞ )
kwkLe1 (Ḃ 1 )
. kwkLe1 (Ḃ 1
T
T
N,∞
It is then possible to conclude to uniqueness by taking advantage of a generalization of
Gronwall lemma: the so called Osgood lemma (see e.g. [13]).
To begin with, let us introduce the functional space in which we are going to prove uniqueness:
N
1
0
e 1 (ḂN,∞
F := L∞ (0, T̄ ; ḂN,∞
) × L∞ (0, T̄ ; Ḃ −1 ) ∩ L
) .
T̄
N,∞
T̄
Let us first state that (δa, δu) ∈ FT̄ . For δa, the proof given in the non critical case still
0
0
0
works. As ḂN,1
,→ ḂN,∞
, we get that δa ∈ L∞ (0, T ; ḂN,∞
).
e 1 (Ḃ −1 ).
For δu, we just have to check that the right-hand side of (4.25) belongs to L
T
N,∞
s ) spaces.
e ρ (Ḃp,r
This is due to propositions 1.5.11, 1.5.12 and exercise 1.22 generalized to L
T
7
If N = 2, the general case is a consequence of the case p = 2 by virtue of Besov embeddings.
82
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
Let us now bound δa. Combining theorem 3.3.1 with propositions 1.5.11 and 1.5.12 and
1
1
the embedding ḂN,1
,→ ḂN,∞
∩ L∞ gives:
R
Z t
C 0t ku2 kḂ 2 dτ
2
1
N,1
kδakL∞ (Ḃ 0 ) ≤ Ce
kδakḂ 0 kdivu kḂ 1 +kδukḂ 1 1+ka kḂ 1
dτ,
t
N,∞
N,∞
0
N,1
N,1
N,1
whence, according to Gronwall inequality,
C
kδakL∞ (Ḃ 0
N,∞ )
t
Rt
0
≤ Ce
ku2 kḂ 2
dτ
1 + ka1 kL∞ (Ḃ 1
N,1
N,1 )
t
kδukL1 (Ḃ 1
N,1 )
t
.
Making use of inequality (4.29) with w = δu, we end up with
kδakL∞ (Ḃ 0
t
N,∞ )
≤ CT kδukLe1 (Ḃ 1
N,∞ )
t
log e +
kδukLe1 (Ḃ 0
N,∞ )
t
+kδukLe1 (Ḃ 2
t
N,∞ )
kδukLe1 (Ḃ 1
N,∞ )
t
for some constant CT depending only on the bounds for a1 and u2 on [0, T ].
0 ) ,→ L1 (Ḃ 0 ) for finite t, we have
e∞
Remark that since L
t (Ḃ
t
N,1
N,1
∀t ∈ [0, T ], kδukLe1 (Ḃ 0
N,∞ )
t
with
i
V (t) :=
+kδukLe1 (Ḃ 2
Z t
0
N,∞ )
t
≤ V (t) := V 1 (t) + V 2 (t) < ∞
kui (τ )kḂ 0 +kui (τ )kḂ 2
dτ.
N,1
N,1
Therefore,
kδakL∞ (Ḃ 0
(4.30)
N,∞ )
t
V (t)
≤ CT kδukLe1 (Ḃ 1 ) log e +
t
N,∞
kδukLe1 (Ḃ 1
t
N,∞ )
with V a non-decreasing function of t ∈ [0, T ] with bounded first derivative.
Let us now turn to the proof of estimates for δu. According to theorem 2.2.3, we have
kδukL∞ (Ḃ −1
t
N,∞
e 1 (Ḃ 1
) + kδukL
N,∞ )
t
.
5
X
kδGi kLe1 (Ḃ −1 ) .
t
i=1
N,∞
−1
e 1t (Ḃ −1 ), we get for
Using proposition 1.5.11, exercise 1.22 and that L1 (0, t; ḂN,∞
) ,→ L
N,∞
t≤T:
kδukL∞ (Ḃ −1 ) +kδukLe1 (Ḃ 1 ) . ku2 kLe2 (Ḃ 1 ) kδukLe2 (Ḃ 0 ) +kδukLe1 (Ḃ 1 ) ka1 kLe∞ (Ḃ 1 )
t
t
t
t
t
t
N,∞
N,∞
N,1
N,∞
N,∞
N,1
Z th
i
1+ku2 kḂ 2
dτ.
