E quations aux

S E M I N A I R E

E quations aux

D ´e r i v ´e e s

P a r t i e l l e s

2005-2006

Jean-Yves Chemin and Ping Zhang

The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations

Séminaire É. D. P.(2005-2006), Exposé n^oVIII, 18 p.

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The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations

Jean-Yves CHEMIN ^∗ and Ping ZHANG ^†

∗ Laboratoire J.-L. Lions, Case 187

Universit´e Pierre et Marie Curie, 75230 Paris Cedex 05, FRANCE chemin@ann.jussieu.fr

† Academy of Mathematics & Systems Science, CAS Beijing 100080, CHINA.

E-mail: zp@amss.ac.cn

April 14, 2006

Abstract

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equa- tions (N Sν) with initial data in the scaling invariant Besov space, B⁻¹⁺

3

p,∞p, here we con- sider a similar problem for the 3-D anisotropic Navier-Stokes equations (AN Sν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces,B⁻

1 2,¹2

4 andB⁻

1 2,¹2

4 (T).Then with initial data in the scaling invariant spaceB⁻

1 2,¹2

4 ,

we prove the global wellposedness for (AN Sν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (AN Sν) with high oscillatory initial data (1.2).

1 Introduction

1.1 Introduction to the anisotropic Navier-Stokes equations

Let us first recall the classical (isotropic) Navier-Stokes system for incompressible fluids in the whole space:

(N S_ν)





∂tu+u· ∇u−ν∆u=−∇p, divu= 0,

u|t=0 =u₀,

whereu(t, x) denote the velocity,p(t, x) the pressure andx= (x_h, x₃) a point ofR³ =R²×R.

In this text, we are going to study a version of the system (N S_ν) where the usual Laplacian is substituted by the Laplacian in the horizontal variables, namely

(AN S_ν)





∂_tu+u· ∇u−ν∆_hu=−∇p, divu= 0,

u|t=0 =u₀.

Systems of this type appear in geophysical fluids (see for instance [5]). It has been studied first by J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier in [6] and D. Iftimie in [10]

where it is proved that (AN S_ν) is locally wellposed for initial data in the anisotropic space H^0,12+εdef= n

u∈L²(R³)/ kuk²_˙

H^{12 +ε} def=

Z

R³|ξ₃|^1+2ε|u(ξˆ _h, ξ₃)|²dξ <+∞o ,

for someε >0.Moreover, it is also proved that small enough datau0 is in the sense that ku₀k^εL²ku₀k^1−ε_˙

H^{0,12 +ε} ≤cν (1.1)

for some sufficiently small constant c, then we have a global wellposedness result. Let us notice that the space in which uniqueness is proved is the space of continuous functions with value in H^0,12+ε and the horizontal gradient of which belongs to L²([0, T];H^0,12+ε).

Let us observe that, as classical Navier-Stokes system, the system (AN S_ν) has a scaling.

Indeed, if u is a solution of (AN Sν) on a time interval [0, T] with initial data u0, then the vector field u_λ defined by u_λ(t, x) ^def= λu(λ²t, λx) is also a solution of (AN S_ν) on the time interval [0, λ⁻²T] with the initial data λu0(λx). The smallness condition (1.1) is of course scaling invariant. But the norm k · k_H˙0,1

2 +ε is not and this norm determines the level of regularity required to have wellposedness.

For classical Navier-Stokes system a lot of results of global wellposedness in scaling in- variant space are available. The first one is the theorem of Fujita-Kato (see [9]) in which it is proved that the system (N S_ν) is globally wellposed for small initial data in the Sobolev space ˙H¹² which is the space of tempered distributions u such that

kuk²_˙

H¹² def=

Z

R³|ξ| |bu(ξ)|²dξ <∞.

M. Paicu proved in [12] a theorem of the same type for the system (AN S_ν) in the case when the initial data u0 belongs to the scaling invariant spaceB^0,12 (see Definition 1.2 below).

