Global Well-Posedness of Incompressible Navier-Stokes Equations with Two Slow Variables?

Weimin PENG Yi ZHOU

1 Introduction

The classical three-dimensional incompressible Navier-Stokes equations are given by

where

and p(x,t)stand for the velocity vector and the pressure function of the flow at the point(x,t)∈R3×R+,respectively,and for simplicity,the kinematic viscosity of the fluid is taken to be equal to one.The initial data u0should also be a divergence-free vector field.

The study on Navier-Stokes equations dates back long time ago.The first important result was obtained by Leray[12].Despite much effort by many mathematicians and physicists,however,our understanding of Navier-Stokes equations remains minimal(see[7]).We can explain the main difficulty by means of the scaling property of the incompressible Navier-Stokes equations from a purely mathematical viewpoint.If(u(x,t),p(x,t))solves the Navier-Stokes equations on the time interval[0,T],then we can form a new solution(uλ(x,t),pλ(x,t))to the Navier-Stokes equations on the time interval[0,λ?2T],by the formula

with the initial data

It can be easily checked that the unique conserved quantity—the energy E(u)(t)is

If the dimension d is equal to 3,then

Thus,the energy is “super-critical”,which makes that the energy becomes increasingly useless for controlling the solution as one moves to finer and finer scales.

For the problem with small initial data,we can see that the initial data should belong to scale-invariant spaces in the following sense:There exists a constant C such that for any given positive λ,we have

The corresponding result inis due to Fujita and Kato[8](see also Leray[12],where the smallness of the initial data is measured by ∥u0∥L2∥?u0∥L2).The study of Navier-Stokes equations in critical spaces was done by many authors,for example,Weissler[13],Kato[10],Giga and Miyakawa[9],Cannone,Meyer and Planchon[2],in particular,Koch and Tataru[11]proved the global well-posedness of the Navier-Stokes equations with small initial data in the space BMO?1.For more information about the classical results,the reader can consult the book by Cannone[1]and the references therein.

There are some results for large initial data,for example,in a series of recent papers[3–6],Chemin et al.constructed some classes of large anisotropic initial data for the Navier-Stokes system.We will not describe them in details,but we will just mention their work related to our paper.In[4],Chemin,Gallagher and Paicu considered the well-poedness of the threedimensional Navier-Stokes equations with initial data slowly varying in one direction:

where ?>0 is a small parameter,(x2,x2)belongs to the torus T2and x3belongs to R.After a change of scale in the vertical variable,the system is not uniformly elliptic,which leads to lose the control on one derivative in the vertical variable.To compensate the loss of derivative,by working in the class of analytical functions and using some kind of the global Cauchy-Kowalewski result,the authors in[4]got the global well-posedness without small assumption on the norm of initial data for the three-dimensional incompressible Navier-Stokes equations.

Inspired by their work,this paper is devoted to the study of the following system with a linear damping:

on the domain

with the initial data slowing varying in two directions

where ?>0 is a small parameter.

For Uj(x1,x2,x3)(j=1,2,3),we take the Fourier transformation with respect to x1,x2∈R2,and also take the coefficients in the Fourier expansion with respect to x3∈ [?π,π].More precisely,let

We give the following assumptions.

(1)Uj(x1,x2,x3)(j=1,2,3)are analytic functions of x1,x2,Uj(x1,x2,x3)(j=1,2)are even functions with respect to x3,and U3(x1,x2,x3)is an odd function with respect to x3.

(2)For(ξ1,ξ2)(j=1,2),the following inequality holds:

where a is a positive number and δ>0 is a small constant.

(3)For(ξ1,ξ2),the following inequality holds:

where M is a bounded constant.

Our result for system(1.8)is the following.

Theorem 1.1Under assumptions(1)–(3),there exists δ= δ(a)>0 so small that if ?M ≤δ,then the three-dimensional Navier-Stokes problem with damping(1.8)and with the initial data given by(1.10)generates a unique global solution for any given small ?>0.

Remark 1.1The linear damping term is put in system(1.8)to deal with the zero Fourier mode.Comparing with[4],we give a much simpler proof at the cost of all initial data having some analytical condition.

2 Proof of Theorem 1.1

Noting that the divergence free condition recovers p from u through the following formula:

we can put the system(1.8)in the following formula:

Let

The rescaled Navier-Stokes equations can be obtained from(1.8)as follows(to simplify the notation,we will drop ? in the rest of this section):

on the domain

with the initial data

For any given point

we denote its horizontal coordinates by

Similarly,the horizontal components of any given vector field

will be denoted by

and ξ=(ξ1,ξ2)will be the frequency variable with respect to

Let

Thanks to the relationship between Fourier transformation,Fourier series and the derivative with respect to x,the first equation in(2.3)becomes

The incompressible condition in(2.3)turns to

Moreover,for j=1,2 the initial data(2.5)becomes

while,for j=3 we have

Multiplyingon both sides of(2.7),we get

Here and hereafter C denotes a positive constant.Since u3is an odd function with respect to x3,we have

The combination of(2.8)with(2.12)leads to

Plugging(2.13)into(2.11)gives

Suppose that there exists a C1function θ(t)satisfying

Multiplying e(a?θ(t))|ξ|on both sides of(2.14),and using the triangle inequality,we obtain

Integrating both sides of(2.16)with respect to t,we get

Now,setting

we have

According to the original assumptions(2)–(3),it is obvious that

then,owing to ?M ≤ δ,we have

It follows from(2.19)that

By the de finition of θ(t),obviously we have

then we can choose δ so small that

Recalling(2.15)and(2.18),the bootstrap argument ensures the existence of θ(t).

Thus,according to(2.19),it follows that

for all time t≥0.This means the global existence of the solution.

AcknowledgementThe authors would like to express their gratitude to Professor Ta-Tsien Li for his helpful advice.

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