+
ku1 kḂ 2 kδukḂ −1 + 1+ka1 kḂ 1 +ka2 kḂ 1 ) kδakḂ 0
0
N,1
N,1
N,∞
N,∞
N,1
N,1
Let us assume that the constant η in inequality (4.23) is small enough so that the second
term in the right-hand side may be absorbed by the left-hand side. Next, remark that
ku2 kLe2 (Ḃ 1 ) tends to 0 when t goes to 0 hence there exists a positive T̄ such that the
t
N,1
first term may also be absorbed8 for all t ∈ [0, T̄ ].
We end up with the following inequality:
Z t
ku1 kḂ 2 kδukḂ −1 +(1+ku2 kḂ 2 )kδakḂ 0
kδukL∞ (Ḃ −1 ) + kδukLe1 (Ḃ 1 ) .
t
8
N,∞
t
N,∞
N,1
N,∞
≤ kδukLe2 1 (Ḃ 1
kδuk e2 ∞
0
1
By interpolation, one easily gets kδukLe 2 (Ḃ 0
t
)
N,∞
N,1
1
t
)
N,∞
−1
Lt (ḂN,∞
)
.
N,∞
dτ.
4.3. FURTHER RESULTS
83
One can now plug (4.30) in the above inequality. Denoting X(t) := kδukL∞ (Ḃ −1
t
kδukLe1 (Ḃ 1
t
N,∞ )
N,∞ )
+
, we eventually get
Z
∀t ∈ [0, T̄ ], X(t) ≤ C
0
t
V (T )
(1+V 0 (τ ))X(τ ) log e+
dτ.
X(τ )
As
0
1
Z
1
V ∈ L (0, T )
and
0
dr
V (T ) r log e +
r
= +∞,
Osgood lemma (see e.g. lemma 3.1 in [13]) entails that X ≡ 0 on [0, T̄ ]. This means that
(a1 , u1 ) and (a2 , u2 ) coincide on [0, T̄ ]. Then appealing to the usual connectivity argument
completes the proof.
Remark. Let us stress the fact that in the statement of proposition 4.2.3, the space ETp may be
s ) spaces instead of L
s ). This is due to the
e ρ (Ḃp,r
replaced by its counterpart with Lρ (0, T ; Ḃp,r
T
fact that theorems 2.2.3 and 3.3.1 provide missing tildes for free.
The critical case is the only case where keeping a tilde in condition (4.23) is fundamental.
Note however that this assumption involves only the size of a1 , not the regularity. We do not
know how to prove uniqueness in theorem 4.2.1 in the case N = 2 or p = N in a functional
s ) spaces.
e ρ (Ḃp,r
framework with no L
T
4.3
Further results
Let us first point out that as a by-product of our proof of existence and of inequality (2.5), we
get the following lower bound for the existence time:
T ≥ sup t > 0, f1 (t) ≤ η and f2 (t) ≤ η
with
f1 (t) :=
P
q( N
−1)
p
˙ q u0 kLp + k∆
˙ qf k 1 p
k∆
Lt (L )
f2 (t) :=
P
q( N
−1)
p
˙ q u0 kLp + k∆
˙ qf k 1 p
k∆
Lt (L )
q2
q
2
1−e−κµt2
κµ22q
2q
1−e−2κµt2
2κµ22q
2q
,
1
2
and η such that
2Cη ≤ µ and 16(e
Cη
µ
− 1) ≤ ka0 k
N
p
Ḃp,1
.
Above C is the constant appearing in the proof of existence.
This lower bound is rather inexplicit unless u0 and f have more regularity.
Let us finally mention that theorem 4.2.1 may be extended or improved in many ways:
• One can show a small data global existence and uniqueness result in critical spaces under
the condition that
kρ0 − ρ̄k
N
N −1
2 ∩Ḃ 2
Ḃ2,1
2,1
+ ku0 k
N −1
2
Ḃ2,1
+ kf k
N
2 )
L1 (Ḃ2,1
1
for some constant positive density ρ̄ such that P 0 (ρ̄) > 0 (see [17] for more details).