On the other hand, the classical isotropic system (N S_ν), is globally wellposed for small initial data in Besov norms of negative index. Let us first recall the definition of the Besov norms of negative index.

Definition 1.1 Let f be inS⁰(R³). Then we state, for positive s, and for (p, q) in[1,∞]², kfkB˙p,q^−s

def=

kt^s2e^t∆fkL^p

L^q(R⁺,^dt_t).

In [2], M. Cannone, Y. Meyer and F. Planchon proved that, if the initial data satisfies, for some p greater than 3, ku₀k

B˙

−1+ 3p p,∞

≤ cν for some constant c small enough, then the incompressible Navier-Stokes system is globally wellposed. Let us mention that H. Koch and D. Tataru generalized this theorem to the∂BM O norm (see [11]).

In particular, this theorem implies that, for any functionφin the Schwartz space S(R³), if we consider the family of initial datau^ε₀ defined by

u^ε₀(x) = sinx1

ε

(0,−∂3φ, ∂2φ,), (1.2) the system (N S_ν) is globally wellposed for such initial data whenεis small enough. The goal here is to prove the same for the anisotropic Navier-Stokes system (AN S_ν).

1.2 Statement of the results

Let us begin by the definition of the spaces we are going to work with. It requires an anisotropic version of dyadic decomposition of the Fourier space, let us first recall the following operators of localization in Fourier space, for (k, `)∈Z²,

∆^h_ka=F⁻¹(ϕ(2^−k|ξ_h|)ba), and ∆^v<sub>`</sub>a=F<sup>−1</sup>(ϕ(2<sup>−`</sup>|ξ₃|)ba), S^h_ka= X

k⁰≤k−1

∆^h_k⁰a, and S_`^va= X

`<sup>0</sup>≤`−1

∆^v_`⁰a, (1.3)

whereFaand badenote the Fourier transform ofa, and ϕa function inDh3 4, 8

3 i

such that

∀τ >0, X

j∈Z

ϕ(2^−jτ) = 1.

Before we present the space we are going to work with, let us first recall the Besov-Sobolev type space B^0,12 defined by M. Paicu in [12].

Definition 1.2 We denote by B^0,12 the space of ainS(R³) such that ba∈L¹_loc and kak_B0,1

2

def= X

`∈Z

2<sup>`2</sup>k∆<sup>v</sup><sub>`</sub>akL²(R³)<∞.

In [12], M. Paicu proved the global wellposedness of (AN S_ν) for small initial data in B^0,12 . In order to state Paicu’s Theorem, let us introduce the following space.

Definition 1.3 We denote by B^0,12(T) the space of ainC^∞([0, T],B^0,12 such that kak_B0,1

2(T)

def= X

`∈Z

2^2`

k∆^v_{`</sub>akL<sup>∞</sup><sub>T</sub>(L<sup>2</sup>(R<sup>3</sup>))+ν<sup>12</sup>k∇h∆<sup>v</sup><sub>`}akL²_T(L²(R³))

<∞. Now let us recall M. Paicu’s theorem.

Theorem 1.1 If u₀ ∈ B^0,12, then a positive time T exists such that the system(AN S_ν) has a unique solution u inB^0,12(T). Moreover, a constant c exists such that

ku0k_B0,1

2 ≤cν=⇒T = +∞.

Let us note that u^ε₀ defined in (1.2) is not small in this space B^0,21 no matter how small the parameterεis. Our motivation to introduce the following spaces is to find a scaling invariant space such that in particular u^ε₀ is small in this space forεsufficient small.

Definition 1.4 We denote by B⁻

1 2,₂¹

4 the space ofa inS⁰(R³)such that b

a∈L¹_loc and kak

B⁻₄ ^12,12

def= X

`∈Z

2^2` X∞

k=`−1

2^−kk∆^h_k∆^v_`ak²_L⁴

h(L²_v)

¹₂

+X

j∈Z

2^j2kS_j−1^h ∆^v_jakL²(R³).

