Almost sure existence of global weak solutions for the ... · 1 Navier Stokes Equations:...

Almost sure existence of global weak solutions forthe supercritical Navier-Stokes equations

Gigliola Staffilani

Massachusetts Institute of Technology

June , 2012

HYP2012, Padova

Gigliola Staffilani (MIT) a.s. global supercritical Navier-Stokes June , 2012 1 / 41

1 Navier Stokes Equations: Introduction

2 Periodic Navier-Stokes Below L2

3 Randomization Setup

4 Main Results

5 Heat Flow on Randomized Data

6 Equivalent Formulations

7 Energy Estimates

8 Construction of Weak Solutions to the Difference Equation

9 Uniqueness in 2D

10 Proof of the Main Theorems


Goal of the talk:

We show that after suitable data randomization there exists a large set ofsuper-critical periodic initial data, in H−α(Td ) for some α(d) > 0, for both 2dand 3d Navier-Stokes equations for which global energy bounds are proved.

We then obtain almost sure super-critical global weak solutions.

We also show that in 2d these global weak solutions are unique.

Joint with:

Andrea R. Nahmod (UMass Amherst).

Natasa Pavlovic (UT Austin).


The Navier-Stokes EquationsConsider a viscous, homogenous, incompressible fluid with velocity ~u onΩ = Rd or Td , d=2, 3 and which is not subject to any external force. Thenthe initial value problem for the Navier-Stokes equations is given by

(NSEp)

~ut + ~v · ∇~u = −∇p + ν∆~u; x ∈ Ω t > 0∇ · ~u = 0~u(x ,0) = ~u0(x),

where 0 < ν =inverse Reynols number (non-dim. viscosity);~u : R+ × Ω→ Rd , p = p(x , t) ∈ R and ~u0 : Ω→ Rd is divergence free.

For smooth solutions it is well known that the pressure term p can beeliminated via Leray-Hopf projections and view (NSEp) as an evolutionequation of ~u alone1,

the mean of ~u is easily seen to be an invariant of the flow (conservation ofmomentum) so can reduce to the case of mean zero ~u0.

1Although understanding the pressure term might be important.Gigliola Staffilani (MIT) a.s. global supercritical Navier-Stokes June , 2012 4 / 41

Then the incompressible Navier-Stokes equations (NSEp) (assume ν = 1)can be expressed as

(NSE)

~ut = ∆~u − P∇ · (~u ⊗ ~u); x ∈ Ω, t > 0∇ · ~u = 0~u(x ,0) = ~u0(x),

where P is the Leray-Hopf projection operator into divergence free vectorfields given via

P~h = ~h −∇ 1∆

(∇ · ~h) = (I + ~R ⊗ ~R)~h

(~R = Riesz transforms vector) and ~u0 is mean zero and divergence free.

By Duhamel’s formula we have

(NSEi) ~u(t) = et∆~u0 +

∫ t

0e(t−s)∆P∇ · (~u ⊗ ~u) ds

In fact, under suitable general conditions on ~u the three formulations(NSEp), (NSE) and (NSEi) can be shown to be equivalent (weaksolutions, mild solutions, integral solutions. Work by Leray, Browder,Kato, Lemarie, Furioli, Lemarie and Terraneo, and others. )


Recall if the velocity vector field ~u(x , t) solves the Navier-Stokesequations in Rd or Td then ~uλ(x , t) with

~uλ(x , t) = λ~u(λx , λ2t),

is also a solution to the system (NSE) for the initial data

~u0λ = λ~u0(λx) .

In particular,

‖~u0λ‖Hsc = ‖~u0‖Hsc , sc =d2− 1.

The spaces which are invariant under such a scaling are called criticalspaces for Navier-Stokes. Examples:

Hd2−1 → Ld → B

−1+ dp

p,∞ → BMO−1 (1 < p <∞).

v ∈ BMO−1 iff ∃ f i ∈ BMO such that v =∑∂i f i (Koch-Tataru)


Classical solutions to the (NSE) satisfy the decay of energy which can beexpressed as:

‖u(x , t)‖2L2 +

∫ t

0‖∇u(x , τ)‖2

L2 dτ = ‖u(x ,0)‖2L2 .

