RESEARCH STATEMENT
ZACHARY BRADSHAW
ZBRADSHAW@MATH.UBC.CA
1. O VERVIEW
My research is in the field of partial differential equations. I am primarily interested
in the three dimensional non-stationary Navier-Stokes equations (3D NSE). This system
describes the movement of an incompressible, viscous fluid’s velocity field u as it evolves
from some initial state u0 at time t = 0. In R3 × [0, ∞), the unknowns are the velocity field
u and pressure p which solve
(1)
∂t u − ν∆u + (u · ∇)u + ∇p = f
∇·u=0
in R3 × (0, T )
for a prescribed viscosity coefficient ν and body force f . This model provides the foundation for our mathematical understanding of fluids in the real world and thus has many
important applications. Remarkably, fundamental mathematical questions remain unanswered, most notably the problem of global regularity which is one of the Clay Mathematics Institute’s Millennium Problems.
In this research statement I will summarize two projects which I have contributed to following the completion of my doctoral work.
The first project is a new construction of an historically important class of solutions – the
self-similar solutions and a generalization thereof – under significantly weaker assumptions than were previously available. Past constructions can be grouped in two categories,
one for initial data which is small in a critical functions space like the Lebesgue space L3
or the weak Lebesgue space L3w and another for large initial data which is required to be
locally Hölder continuous away from the origin. In a collaboration with Tai-Peng Tsai (see
[5]) we construct self-similar solutions (and a generalized class of discretely self-similar
solutions) for any self-similar (or discretely self similar) data in L3w . No smoothness or
smallness assumptions are needed; our construction allows the data to be discontinuous
and even singular at points other than the origin (the latter is only relevant in the discretely self-similar case).
The second project aims to illustrate the essential role of high frequencies in singularity
formation by refining an existing Prodi-Serrin-type regularity criteria to include only a
range of Littlewood-Paley frequencies above some threshold which diverges to positive
infinity at a possible blow-up time. This work is in collaboration with Zoran Grujić (see
[1]). Thematically similar results such as the partial regularity theory of Caferelli, Kohn,
and Nirenberg [6] are well known and generally involve small scale dynamics realized
through parabolic Morrey space norms, e.g. weighted averages over parabolic cylinders.
Date: October 29, 2015.
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RESEARCH STATEMENT // Z. BRADSHAW (ZBRADSHAW@MATH.UBC.CA)
Our result confirms that high frequencies in the Littlewood-Paley formalism play an analogous role in possible singularity formation.
2. S ELF - SIMILAR SOLUTIONS TO 3D NSE
The Navier-Stokes equations satisfy a natural scaling: given a solution pair (u, p) for the
initial data u0 and any λ > 0, it follows that
uλ (x, t) := λu(λx, λ2 t),
is also a solution with associated pressure
pλ (x, t) := λ2 p(λx, λ2 t),
and initial data
uλ0 (x) := λu0 (λx).
A solution is self-similar if it is invariant with respect to this scaling, i.e. if uλ (x, t) = u(x, t)
for all λ > 0. If this holds for a particular λ > 1, then u is discretely self-similar with
factor λ (i.e. u is λ-DSS). Similarly u0 can be self-similar (a.k.a. −1-homogeneous) or λDSS. Any self-similar solution u is thus λ-DSS for any λ > 1. Self-similar solutions were
first mentioned by Leray in [15] and provide a rich class of solutions in which to prove or
disprove pathological behavior (see, e.g., [16, 19]).
Self-similar solutions are determined by the behavior at any fixed time. This leads to an
ansatz of u in terms of a time-independent profile U where
1
x
u(x, t) = √ V √
,
2t
2t
where V solves the Leray equations
−∆V − V − y · ∇V + V · ∇V + ∇P = 0 in R3
∇·V =0
in R3 ,
√
in the variable y = x/ t. Similarly, λ-DSS solutions are decided by their behavior on the
time interval 1 ≤ t ≤ λ2 and we have
1
u(x, t) = √ V (y, s),
2t
for
√
x
y = √ , s = log( 2t),
2t
where V is time-periodic with period log(λ) and solves the time-dependent Leray equations
∂s V − ∆V − V − y · ∇V + V · ∇V + ∇V = 0 in R3 × R
(3)
∇·V =0
in R3 × R.
(2)
Note that the discretely self-similar transform gives a one-to-one correspondence between
weak solutions of (1) and time-periodic weak solutions of (3).