Definition 1.5 We denote by B⁻4 ^12,21(T) the space ofainC([0, T],B⁻4 ^12,12)such that kak

B⁻

12,1 4 2(T)

def= X

`∈Z

2^2`

X∞

k=`−1

2^−kk∆^h_k∆^v_`ak²_L^∞

T(L⁴_h(L²v))

¹₂

+ν¹² X∞

k=`−1

2^kk∆^h_k∆^v_`ak²_L²

T(L⁴_h(L²_v))

¹₂!

+X

j∈Z

2^2j

kS_j−1^h ∆^v_jakL^∞_T(L²(R³))+ν¹²k∇hS_j−1^h ∆^v_jakL²_T(L²(R³))

. In the following section, we shall use Littlewood-Paley theory to study the inner relations betweenB4^−12,12(T) and B^0,12(T). Now, we present the main results of this paper.

Theorem 1.2 A constant c exists such that, ifu0∈ B⁻

1 2,¹₂

4 and ku0k

B⁻

12,1 4 2

≤cν, then, with initial data u₀, the system (AN S_ν) has a unique global solution u inB⁻

1 2,¹₂ 4 (∞).

This theorem can be applied to initial data given by (1.2) thanks to the following proposition, proved in section 2.

Proposition 1.1 Letφ∈ S(R³). Ifφε(x)^def= e^ix1/εφ(x), thenkφεk

B

−12,1 4 2

=O(ε¹²).

Classically, a global wellposedness theorem with small data in a space where smooth functions are dense corresponds to a version concerning local wellposedness for large data.

Theorem 1.3 Ifu₀ belongs toB4^−21,12, then a positiveT exists such that the system (AN S_ν) has a unique solution in the space B⁻

1 2,¹₂ 4 (T).

1.3 Structure of the text

This text does not contain all the details of the proofs and we refer to [7] for complete proofs.

The purpose of the second section is to state some results about anisotropic Littlewood-Paley theory which will be of a constant use in what follows.

The third section will be devoted to the proof of the existence of a solution of (AN S_ν).

In order to do it, we shall search for a solution of the form u=uF +w with uF

def= e^νt∆hu_hh, u_hh^def= X

k≥`−1

∆^h_k∆^v_`u0 and w∈ B^0,12(∞). (1.4) In the last section, we shall prove the uniqueness in the following way. First, we shall establish a regularity theorem claiming that if u ∈ B4^−12,12(T) is a solution of (AN S_ν) with initial data in B⁻

1 2,¹₂

4 , then w =u−uF ∈ B^0,12(T). Therefore, looking at the equation of w, we shall prove the uniqueness of the solution u in the spaceu_F +B^0,12(T). Let us point out that the uniqueness result we prove is surprinsingly not a stability result. We should mention that the method introduced by M. Paicu in [12] will play a crucial role in our proof here.

We present only the sketch of the proof and refer to [7] for the details.

Notations: Let A, B be two operators, we denote [A;B] = AB−BA, the commutator between A and B, a . b, we means that there is a uniform constant C, which may be different on different lines, such that a ≤ Cb. Finally, we denote L^r_T(L^p_h(L^q_v)) the space L^r([0, T];L^p(R_x1×R_x2;L^q(R_x3))).

2 Some properties of anisotropic Littlewood-Paley theory

As we shall constantly use the anisotropic Littlewood-Paley theory, and in particular aniso- tropic Bernstein inequalities. We list them as the following:

Lemma 2.1 Let Bh (resp. Bv) a ball of R²_h (resp. R_v), and Ch (resp. Cv) a ring of R²_h (resp. R_v); let 1≤p₂ ≤p₁≤ ∞and 1≤q₂ ≤q₁ ≤ ∞.Then there holds:

If the support of bais included in2^kBh, then k∂_x^α

hak_L^p1

h (L^qv¹) .2^k

“

|α|+2“

1 p2−_p¹

1

””

kak_L^p2

h (L^qv¹). If the support of bais included in2^`Bv, then

k∂₃^βakL^p_h¹(L^qv¹) .2^{`(β+(q12−q11))}kakL^p_h¹(L^qv²). If the support of bais included in2^kCh, then

kakL^p_h¹(L^qv¹).2^−kN sup

|α|=Nk∂_h^αakL^p_h¹(L^qv¹). If the support of bais included in2^kCv, then

kakL^p_h¹(L^qv¹).2^−kNk∂₃^NakL^p_h¹(L^qv¹).