When d = 2: the energy ‖u(x , t)‖L2 , which is globally controlled, isexactly the scaling invariant Hsc = L2-norm. In this case the equationsare said to be critical. Classical global solutions have been known toexist; see Ladyzhenskaya (1969).

When d = 3: the global well-posedness/regularity problem of (NSE) is along standing open question!

I The energy ‖u(x , t)‖L2 is at the super-critical level with respect to the scalinginvariant H

12 -norm, and hence the Navier-Stokes equations are said to be

super-criticalI The lack of a known bound for the H

12 contributes in keeping the large data

global well-posedness question for the initial value problem (NSE) still open.


Some Background

One way of studying the initial value problem (NSE) is via weak solutionsintroduced by Leray (1933-34).

Leray (1934) and Hopf (1951) showed existence of a global weak solutionof the Navier-Stokes equations corresponding to initial data in L2(Rd ).Lemarie extended this construction and obtained existence of uniformlylocally square integrable weak solutions.

Questions addressing uniqueness and regularity of these solutions whend = 3 have not been answered yet. But important contributions inunderstanding partial regularity and conditional uniqueness of weaksolutions by many; see e.g.

I Caffarelli-Kohn-Nirenberg (82’); Struwe (88’-07’); Lin (98’); P.L.Lions-Masmoudi (98’), Seregin-Sverak (02’) Escauriaza-Seregin-Sverak(03’); Vasseur (07’), Kenig-G. Koch (11’), and many others.


Another approach is to construct solutions to the corresponding integralequation (‘mild’ solutions) pioneered by Kato and Fujita (1961).

Mild solutions to the Navier-Stokes equations for d ≥ 3 has been studiedlocally in time and globally for small initial data in various sub-critical orcritical spaces. Many references, see e.g.

I T. Kato (84’), Giga-Miyakawa (89’), Taylor (92’), Planchon (96’), Cannone(97’), H.Koch-Tataru (01’), Gallagher-Planchon (02’),Gallagher-Iftimie-Planchon(05’), Germain-Pavlovic-Staffilani(07’), Kenig-G.Koch (09’), others.


Periodic Navier-Stokes Below L2

We consider the periodic Navier-Stokes problem (NSE)

(NSE)

~ut = ∆~u − P∇ · (~u ⊗ ~u); x ∈ Td t > 0∇ · ~u = 0~u(x ,0) = ~u0(x),

where d = 2,3 and ~u0 is divergence free and mean zero and P is the Lerayprojection into divergence free vector fields.

We address the question of long time existence of weak solutions forsuper-critical randomized initial data both in d = 2, 3.For d = 2 we address uniqueness as well.


Periodic setting: similar supercritical randomized well-posedednessresults were obtained for the 2D cubic NLS by Bourgain (96’) and for the3D cubic NLW by Burq and Tzvetkov (11’).

This approach was applied in the context of the Navier-Stokes to obtain:I Local in time solutions to the corresponding integral equation for

randomized initial data in L2(T3) by Zhang and Fang (2011) and by Dengand Cui (2011). Also global in time solutions to the corresponding integralequation for randomized small initial data.

I Deng and Cui (2011) obtained local in time solutions to the correspondingintegral equation for randomized initial data in Hs(Td), for d = 2, 3 with−1 < s < 0.

We are concern with existence of global in time weak solutions(NSE)for randomized initial data (without any smallness assumption) innegative Sobolev spaces H−α(Td ),d = 2,3, for some α = α(d) > 0.


Main Ideas

We start with a divergence free and mean zero initial data~f ∈ (H−α(Td ))d ,d = 2,3 and suitably randomize it to obtain~fω preservingthe divergence free condition.

Key point: although the initial data is in H−α for some α > 0, therandomized data and its heat flow have almost surely improved Lp

bounds. These bounds yield improved nonlinear estimates arising in theanalysis of the difference equation for ~w almost surely.