The fact that (2) is time-independent motivates an analogy between the self-similar profile and solutions to the steady state Navier-Stokes equations. It is known for certain
RESEARCH STATEMENT // Z. BRADSHAW (ZBRADSHAW@MATH.UBC.CA)
3
large data and appropriate forcing that solutions to the stationary Navier-Stokes boundary value problem are non-unique [9, 18]. In [11], Jia and Šverák conjecture that similar
non-uniqueness results might hold for solutions to (2). These solutions would necessarily involve large data but, until recently, existence results for self-similar solutions
were only known for small data (for small data existence of forward self-similar solutions see [7, 8, 10, 12]). Jia and Šverák addressed this in [11] where they proved the existence of a forward self-similar solution using Leray-Schauder degree theory for large
−1-homogeneous initial data which is locally Hölder continuous away from the origin (see
also remarks in [20]). Similar results were later proven in [20] for λ-DSS solutions with
factor close to one where closeness is determined by the local Hölder norm of u0 away from
the origin. It is also shown in [20] that the closeness condition on λ can be eliminated if the
initial data is axisymmetric with no swirl. In [13], the existence of self-similar solutions on
the half space (and the whole space) is established for appropriately smooth initial data.
The approach of [13] differs from [11] and [20] in that the existence of a solution to the
stationary Leray equations (2) is established directly. It also gives a second proof of the
main result of [11].
In a recently submitted work, Z. B. and T.-P. Tsai offer a new construction of λ-DSS solutions for any data u0 ∈ L3w (R3 ) which is λ-DSS for some λ > 1. No smoothness or
boundedness is assumed and λ is allowed to be large. A key difference between [5] and
[11, 20, 13] is the lack of local compactness, which is required by the Leray-Schauder theorem and is provided by the regularity theory for self-similar solutions or λ-DSS solutions
with λ close to one. In contrast, regularity is unavailable for general DSS solutions and a
new approach is needed. Our main result is the following theorem.
Theorem 1 (Z. B. and T.-P. Tsai). Let u0 be a divergence free, λ-DSS vector field for some λ > 1
which belongs to L3w . Then, there exists a local Leray solution u to (1) which is λ-DSS and
additionally satisfies
ku(t) − et∆ u0 kL2 (R3 ) ≤ C0 t1/4
for any t ∈ (0, ∞) and a constant C0 = C0 (u0 ).
Note that et∆ is the solution operator for the heat equation and the local Leray class is
a class of solutions which have only locally finite energy and contains the weak Lebesgue
space L3w , which is the natural function space for self-similar solutions.
To prove this result we provide a priori estimates for (3). The key observation leading to
the explicit a priori bound is the following: If a√solution V (y, s) of the Leray equations (3)
asymptotically agrees with a given U0 (y, s) = 2tet∆ u0 (x) (e.g. V − U0 ∈ L∞ (R, L2 (R3 ))),
then the difference U = V − U0 formally satisfies
Z TZ Z TZ 1 2
2
|∇U | + |U | dy ds =
(U · ∇)U · U0 − R(U0 ) · U dy ds,
2
0
0
RR
where R(U0 ) is a source term determinedRR
by U0 . The integral
(U ·∇)U ·U0 is usually out
2
of control for large U0 , but now we have
|V | on the left side (which is not available for
Navier-Stokes). For initial data as in the theorem’s statement, U0 decays at spatial infinity.
Hence the troublesome term can be controlled if the local part of U0 is suitably “cutout.”
Then, a standard construction using a mollified system and a Galerkin scheme leads to
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RESEARCH STATEMENT // Z. BRADSHAW (ZBRADSHAW@MATH.UBC.CA)
the existence of a time periodic solution U to a perturbed version of the ansatz (3) as well
as energy estimates for U . This allows us to recover a discretely self-similar solution NSE
in the physical variables. Since L3w embeds in the uniformly local L2 spaces we can show
moreover that these solutions are local Leray solutions in the sense of [14].
If u0 is −1-homogeneous, then we can use Theorem 1 to construct a self-similar solution
by considering the solutions obtained from Theorem 1 treating the data as λk -DSS for an
appropriate sequence λk which decreases to 1 as k → ∞. Alternatively, we can also construct a self-similar solution directly by modifying our approach to the stationary ansatz
(2). We are thus able to provide two proofs of the following theorem.
Theorem 2 (Z. B. and T.-P. Tsai). Let u0 be a (−1)-homogeneous divergence free vector field
such that u0 ∈ L∞ ({|x| = 1}). Then, there exists a smooth local Leray solution u to (1) which is
self-similar and additionally satisfies
ku(t) − et∆ u0 kL2 (R3 ) ≤ C0 t1/4
for any t ∈ (0, ∞) and a constant C0 = C0 (u0 ).