Let us state two corollaries of this lemma, the proof of which are obvious and thus omitted.

Corollary 2.1 The space B^0,12 is included in B⁻4 ^12,12 and so is B^0,12(T) inB4^−12,12(T) for any positive T. Moreover, the spaceB^0,12(T) is included inL^∞_T (L²_h(L^∞_v )).

Corollary 2.2 If abelongs toB⁻

1 2,¹₂

4 (T), then we have X

`∈Z

2^2`X

k∈Z

2^−kk∆^h_k∆^v_`a(0)k²_L⁴

h(L²_v)

¹₂

.ka(0)k

B⁻

12,1 4 2

and X

`∈Z

2^`2X

k∈Z

2^−kk∆^h_k∆^v_`ak²_L^∞

T(L⁴_h(L²_v))+ν2^kk∆^h_k∆^v_`ak²_L²

T(L⁴_h(L²_v))

¹2

.kak

B

−12,1 4 2(T). Proposition 1.1 tells us how large is the difference between the normsk · k_B0,1

2 andk · k

B

−12,1 4 2

.

Proof of Proposition 1.1 By definition of the normk · k

B⁻

12,1 4 2

, we have, as thek · k`<sup>2</sup> norm is less than or equal to thek · k`¹ norm,

kφ_εk

B⁻₄ ^12,12 ≤ X4

j=1

Φ^(j)_ε with Φ⁽¹⁾_ε ^def= X

ε2^k>1 k≥`−1

2<sup>−k−2`</sup>k∆<sup>h</sup><sub>k</sub>∆<sup>v</sup><sub>`</sub>φ_εkL⁴_h(L²_v),

Φ⁽²⁾_ε ^def= X

ε2^k≤1 k≥`−1

2<sup>−k−2`</sup>k∆<sup>h</sup><sub>k</sub>∆<sup>v</sup><sub>`</sub>φεkL⁴_h(L²_v),

Φ⁽³⁾_ε ^def= X

ε2^j>1

2^j2kS_j−1^h ∆^v_jφεkL² and Φ⁽⁴⁾_ε ^def= X

ε2^j≤1

2^j2kS_j−1^h ∆^v_jφ_εkL².

In order to estimate Φ⁽¹⁾_ε , let us notice that Φ⁽¹⁾_ε ≤ X

ε2^k>1

2^−k2 X

`∈Z

2^`2 sup

k∈Zk∆^h_k∆^v_`φ_εkL⁴_h(L²_v) ≤ε¹² X

`∈Z

2^`2sup

k∈Zk∆^h_k∆^v_`φ_εkL⁴_h(L²_v). Using Lemma 2.1, we have, by definition of φ_ε,

sup

k∈Zk∆^h_k∆^v_`φ_εkL⁴_h(L²v).kφ_εkL⁴_h(L²v).kφkL⁴_h(L²v). and sup

k∈Zk∆^h_k∆^v<sub>`</sub>φεkL<sup>4</sup><sub>h</sub>(L<sup>2</sup><sub>v</sub>).2<sup>−`</sup>k∂3φεkL⁴_h(L²_v) .2<sup>−`</sup>k∂3φkL<sup>4</sup><sub>h</sub>(L<sup>2</sup><sub>v</sub>). Thus, taking the sum over `≤N and` > N and choosing the best N gives