This induced ‘smoothing’ phenomena -akin to the role of Kintchineinequalities in Littlewood-Paley theory- stems from classical results ofRademacher, Kolmogorov, Paley and Zygmund proving that randomseries on the torus enjoy better Lp bounds than deterministic ones.


For example, consider Rademacher Series

f (τ) :=∞∑

n=0

an rn(τ) τ ∈ [0,1), an ∈ C

rn(τ) := sign sin(2n+1π τ), n ≥ 0

rk,j (τ) := rk (τ)rj (τ), 0 ≤ k < j <∞ is o.n. over(0,1)

If an ∈ `2 the sum f (τ) converges a.e.

Classical Theorem (cf. Zygmund Vol I)If an ∈ `2 then the sum f (τ) belongs to Lp([0,1)) for all p. More precisely,

(

∫ 1

0|f |p dτ )1/p ≈p ‖an ‖`2

In particular, p > 2 !


These ideas were already exploited in Bourgain’s work on NLS, KdV,mKdV, Zakharov system.

I Almost surely global well-posedness on the statistical ensemble via theexistence and invariance of the Gibbs measure (after Lebowitz, Rose andSpeer’s and Zhidkov’s works).

I Well-posedness’ failure might come from certain ‘exceptional’ initial data set.The virtue of the Gibbs measure-or weighted Wiener measure- is that it doesnot see that exceptional set.

The starting point of this method is a good local theory on the statisticalensemble (support of the measure) of randomized data of the form

φ = φω(x) =∑ gn(ω)

|n|αei〈x,n〉,

where gn(ω)n are independent standard (complex/real) Gaussianrandom variables on a probability space (Ω,F ,P) ( morally ‘φn = gn

|n|α ’)and αdepends on the equation and dimension.If S(t)φω is the linear evolution of the problem at hand and u the solution,one shows that w = u − S(t)φω almost surely in ω is smoother than thelinear part.


The invariance of the measure, just like the usual conserved quantities, isused to control the growth in time of those solutions in its support andextend the local in time solutions to global ones almost surely.

Some recent works (after Bourgain’s ):I Tzvetkov for subquintic radial nonlinear wave equation on the disc.I Burq-Tzvetkov for subcubic and subquartic (radial if via measure) nonlinear

wave equations on 3D ball.I T. Oh in his thesis for the periodic KdV-type coupled systems. Then for white

noise for the KdV equation, and Schrodinger-Benjamin-Ono system.I Nahmod- Oh - Rey Bellet- S.– and by Nahmod - Rey Bellet - Sheffield - S.–

for the derivative NLS equation on T. Need to understand how gaussianmeasures and their supports transform under gauge transformations(periodic setting).

I Colliander-Oh for the cubic NLS below L2(T) (no measure).


Back to Navier-Stokes: Main Steps

We start with an initial data ~f ∈ H−α, α > 0, hence supercritical. Assume~an are the Fourier coefficients of ~f .

Randomizing ~f means that we replace ~an by ln(ω)~an , where ln(ω)are independent random variables, and we take its Fourier series ~fω asthe new randomized initial data.

We seek a solution to the initial value problem (NSE) in the form~u = et∆~fω + ~w and identify the difference equation that ~w should satisfy.

The heat flow of the randomized data gives almost surely improved Lp

bounds. These bounds yield improved nonlinear estimates arising in theanalysis of the difference equation for ~w almost surely.

We revisit the proof of equivalence between the initial value problem forthe difference equation and the integral formulation of it in our context(similar to Lemarie and Furioli, Lemarie and Terraneo).


We prove a priori energy estimates for ~w . The integral equationformulation is used near time zero and the other one away from zero.

A construction of a global weak solution to the difference equation via aGalerkin type method is thus possible.

We prove uniqueness of weak solutions when d = 2. Our proof is done‘from scratch’ for the difference equation (in spirit ofLadyzhenskaya-Prodi-Serrin condition).

Put all ingredients together to conclude.