84
CHAPTER 4. A SHORT INSIGHT INTO COMPRESSIBLE FLUID MECHANICS
• The condition that the density is a small perturbation of a constant state may be relaxed
provided some extra regularity is available (see [20]). It is however fundamental for the
initial density to be bounded away from 0.
In a recent paper however, it has been stated that, for suitably smooth data, existence
and uniqueness results may be proved in some cases where vacuum occurs (see [15]).
• Similar results may be proved for polytropic heat conducting compressible fluids (see [20]
and [23]).
• After a suitable rescaling, one can state results of convergence for the barotropic NavierStokes equations to the incompressible Navier-Stokes equations when the Mach number ε
tends to 0 (see [22] and [21]).
Finally, we want to stress that combining a priori estimates in Besov spaces for the heat equation
and for the transport equation may give local well-posedness results for many other evolution
PDE’s coming from fluid mechanics or other fields of physics. One can mention for example that
this method leads to such results for the system of density-dependent incompressible NavierStokes equations (see [19]) and for visco-elastic fluids (see [14]).
4.4
Exercises
Exercise 4.1. Prove lemma 4.1.1 for smooth u, b, and Ω.
Exercise 4.2. Let XTp and YTp be defined according to page 76. Prove that both XTp and YTp
are Banach spaces, that XTp is separable and that YTp is the dual of XTp .
Exercise 4.3. Prove the passage to the limit from system (4.20) to system (4.21).
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List of symbols
[r], 12
≈, 7
., 7
B(0, R), 7
B(0, R), 7
s , 13
Bp,r
s , 31
Ḃp,r
C(0, R1 , R2 ), 7
C0 , 16
Cb∞ , 59
Cc∞ , 15
C r , 12
∆q , 9
˙ q , 29
∆
∆per
q , 10
F, 7
s , 15
Fp,r
H s , 11
QN
a , 10
R(u, v), 21
Ṙ(u, v), 35
S, 7
S m , 20
S0 , 7
Sh0 , 29
Sq , 9
Ṡq , 29
Sqper , 10
TN
a , 10
Tu v , 21
Ṫu v , 35
e N , 10
Z
a
89
90
LIST OF SYMBOLS
Index
Angular momentum, 70
Logarithmic interpolation, 19, 30
Ball, 7
Barotropic, 72
Bernstein lemma, 8
Besov space, 13, 31
Bony decomposition, 21, 35
Mass, 69
Material derivative, 53
Meyer multiplier, 27
Mild solution, 46
Momentum, 69
Multiplier, 7, 20
Camassa-Holm equation, 60
Compact embedding, 24
Complex interpolation, 16
Convective term, 44
Critical space, 47
Deformation tensor, 71
Density, 69
Duality, 15
Dyadic block, 9, 10, 29
Embeddings, 15
Energy, 69
equality, 45
inequality, 45
Entropy, 69
Fatou property, 16
flow, 53
Fourier
law, 71
series, 10
transform, 7
Fujita-Kato theorem, 46
Gibbs relation, 71
Holder space, 12
Heat
equation, 39
flux, 70
Internal energy, 69
Inviscid, 71
Navier-Stokes equations, 44
Paradifferential calculus, 21
Paraproduct, 21
paraproduct, 35
Poisson formula, 10
Pressure, 44, 71
Real interpolation, 16
Remainder, 21
remainder, 35
Scaling invariance, 46
Schauder-Tychonoff
theorem, 73
Schwartz space, 7
Separable, 15
Shell, 7
Sobolev space, 11
State relation, 71
Strain tensor, 70
Summation convention, 45
Temperate distribution, 7
transport equation, 53
Triebel-Lizorkin spaces, 15
Velocity, 69
Viscosity, 44, 71
Viscous stress tensor, 71
Wavefront, 21
Weak solution, 45
Leray theorem, 45
Littlewood-Paley decomposition, 9, 10, 28
91