Φ⁽¹⁾_ε ≤ε¹² X

`∈Z

2^`2 sup

k∈Zk∆^h_k∆^v_`φεkL⁴_h(L²_v)≤ε¹²kφk

1 2

L⁴_h(L²v)k∂3φk

1 2

L⁴_h(L²v). The estimate of Φ⁽²⁾_ε uses the oscillations. We have ∆^h_k∆^v_{`</sub>φ<sub>ε</sub>=φ<sup>1,ε</sup><sub>k,`}+φ^2,ε_k,` with

φ^1,ε_{k,`</sub>(x) <sup>def</sup>= iε∆<sup>h</sup><sub>k</sub>∆<sup>v</sup><sub>`}(e^iyε1∂₁φ) and φ^2,ε_{k,`</sub>(x) <sup>def</sup>= −iε2<sup>k</sup>φe<sup>2,ε</sup><sub>k,`}(x) with φe^2,ε<sub>k,`</sub>(x) <sup>def</sup>= 2<sup>2k</sup>2<sup>`</sup>

Z

(∂₁eg)(2^k(x_h−y_h))eh(2^`(x₃−y₃))e^iyε1φ(y)dy,

where (eg,eh) ∈ S(R²) × S(R) such that Fg(ξe _h) = ϕ(e |ξ_h|) and Feh(ξ₃) = ϕ(ξe ₃). Using Lemma 2.1, we get

2^−k2 X

`≤k+1

2<sup>`2</sup>kφ<sup>1,ε</sup><sub>k,`</sub>kL⁴_h(L²v) .εsup

`∈Zk∆<sup>h</sup><sub>k</sub>∆<sup>v</sup><sub>`</sub>(e^iyε1∂₁φ)kL⁴_h(L²v).ε2^k2k∂₁φkL²(R³).

Moreover, we have

2^−k2 X

`≤k+1

2<sup>`2</sup>kφ<sup>2,ε</sup><sub>k,`</sub>kL⁴_h(L²_v) ≤ε2^k2 X

`∈Z

2<sup>`2</sup>kφe<sup>2,ε</sup><sub>k,`</sub>kL⁴_h(L²_v). Using Lemma 2.1, we get

kφe^2,ε_{k,`</sub>kL<sup>4</sup><sub>h</sub>(L<sup>2</sup><sub>v</sub>).kφkL<sup>4</sup><sub>h</sub>(L<sup>2</sup><sub>v</sub>) and kφe<sup>2,ε</sup><sub>k,`}kL⁴_h(L²_v).2<sup>−`</sup>k∂<sub>3</sub>φkL<sup>4</sup><sub>h</sub>(L<sup>2</sup><sub>v</sub>). Again taking the sum over `≤N and ` > N and choosing the best N, we get

X

`∈Z

2<sup>`2</sup>kφe<sup>2,ε</sup><sub>k,`</sub>kL⁴_h(L²v) .kφk_L¹²4

h(L²_v)k∂₃φk_L¹²4 h(L²_v). We get that Φ⁽²⁾_ε ≤C_φε X

ε2^k≤1

2^k2 ≤C_φε¹². The estimates on Φ⁽³⁾_ε and Φ⁽⁴⁾_ε are analogous.

Notations In that follows, we make the convention that (c_k)_k∈Z (resp. (d_j)_j∈^Z) denotes a generic element of the sphere of`<sup>2</sup>(Z) (resp.`¹(Z)). Moreover, (c_{k,`</sub>)<sub>(k,`)∈}Z2 denotes a generic element of the sphere of `<sup>2</sup>(Z<sup>2</sup>) and (d<sub>k,`</sub>)_(k,`)∈Z2 denotes a generic sequence indexed by Z² such that X

`∈Z

X

k∈Z

d²_k,`¹₂

= 1. Let us notice that we shall often use the following property, the easy proof of which is omitted.

Lemma 2.2 Let αbe a positive real number and N₀ an integer. Then we have X

(k,`)∈Z2

`≥j−N0

2<sup>−α(`−j)</sup>d<sub>k,`</sub>c_k.d_j.

The following lemma will be of a frequent use in this work.