Self-similar solutions evolving from −1-homogeneous data were first constructed for
small data (in an appropriate function space, e.g. L3w , see for example [7]). The first
construction of large self-similar solutions was recently given in [11] using local Hölder
estimates and the Leray-Schauder theorem. A second construction appears in [13] using
an a priori bound for Leray equations derived by a contradiction argument and a study
of the Euler equations. The construction of T.-P. Tsai and Z. B. provides a new (third)
construction based on an explicit a priori bound. This is a significant improvement upon
earlier works as it accommodates all −1-homogeneous data with an isolated singularity at the origin; in contrast, previous constructions assumed smoothness away from the
origin.
Future research plans: I am interested and actively pursuing several open problems related to discretely self-similar solutions:
• There is room to improve the main result of [5]. Our construction assumes u0 ∈ L3w
and it is natural to try to generalize this to larger critical spaces such as BM O−1 or
−1
even the Besov space Ḃ∞,∞
. The perturbation approach in [5] breaks down in each
of these cases because solutions to the heat equation with DSS data do not decay
rapidly at spatial infinity. This has two consequences. First, we can no longer
ensure that
Z
1
W · ∇U · W dx ≤ k∇U k2L2
4
by “cutting-out” the local part of U0 in our definition of the perturbation term W
to ensure it is small. Second we no longer have a priori estimates for
Z
(et∆ u0 ) · ∇(et∆ u0 ) · Uk dx,
where {Uk } is a sequence of approximating solutions.
RESEARCH STATEMENT // Z. BRADSHAW (ZBRADSHAW@MATH.UBC.CA)
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• Smoothness of DSS solutions is only known a priori if λ is close to 1. For selfsimilar solutions on the other hand, smoothness follows from the Cafarelli-KohnNirenberg partial regularity theory. For finite energy weak solutions in the sense
of Leray, the singular set must belong to a compact set of space-time. This is not
known for solutions in the local Leray class, i.e. those with only locally finite energy, and these solutions might possess irregularities at arbitrarily large times. The
existence of irregular DSS solutions for large L3w data would prove that the singular
set of local Leray solutions is not compact.
3. R EFINED P RODI -S ERRIN - TYPE REGULARITY CRITERIA
A general regularity theory is unavailable for solutions to the Navier-Stokes equations
belonging to the energy class and regularity criteria only exist in stronger classes. For
example, if u is a weak solution to (1) on R3 × [0, T ] belonging to the energy class and
satisfying
Z T
kukqLp dt < ∞,
0
for certain pairs (p, q) where
2 3
+ = 1,
q p
then u is smooth. For 3 < p < ∞ and 2 < q < ∞ this is the Prodi-Serrin class. Smoothness
is also known in the endpoint cases (p, q) = (∞, 2) and (p, q) = (3, ∞). More general
results can often be formulated by replacing Lp with larger function spaces possessing
identical scaling, e.g. the weak Lebesgue spaces Lpw for p < ∞ or certain homogeneous
Besov spaces [14]. In a recent collaboration between Z. B. and Z. Grujić, we prove that a
Prodi-Serrin-type criteria involving Besov spaces can be reformulated to emphasize high
frequencies. To make this matter precise we briefly introduce Besov spaces.
The homogeneous Littlewood-Paley decomposition
X
˙ j u,
∆
u=
j∈Z
˙ ju
is essentially a smooth version of the classical freqency decomposition in that each ∆
3
j
is, in the Fourier variable ξ, the localization of û to the dyadic shell {ξ ∈ R : |ξ| ∼ 2 }. We
s
say u belongs to the Besov space Ḃp,q
where s ∈ R and 1 ≤ p, q ≤ ∞ if
˙ j ukLp (Rn ) q < ∞.
||u||Ḃp,q
:= λsj k∆
s
l
Roughly speaking, these spaces allow us to measure the varying importance of different frequencies. A Prodi-Serrin-type regularity criteria states that weak solutions in the
energy class which are regular on (0, T ) remain regular at time T if
Z T
2/(1−)
ku(t)kḂ − dt < ∞,
0
∞,∞
for some ∈ [0, 1) [14]. In [1], Z. B. and Z. Grujić refine this criteria by considering only
Littlewood-Paley blocks above a time-dependent threshold which is diverging to +∞ at
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RESEARCH STATEMENT // Z. BRADSHAW (ZBRADSHAW@MATH.UBC.CA)
an initial blow-up time – i.e. the window of relevant frequencies is vanishing. The precise
statement follows.