RemarkWe should immediately notice that although in our paper we use improvedproperties for et∆~fω, one can show that already ~fω belongs to certain criticalBesov spaces for wich Gallagher-Planchon already proved in 2d globalwell-posedness. On the other hand while their proof is based on acombination of the high-low argument of Bourgain and the H. Kock-Tatarusmall BMO−1 data result, ours is much more self contained and gives moreprecise energy estimate. Moreover our existence result extends to 3d, asmentioned.


Randomization SetupLemma [Large deviation bound] Burq-Tzevtkov 08’

Let (lr (ω))∞r=1 be a sequence of real, 0 mean, independent random variableson a probability space (Ω,A,p) with associated sequence of distributions(µr )∞r=1. Assume that µr satisfy the property

(†) ∃c > 0 : ∀γ ∈ R,∀r ≥ 1, |∫ ∞−∞

eγx dµr (x)| ≤ ecγ2.

Then there exists α > 0 such that for every λ > 0, every sequence(cr )∞r=1 ∈ `2 of real numbers,

p

(ω : |

∞∑r=1

cr lr (ω)| > λ

)≤ 2e

− αλ2∑r c2

r .

As a consequence there exists C > 0 such that for every q ≥ 2 and every(cr )∞r=1 ∈ `2, ∥∥∥∥∥

∞∑r=1

cr lr (ω)

∥∥∥∥∥Lq(Ω)

≤ C√

q

( ∞∑r=1

c2r

) 12

.


Burq and Tzvetkov showed that the standard real Gaussian as well asstandard Bernoulli variables satisfy the assumption (†)

Definition [Diagonal randomization]Let (ln(ω))n∈Zd be a sequence of of real, independent, random variables on aprobability space (Ω,A,p) For ~f ∈ (Hs(Td ))d , let (ai

n), i = 1,2, . . . ,d , be itsFourier coefficients. We introduce the map from (Ω,A) to (Hs(Td ))d equippedwith the Borel sigma algebra, defined by

(DR) ω −→ ~fω, ~fω(x) =

∑n∈Zd

ln(ω)a1nen(x), . . . ,

∑n∈Zd

ln(ω)adn en(x)

,

where en(x) = ein·x and call such a map randomization.


Remarks

The map (DR) is measurable and ~fω ∈ L2(Ω; (Hs(Td ))d ), is an(Hs(Td ))d -valued random variable.

The diagonal randomization defined in (DR) commutes with the Lerayprojection P.

No Hs regularization ‖~fω‖Hs ∼ ‖~f‖Hs (Burq-Tzvetkov).

But randomization gives improved Lp estimates (almost surely).


Main ResultsTheorem [Existence and Uniqueness in 2D]

Fix T > 0, 0 < α < 12 and let ~f ∈ (H−α(T2))2, ∇ ·~f = 0 and of mean zero.

Then there exists a set Σ ⊂ Ω of probability 1 such that for any ω ∈ Σ theinitial value problem (NSE) with datum ~fω has a unique global weak solution ~uof the form

~u = ~u~fω + ~w

where ~u~fω = et∆~fω and ~w ∈ L∞([0,T ]; (L2(T2))2) ∩ L2([0,T ]; (H1(T2))2).

Theorem [Existence in 3D]

Fix T > 0, 0 < α < 13 and let ~f ∈ (H−α(T3))3, ∇ ·~f = 0, and of mean zero.

Then there exists a set Σ ⊂ Ω of probability 1 such that for any ω ∈ Σ the initialvalue problem (NSE) with datum ~fω has a global weak solution ~u of the form

~u = ~u~fω + ~w ,

where ~u~fω = et∆~fω and ~w ∈ L∞([0,T ]; (L2(T3))3) ∩ L2([0,T ]; (H1(T3))3).


Free Evolution of the Randomized Data

Deterministic estimates.For 0 < α < 1, k ≥ 0 integer and ~u~fω = et∆~fω, ~fω ∈ (H−α(Td ))d , we have:

‖∇k~u~fω (·, t)‖L2x

. (1 + t−α+k

2 ) ‖~f‖H−α .