Lemma 2.3 For any a∈ B⁻

1 2,¹₂

4 (T), one has

kS_k^h∆^v_{`</sub>akL<sup>∞</sup><sub>T</sub>(L<sup>4</sup><sub>h</sub>(L<sup>2</sup>v))+ν<sup>12</sup>k∇hS<sub>k</sub><sup>h</sup>∆<sup>v</sup><sub>`}akL²_T(L⁴_h(L²v)).d<sub>k,`</sub>2<sup>k2</sup>2<sup>−`2</sup>kak

B⁻₄ ^12,12(T) and kS_k^hakL^∞_T(L⁴_h(L^∞_v ))+ν¹²k∇hS_k^hakL²_T(L⁴_h(L^∞_v )).c_k2^k2kak

B⁻₄ ^12,12(T).

With Lemma 2.3, we are going to state a result which is very close to Sobolev embedding and will be of a constant use in the existence proof of Theorem 1.2.

Lemma 2.4 The space B⁻4 ^12,12(T) is included in L⁴_T(L⁴_h(L^∞_v )). More precisely, let a be a function inB⁻

1 2,¹₂

4 (T), then we have k∆^v_jakL⁴_T(L⁴_h(L²v)). d_j

ν¹⁴2^−j2kak

B⁻₄ ^12,12(T) and kakL⁴_T(L⁴_h(L^∞v)) . 1 ν¹⁴kak

B⁻¹₄^2,12(T).

Proof of Lemma 2.4 Let us first notice thatk∆^v_jak²_L⁴

T(L⁴_h(L²v))=k(∆^v_ja)²kL²_T(L²_h(L¹_v)). Then using Bony’s decomposition in the horizontal variables, we write

(∆^v_ja)² =X

k∈Z

S^h_k−1∆^v_ja∆^h_k∆^v_ja+X

k∈Z

S^h_k+2∆^v_ja∆^h_k∆^v_ja

These two terms are estimated exactly in the same way. Applying H¨older inequality, we get kS_k−1^h ∆^v_ja∆^h_k∆^v_jakL²_T(L²_h(L¹v))≤2^−k2kS_k−1^h ∆^v_jakL^∞_T(L⁴_h(L²v))2^k2k∆^h_k∆^v_jakL²_T(L⁴_h(L²v)). Using the first inequality of Lemma 2.3 and Corollary 2.2, we infer

kS_k−1^h ∆^v_ja∆^h_k∆^v_jakL²_T(L²_h(L¹v)). d²_k,j

ν¹² 2^−jkak²

B

−12,1 4 2(T)

. Taking the sum over kand using Lemma 2.2, we deduce

k(∆^v_ja)²kL²_T(L²_h(L¹_v)). d²_j

ν¹²2^−jkak²

B₄^−12,12(T)

,

which is exactly the first inequality of the lemma. Now, using Lemma 2.1, we have k∆^v_jakL⁴_T(L⁴_h(L^∞v)) .2^j2k∆^v_jakL⁴_T(L⁴_h(L²v)).

This proves the whole lemma.

Now let us use Lemma 2.1 to study the free evolution u_F to the high horizontal fre- quency part of the initial data u₀, as defined in (1.4). In order to do so, let us first recall a lemma from [3] or [4], which describes the action of the semi-group of the heat equation on distributions, the Fourier transform of which are supported in a fixed ring.

Lemma 2.5 Let u₀∈ B⁻4 ^12,12 and u_F be as in (1.4), α∈N³,1≤p≤ ∞.Then, there holds k∆^h_k∆^v_`u_FkL^p_T(L⁴_h(L²v)).



 d_k,`

ν^1p 2^k

“1 2−²_p”

2^−2`ku₀k

B₄^−12,12, for k≥`−1, 0, otherwise.

Moreover, u_F belongs to B⁻4 ^12,12(∞), and we have ku_Fk

B

−12,1

4 2(∞).ku₀k

B⁻

12,1 4 2

.