Theorem 3. [Z. B. and Z. Grujić] Assume u is a weak solution to 3D NSE on [0, T ] belonging
to the energy class which is regular on (0, T ). Then, there exists a function J : [0, T ) → R which
satisfies lim− J(t) = +∞ so that, if
t→T
Z
T
sup
0
˙
λ−
j k∆j u(t)k∞
2/(1−)
dt < ∞,
j≥J(t)
then u is regular on (0, T ].
The novelty here is that only a finite range of frequencies need to remain subdued to
guarantee the solution remains finite. The upper and lower cutoffs are asymptotically
comparable to
2/(3−2) −
||u(t)||Ḃ∞,∞
,
Jlow (t) = log2
c||u||L∞ (0,T ;L2 )
and
1/(1−)
Jhigh (t) = log2 c||u(t)||Ḃ −
.
∞,∞
Although it would be desirable to get comparable upper and lower cutoffs, these appear
to be sharp (see discussion in [1]).
R EFERENCES
[1] Bradshaw, Z. and Grujić, Z. Navier-Stokes equations and scaling: a vignette. submitted, arxiv:
[2] Bradshaw, Z. and Grujić, Z. Energy cascades in physical scales of 3D incompressible magnetohydrodynamic turbulence. J. Math. Phys. 54 (2013), no. 9, 093503, 18 pp.
[3] Bradshaw, Z. and Grujić, Z. A note on the surface quasi-geostrophic temperature variance cascade.
Commun. Math. Sci. 13 (2015), no. 2, 557-564.
[4] Bradshaw, Z. and Grujić, Z. On the transport and concentration of enstrophy in 3D magnetohydrodynamic turbulence. Nonlinearity 26 (2013), no. 8, 2373-2390.
[5] Bradshaw, Z. and Tsai, T.-P. Forward discretely self-similar solutions of the Navier-Stokes equations
II. submitted (available online at http://arxiv.org/abs/1510.07504).
[6] Caffarelli, L., Kohn, R. and Nirenberg, L. Partial regularity of suitable weak solutions of the NavierStokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771-831.
[7] Cannone, M., Meyer, Y., and Planchon, F., Solutions auto-similaires des équations de Navier-Stokes
dans R3 , Exposé n. VIII, Séminaire X-EDP, Ecole Polytechnique, 1994.
[8] Cannone, M. and Planchon, F. Self-similar solutions for Navier-Stokes equations in R3. Comm. Partial
Differential Equations 21 (1996), no. 1-2, 179-193.
[9] Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations. Vol. II. Nonlinear
steady problems. Springer Tracts in Natural Philosophy, 39. Springer-Verlag, New York, 1994.
[10] Giga, Y. and Miyakawa, T., Navier-Stokes flows in R3 with measures as initial vorticity and the Morrey
spaces, Comm. Partial Differential Equations 14 (1989), 577-618.
[11] Jia, H. and Šverák, V., Local-in-space estimates near initial time for weak solutions of the NavierStokes equations and forward self-similar solutions. Invent. Math. 196 (2014), no. 1, 233-265.
[12] Koch, H., Tataru, D., Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (1), 22–35 (2001)
[13] Korobkov, M. and Tsai, T.-P., Forward self-similar solutions of the Navier-Stokes equations in the half
space, submitted.
[14] Lemarié-Rieusset, P. G., Recent developments in the Navier-Stokes problem. Chapman Hall/CRC Research
Notes in Mathematics, 431. Chapman Hall/CRC, Boca Raton, FL, 2002.
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[15] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace. (French) Acta Math. 63 (1934),
no. 1, 193-248.
[16] Nečas, J., Ružička, M., and Šverák, V., On Leray’s self-similar solutions of the Navier-Stokes equations,
Acta Math. 176 (1996), 283–294.
[17] Šverák, V. On Landau’s solutions of the Navier-Stokes equations. J. Math. Sci., 179 (2011), no. 1, 208228.
[18] Temam, R., Navier-Stokes equations. Theory and numerical analysis. Reprint of the 1984 edition. AMS
Chelsea Publishing, Providence, RI, 2001.
[19] Tsai, T.-P.: On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy
estimates, Archive for Rational Mechanics and Analysis 143 (1998), 29–51.
[20] Tsai, T.-P., Forward discretely self-similar solutions of the Navier-Stokes equations. Comm. Math.
Phys. 328 (2014), no. 1, 29-44.