‖∇k~u~fω‖L∞x .(

maxt−1, t−(k+α+ d2 )) 1

2 ‖~f‖H−α .

Probabilistic estimates.Let T > 0 and α ≥ 0. Let r ≥ p ≥ q ≥ 2, σ ≥ 0 and γ ∈ R be such that(σ + α− 2γ)q < 2. Then there exists CT > 0 such that for every~f ∈ (H−α(Td ))d

‖tγ(−∆)σ2 et∆~fω‖Lr (Ω;Lq([0,T ];Lp

x ) ≤ CT ‖~f‖H−α ,

where CT may depend also on p,q, r , σ, γ and α.


Probabilistic estimates (cont.)Moreover, if we set

Eλ,T ,~f ,σ,p = ω ∈ Ω : ‖tγ(−∆)

σ2 et∆~fω‖Lq([0,T ];Lp

x ) ≥ λ,

then there exists c1, c2 > 0 such that for every λ > 0 and for every~f ∈ (H−α(Td ))d

P(Eλ,T ,~f ,σ,p) ≤ c1 exp

[−c2

λ2

CT‖~f‖2H−α

].


Difference Equation. Equivalent Formulations

Let

H = the closure of ~f ∈ (C∞(Td ))d | ∇ ·~f = 0 in (L2(Td ))d ,

V = the closure of ~f ∈ (C∞(Td ))d | ∇ ·~f = 0 in (H1(Td ))d ,

V ′ = the dual of V .

and recall~u − ~u~fω =: ~w ,

We consider two formulations of the initial value problem for the differenceequation that ~w solves and re-prove in our context an equivalence lemma,which is similar to the version for the Navier-Stokes equations themselves(Lemarie, Furioli-Lemarie, Terraneo).


The Equivalence LemmaLet T > 0. Assume that ∇ · ~g = 0, ‖~g(x , t)‖L2 . (1 + 1

tα2

) and‖~g‖L4([0,T ],L4

x ) ≤ C, if d = 2‖~g‖L6([0,T ],L6

x ) ≤ C, if d = 3,

for some C > 0. Then the following statements are equivalent.(DE) ~w is a weak solution to the initial value problem ∂t ~w = ∆~w − P∇(~w ⊗ ~w) + c1[P∇(~w ⊗ ~g) + P∇(~g ⊗ ~w)] + c2P∇(~g ⊗ ~g)

∇ · ~w = 0,~w(x ,0) = 0.

(IE) The function ~w ∈ L∞((0,T ); H) ∩ L2((0,T ),V ), solves

~w(t) = −∫ t

0e(t−s)∆∇~F (x , s) ds, where

~F (x , s) = −P(~w ⊗ ~w) + c1[P(~w ⊗ ~g) + P(~g ⊗ ~w)] + c2P(~g ⊗ ~g).


Energy Estimates for the Difference Equation

E(~w)(t) = ‖~w(t)‖2L2 + c

∫ t

0

∫Td|∇ ⊗ ~w |2 dx ds

Theorem (Energy Estimates).Let T > 0, λ > 0, γ < 0, and α > 0 be given. Let ~g be s.t. ∇ · ~g = 0 and

‖~g(x , t)‖L2 . (1 +1

t α2), ‖∇k~g(x , t)‖L∞ .

(maxt−1, t−(k+α+ d

2 )) 1

2k = 0,1;

‖tγ~g‖L4([0,T ];L4

x ) ≤ λ, if d = 2‖tγ~g‖L6([0,T ];L6

x ) ≤ λ, if d = 3.

Let ~w ∈ L∞((0,T ); H) ∩ L2((0,T ); V ) be a solution to (DE). Then,

E(~w)(t) . C(T , λ, α), for all t ∈ [0,T ].

‖ ddt~w‖Lp

t H−1x≤ C(T , λ, α),

where p = 2, if d = 2 and p = 43 , if d = 3.


We rely on the equivalence lemma and use the integral equationformulation for ~w near time zero and the other one away from zero.These a priori estimates for ~w are then used in conjunction with Galerkinapproximations to construct weak solutions.Related work by T. Tao (07’).