Proof of Lemma 2.5 The relations (4) and (5) of the proof of Lemma 2.1 of [3] tell us that

∆^h_k∆^v_{`</sub>uF(t) = 2<sup>2k</sup>g(t,2<sup>k</sup>·)?∆<sup>h</sup><sub>k</sub>∆<sup>v</sup><sub>`}u0 with kg(t,·)kL¹(R²)≤Ce^−cνt22k. (2.1) Here the convolution must be understood as the convolution on R². Thus

k∆^h_k∆^v_{`</sub>u<sub>F</sub>(t, x<sub>h</sub>,·)kL<sup>2</sup>v ≤2<sup>2k</sup>|g(t,2<sup>k</sup>·)|?k∆<sup>h</sup><sub>k</sub>∆<sup>v</sup><sub>`}u₀(·)kL²v. Using (2.1) and again Lemma 2.1, we get

k∆^h_k∆^v_{`</sub>u<sub>F</sub>(t)kL<sup>4</sup><sub>h</sub>(L<sup>2</sup><sub>v</sub>).e<sup>−cνt22k</sup>k∆<sup>h</sup><sub>k</sub>∆<sup>v</sup><sub>`}u₀kL⁴_h(L²_v).e^−cνt22kd<sub>k,`</sub>2<sup>k2</sup>2<sup>−`2</sup>ku₀k

B⁻

12,1 4 2

.

By integration, the lemma follows.

Lemma 2.6 Under the assumptions of Lemma 2.5, one has k∆^v_ju_FkL²(R⁺;L^∞_h(L²v)). d_j

√ν2^−j2ku₀k

B₄^−12,12 and ku_FkL²(R⁺;L^∞(R³)). 1

√νku₀k

B₄^−12,12.

3 The proof of an existence theorem

The purpose of this section is to prove the following existence theorem.

Theorem 3.1 A sufficiently small constant c exists which satisfies the following property:

ifu₀ is in B⁻

1 2,¹₂

4 ,and ku₀k

B⁻

12,1 4 2

≤cν, then the system (AN S_ν) has a global solution in the space u_F +B^0,12(∞)where u_F is defined in (1.4).

Proof of Theorem 3.1 As announced in the introduction, we shall look for a solution of the form u=u_F +w. Let us first establish the equation satisfied byw. Actually by substituting the above formula to (AN S_ν), we obtain

(AN S]_ν)







∂_tw+w· ∇w−ν∆_hw+w· ∇u_F +u_F · ∇w=−u_F · ∇u_F − ∇p, divw= 0,

w|t=0 =u_{`h</sub><sup>def</sup>= u<sub>0</sub>−u<sub>hh</sub>. Notice that by (1.4), we have u<sub>`h}=X

j∈Z

S_j−1^h ∆^v_ju₀. Moreover, there holds

∆^v_ju_`h= X

|j−j⁰|≤1

S_j^h0₋₁∆^v_j⁰∆^v_ju₀ and thus k∆^v_ju_`hkL² . X

|j−j⁰|≤1

kS_j^h0₋₁∆^v_j⁰u₀kL².

This implies that, if u₀ belongs to B⁻

1 2,¹₂

4 , then u_{`h</sub> belongs toB<sup>0,12</sup> and ku<sub>`h}k_B0,1

2 .ku₀k

B₄^−12,12. (3.1)

We shall use the classical Friedrichs’ regularization method to construct the approximate solutions to (AN S]_ν). For simplicity, we just outline it here (for the details in this context, see [12] or [4]). Let us define the sequence (P_n)_n∈N by P_na^def= F⁻¹ 1_B(0,n)ba

and

(AN S]_ν,n)













∂twn−ν∆_hwn+Pn(wn· ∇wn) +Pn(wn· ∇uF,n) +Pn(uF,n· ∇wn)

=−Pn(uF,n· ∇uF,n) +Pn∇∆⁻¹∂j∂_k

(u^j_F,n+wn^j)(u^k_F,n+w^k_n) divw_n= 0,

w_n|t=0 =P_n(u_`h)^def= P_n(u₀−u_hh).

where u_F,n^def= (Id−S_jn)u_F withj_n∼ −log₂nand where ∆⁻¹∂_j∂_k is defined precisely by

∆⁻¹∂_j∂_ka^def= F⁻¹(|ξ|⁻²ξ_jξ_kba).