Sketch of proof: consider two cases: t near zero and t away from zero.

Case t near zero: By (IE):

~w(t) = −∫ t

0e(t−s)∆∇~F (x , s) ds,

We use a continuity argument: assume 0 ≤ τ ≤ δ∗, where δ∗ will bedetermined later. Then for τ ∈ [0, δ∗] we have:

‖~w(t)‖L2x. ‖~F‖L2

t∈[0,τ ]L2

x, for all t ∈ [0, τ ].

Also by applying the maximal regularity we obtain:

‖~w‖L2t∈[0,τ ]

H1x. ‖~F‖L2

t∈[0,τ ]L2

x.

Hence it suffices to analyze ‖~F‖L2t∈[0,τ ]

L2x.


We have,

‖~F‖L2t∈[0,τ ]

L2x. ‖~w ⊗ ~w‖L2

t∈[0,τ ]L2

x+‖~w ⊗ ~g‖L2

t∈[0,τ ]L2

x+ ‖~g ⊗ ~w‖L2

t∈[0,τ ]L2

x+‖~g ⊗ ~g‖L2

t∈[0,τ ]L2

x.

‖~w ⊗ ~w‖L2t∈[0,τ ]

L2x

= ‖~w‖2L4

t∈[0,τ ]L4

x. E(~w)(τ),

by interpolating L∞t L2x and L2

t H1x in E(~w)(t).

For the next two terms by Holder’s inequality we have:

‖~w ⊗ ~g‖L2t∈[0,τ ]

L2x

+ ‖~g ⊗ ~w‖L2t∈[0,τ ]

L2x. ‖~g‖Lp

x,t‖~w‖

L2p

p−2x,t

. ‖~g‖Lpx,t‖D

dp ~w‖

L2p

p−2t L2

x

. ‖tγ~g‖Lpx,t

(δ∗)−γ‖Ddp ~w‖

L2pd

t L2x

(δ∗)p−2−d

2p ,

by Sobolev embedding and Holder’s inequality in t (recall p ≥ d + 2.).


By letting p = 4 when d = 2, and p = 6 when d = 3 it follows from theassumptions on ~g, in conjunction with interpolation between the spaces thatappear in E(~w)(t), that

‖~w ⊗ ~g‖L2t∈[0,τ ]

L2x

+ ‖~g ⊗ ~w‖L2t∈[0,τ ]

L2x. λ (δ∗)−γ+β(d)E

12 (~w)(τ),

where

β(d) =

0, if d = 21

12 , if d = 3.

Finally the last term can be estimated as

‖~g ⊗ ~g‖L2t∈[0,τ ]

L2x

= ‖~g‖2L4

t∈[0,τ ]L4

x≤(λ(δ∗)−γ+β(d)

)2,

with β(d) as above.


Combining the estimates we obtain:

E12 (~w)(τ) ≤ C1E(~w)(τ) + C2λ(δ∗)−γ+β(d)E

12 (~w)(τ) + C3

(λ(δ∗)−γ+β(d)

)2.

Hence if we denote E12 (~w)(τ) = X , we obtain the inequality:

X ≤ C1X 2 + C2λ(δ∗)−γ+β(d)X + C3(λ(δ∗)−γ+β(d)

)2.

By a continuity argument X is bounded for all τ ∈ [0, δ∗] providedC3(λ(δ∗)−γ+β(d)

)2 is small enough2. Hence,

E(~w)(τ) ≤ C,

for all τ ∈ [0, δ∗].

2Depending only on C1,C2 and C3Gigliola Staffilani (MIT) a.s. global supercritical Navier-Stokes June , 2012 30 / 41

Case t ∈ [δ∗,T ]. By a standard energy argument for (DE) we have,

ddt

E(~w)(t) =

∫Td

2~w(x , t) · ~wt (x , t) dx + 2∫Td|∇ ⊗ ~w |2(x , t) dx

=

∫Td

2~w(x , t)∆~w dx − 2∫Td

~w · P∇(~w ⊗ ~w) dx + 2∫Td|∇ ⊗ ~w |2(x , t) dx

+ 2(∫

Td

~w · P∇(~w ⊗ ~g) dx +

∫Td

~w · P∇(~g ⊗ ~w) dx)

+ 2∫Td

~w · P∇(~g ⊗ ~g)dx .

The expression on second line equals zero as in the case of solutions tothe Navier-Stokes equations itself.

To estimate the third line note that since ~g is divergence-free,∫Td ~w · P∇(~w ⊗ ~g) dx =

∫Td ~w · P(~g · ∇)~w dx and the last expression

equals zero (skew-symmetry).

Also since ~w is divergence-free too:∫Td

~w · P∇(~g ⊗ ~w) dx =

∫Td

~w · P(~w · ∇~g) dx . ‖~w‖2L2

x‖∇~g‖L∞x .


On the other hand by Holder’s inequality,∫Td

~w · P∇(~g ⊗ ~g) dx ≤ ‖~w‖L2x‖~g‖L2

x‖∇~g‖L∞x .

Combining the above and using the assumptions on ~g we obtain:

ddt

E(~w)(t) . E(~w)(t) ‖∇~g‖L∞x + E12 (~w)(t) ‖~g‖L2

x‖∇~g‖L∞x

. h(t) E(~w)(t) + m(t) E12 (~w)(t),

whence

E(~w)(t) ≤ C(T , δ∗, α) for all t ∈ [δ∗,T ].


Construction of Weak Solutions to the DifferenceEquation

Write~f (x , t) =

∑k

~f (k, t)eik·x ,

where k is the discrete wavenumber:

k =d∑

j=1

(2πnj )ej , nj ∈ Z,

and ej is the unit vector in the j-th direction. By PM we denote the rectangularFourier projection operator:

PM~f (t) =

∑k : |nj |≤M for 1≤j≤d

~f (k, t)eik·x .


Theorem (Existence).Let T > 0, λ > 0, γ < 0 and α > 0 be given. Assume that the function ~gsatisfies ∇ · ~g = 0 and

‖~g(x , t)‖L2 . (1 +1

t α2)

‖∇k PM~g(x , t)‖L∞ .(

maxt−1, t−(k+α+ d2 )) 1

2for k = 0,1.

Furthermore, assume that we have:‖tγ~g‖L4

x ;t∈[0,T ]≤ λ, if d = 2

‖tγ~g‖L6x ;t∈[0,T ]

≤ λ, if d = 3.

Then there exists a weak solution ~w for the initial value problem (DE).

Idea of the proof In the construction of weak solutions, we follow in part theapproach based on Galerkin approximations of Doering and Gibbon and ofConstantin and Foias. The plan is to construct a global weak solution viafinding its Fourier coefficients, which, in turn, will be achieved by solving finitedimensional ODE systems for them.


Uniqueness in 2D

Theorem (Uniqueness in 2D)Assume that ~g satisfies the same conditions as above Then in d = 2 any twoweak solutions to (DE) in L2([0,T ]; V ) ∩ L∞((0,T ); H) coincide.

Our proof is inspired by the proof in Constantin-Foias which establishes arelated uniqueness result for solutions to the Navier-Stokes equations.Let ~wj with j = 1,2 be two solutions of (DE) with ~g as above. Let ~v = ~w1 − ~w2;then ∂t~v = ∆~v − P∇(~w1 ⊗ ~v)− P∇(~w2 ⊗ ~v) + c1

[P∇(~v ⊗ ~g) + P∇(~g ⊗ ~v)

]∇ · ~v = 0,~v(x ,0) = 0.

Pair in (L2(T2))2 the first equation with ~v and estimate using estimates on ~gand ~wj .


Proof of Main Theorems: Gathering all the PiecesWe find solutions ~u to (NSE) by writing

~u = ~uω~f + ~w

where we recall that ~uω~f is the solution to the linear problem with initial datum~fω and ~w is a solution to (DE) with ~g = ~uω~f .