Because of properties of L² and L¹ functions the Fourier transform of which are supported in the ballB(0, n), the system (AN S]_ν,n) appears to be an ordinary differential equation in

L²_n^def= n

a∈L²(R³)/Supp ba⊂B(0, n)o . This ordinary differential equation is globally wellposed because

d

dtkw_n(t)k²L² ≤C_nku_F,n(t)kL^∞kw_nk²L² +C_nku_F,n(t)k²_L⁴_(L²_v)kw_n(t)kL²

and u_F,n belongs to L²(R⁺;L^∞∩L⁴_h(L²_v)). We refer to [4] and [12] for the details. Now, the proof of Theorem 3.1 reduces to the following three propositions, which we admit for the time being.

Proposition 3.1 Letu0 be inB⁻

1 2,¹₂

4 , andainB^0,21(T). WithuF is defined in (1.4), we have

∀j∈Z, I_j(T)^def= Z T

0

∆^v_j(u_F · ∇u_F)|∆^v_jadt. d²_j

ν 2^−jku₀k²

B⁻₄ ^12,12kak_B0,1 2(T). Proposition 3.2 Leta be a divergence free vector inB^0,12(T), and b inB^0,21(T). Then

∀j∈Z, J_j(T)^def= Z T

0

∆^v_j(a· ∇u_F)|∆^v_jbdt. d²_j

ν 2^−jkak_B0,1

2(T)ku₀k

B⁻

12,1 4 2

kbk_B0,1 2(T). Proposition 3.3 Leta be a divergence free vector inB⁻4 ^12,12(T), and b∈ B^0,12(T). Then

∀j∈Z, F_j(T)^def= Z T

0

∆^v_j(a· ∇b)|∆^v_jb dt. d²_j

ν 2^−jkak

B⁻

12,1

4 2(T)kbk²

B^0,12(T).

Conclusion of the proof of Theorem 3.1 Notice from (AN S]_ν,n) that P_nw_n = w_n, we apply the operator ∆^v_j to (AN S]_ν,n) and take theL² inner product of the resulting equation with ∆^v_jw_n to get

d

dtk∆^v_jw_n(t)k²L²+ 2νk∇h∆^v_jw_n(t)k²L² =−2(∆^v_j(w_n· ∇w_n)|∆^v_jw_n)

−2(∆^v_j(u_F,n· ∇w_n)|∆^v_jw_n)−2(∆^v_j(w_n· ∇u_F,n)|∆^v_jw_n)−2(∆^v_j(u_F,n· ∇u_F,n)|∆^v_jw_n).

By integration the above equation over [0, T],we get 2^jk∆^v_jw_nk²L^∞_T(L²)+ 2^j+1νk∇h∆^v_jw_nk²_L²

T(L²) ≤2^jk∆^v_jw_n(0)k²L² + 2 X4

k=1

W_j^k(T) (3.2) with

W_j¹(T) ^def= 2^j Z _T

0

(∆^v_j(w_n(t)· ∇w_n(t))|∆^v_jw_n(t)) dt, W_j²(T) ^def= 2^j

Z T

0

(∆^v_j(uF,n(t)· ∇wn(t))|∆^v_jwn(t)) dt, W_j³(T) ^def= 2^j

Z _T

0

(∆^v_j(w_n(t)· ∇u_F,n(t))|∆^v_jw_n(t)) dt, W_j⁴(T) ^def= 2^j

Z T

0

(∆^v_j(uF,n(t)· ∇uF,n(t))|∆^v_jwn(t)) dt.

Proposition 3.3 applied witha=b=wn together with Corollary 2.1 gives W_j¹(T). d²_j

ν kw_nk³

B^0,12(T). (3.3)