~u is a weak solution for (NSE) if and only if ~w is a weak solution for (DE).We also remark that uniqueness of weak solutions to (DE) is equivalentto uniqueness of weak solutions (NSE).The proof of the existence of weak solutions is the same for both d = 2and d = 3 and it is a consequence existence theorem above. For theuniqueness claimed in d = 2 we invoke the uniqueness theorem above.Now to the details.

Let γ < 0 be such that

0 < α <

12 + 2γ, if d = 213 + 2γ, if d = 3.


By the probabilistic estimates with σ = 0, p = q = 4 when d = 2, andp = q = 6 when d = 3 we have that given λ > 0, if we define the set

Eλ := Eλ,α,~f ,γ,T = ω ∈ Ω / ‖tγ~uω~f ‖Lp

[0,T ],x> λ.

Then there exist C1,C2 > 0 such that

P(Eλ) ≤ C1 exp

−C2

(λ

CT‖~f‖H−α

)2 .

Now, let λj = 2j , j ≥ 0 and define Ej = Eλj . Note Ej+1 ⊂ Ej . Let

Σ := ∪Ecj ⊂ Ω.

Then

1 ≥ P(Σ) = 1− limj→∞

P(Ej ) ≥ 1− limj→∞

exp

−C2

(2j

CT‖~f‖H−α

)2 = 1.


Final Step:

Our goal is now to show that for a fixed divergence free vector field~f ∈ (H−α(Td ))d and for any ω ∈ Σ, if we define ~g = ~uω~f , the initial valueproblem (DE) has a global weak solution. In fact given ω ∈ Σ, there exists jsuch that ω ∈ Ec

j . In particular we then have

‖tγ ~g‖Lpx,T≤ λj .

Hence assumptions on ~g in the previous theorems are satisfied. Thisconcludes the proof.


Appendix

Let B be a Banach space of functions. The space Cweak((0,T ),B)denotes the subspace of L∞((0,T ),B) consisting of functions which areweakly continuous, i.e. v ∈ Cweak((0,T ),B) if and only if φ(v(t)) is acontinuous function of t for any φ ∈ B∗.

Definition (weak solution)

Let ~f ∈ (H−α(Td ))d , α > 0, ∇ ·~f = 0, and of mean zero.A weak solution to (NSE) on [0,T ], is a function~u ∈ L2

loc((0,T ); V ) ∩ L∞loc((0,T ); H) ∩ Cweak((0,T ); (H−α(Td ))d )

satisfyingd~udt∈ L1

loc((0,T ),V ′) and

〈d~u

dt, ~v〉+ ((~u, ~v)) + b(~u, ~u, ~v) = 0 for a.e. t and for all ~v ∈ V ,

limt→0+

~u(t) = ~f weakly in the (H−α(Td ))d topology.


Given two vectors ~u and ~v in Rd we use the notation

〈~u, ~v〉 = ~u · ~v .

In (L2(Td ))d we use the inner product notation

(~u, ~v) =

∫~u(x) · ~v(x) dx .

In (H1(Td ))d we use the inner product notation

((~u, ~v)) =d∑

i=1

(Di~u,Di~v).

Finally we introduce the trilinear expression

b(~u, ~v , ~w) =

∫~ujDj~vi ~wi dx =

∫〈~u · ∇~v , ~w〉dx .

Also we note that when ~u is divergence free, we have

b(~u, ~v , ~w) =

∫〈∇(~v ⊗ ~u), ~w〉dx .


Definition

Assume that ∇ · ~g = 0. A weak solution to (DE) on [0,T ], is a function

~w ∈ L2((0,T ); V ) ∩ L∞((0,T ); H) satisfyingd ~wdt∈ L1((0,T ); V ′) and such that

for almost every t and for all ~v ∈ V ,

〈d~w

dt, ~v〉+ ((~w , ~v)) + b(~w , ~w , ~v) + b(~w , ~g, ~v) + b(~g, ~w , ~v) + b(~g, ~g, ~v) = 0

andlim

t→0+~w(t) = 0 weakly in the H topology.


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