LIU XiaofengZOU Lulu

College of Science,Donghua University,Shanghai 201620,China

Abstract:The generalized Navier-Stokes equations with damping are considered. We will show that the generalized Navier-Stokes equations with damping |u|β-1u have weak solutions for any β>1 and 0 <α<5/4,and we will use the Fourier splitting method to prove the L2 decay of weak solutions for β>2 and 0 <α<3/4.

Key words:generalized Navier-Stokes equation;damping;decay;Fourier splitting method

Introduction

In this paper we consider the following generalized Navier-Stokes equations with damping

(1)

whereu=(u1,u2,u3) is the velocity;pis pressure;βandαare two constants，β>1 and 0<α<5/4;u0denotes the initial velocity;Λ2αis defined through Fourier transform

(2)

The existence of global weak solutions of the initial value problem of the Navier-Stokes equations were proved by Leray[1]and Hopf[2]long before.Since then,the uniqueness and the regularity of weak solutions and the global (in time) existence of strong solutions have been extensively investigated[2-5].However,the uniqueness of weak solutions and the global (in time) existence of strong solutions to the three-dimensional Navier-Stokes equations remain completely open.Cai and Jiu[6]have studied the existence and the regularity of solutions for the three-dimensional Navier-Stokes equations with damping,and they have obtained the global weak solutions forβ≥1 andα=5/4,the global strong solutions forβ≥7/2 andα=5/4,and the strong solutions with unique for any 7/2≤β≤5 andα=5/4.Based on it,Song and Hou[7-8]considered the global attractor.

So far,it has been proved that whenα≥5/4,the three-dimensional generalized Navier-Stokes equations have a global and unique regular solution[9].However,whenα≤5/4,the global well-posedness theories of the three-dimensional fractional Navier-Stokes system remain open.Recently,using the classic Fourier splitting method,Jiaetal.[10]considered theL2decay of weak solutions forβ≥10/3 andα=5/4,the optimal upper bounds of the higher-order derivative of the strong solution for 7/2≤β≤5,and the asymptotic stability of the large solution to the system forβ≥7/2 andα=5/4.Jiu and Yu[11]considered the supercritical case,i.e.,α<5/4,and they showed that the weak solutions to the generalized Navier-Stokes equations subject to large initial data,decay inL2at a uniform algebraic rate.

In this paper,we will prove the weak solutions of Eq.(1) for anyβ>1 and 0<α<5/4,and prove theL2decay of weak solutions to Eq.(1) forβ>2 and 0<α<3/4.

1 Preliminaries

First of all,let us give the definition of weak solutions for Eq.(1) with the initial datau0∈L2.

Definition1The function pair (u(x,t),p(x,t)) is called a weak solution of Eq.(1) if for anyT>0,and the following conditions are satisfied:

(1)u∈L∞(0,T;L2(R3)) ∩L2(0,T;Hα(R3))∩Lβ+1(0,T;Lβ+1(R3)).

(3)

(3) divu(x,t)=0 for a.e.(x,t)∈R3×[0,T).

Next,we list some notations that will be used in this paper.Lp(R3) with 1≤p≤∞ denotes the usual Lebesgue space of allLpintegral functions associated with the norm,

andHs(R3) withs∈Ris the fractional Sobolev space with

In this paper,we will prove the existence of weak solutions for Eq.(1) with initial datau0∈L2whenβ>1 and 0<α<5/4.More precisely,the result can be stated as follows.

Theorem1Suppose thatu0∈L2(R3),β>1 and 0<α<5/4.Then for any givenT>0,there exists a weak solutionuto Eq.(1) such that

u∈L∞(0,T;L2(R3)) ∩L2(0,T;Hα(R3))∩Lβ+1(0,T;Lβ+1(R3)).

Remark1When 0<α<5/4,it has been proved that the generalized Navier-Stokes equations have a weak solution[11].We will consider the generalized Navier-Stokes equations with damping whenβ>1 and 0<α<5/4 in this paper.

Remark2We will use Aubin-Lions lemma[12]and Galerkin approximation in the proof process of the theorem 1 .

Jiu and Yu[11]first studied the decay of solutions to the generalized Navier-Stokes equations when 0<α<5/4.In the following,we will consider the time decay rate of solutions to the the generalized Navier-Stokes equations with damping forβ>2 and 0<α<3/4.

Theorem2Assume thatu0∈L2(R3)∩Lp(R3) with divu= 0,0<α<3/4 andβ>2.Then Eq.(1) admits a weak solutionu:

where,1≤p≤2;the constantCdepends onα,theLpnorm ofu0and theL2norm ofu0.

Remark4Similar results have been established respectively[13]for the classical Navier-Stokes equations.One of the powerful tools is the Fourier splitting method,which is introduced by Schonbek[14-15].

In order to proof the theorem 2,we need the following lemma[11].

Lemma1Letf∈Lp(R3) with 1≤p≤2,and then

where

and the constantCdepends onγ，αand theLpnorm off.

The remainder of this paper is organized as follows.In section 2,we will prove the theorem 1.In section 3,we will prove the theorem 2 by developing the Fourier splitting method.

2 Proof of Theorem 1

In this section,we consider the existence of weak solutions for Eq.(1),and we will prove the theorem 1.

ProofForN≥1,letJNbe the spectral cutoff defined by

LetPdenote the Leray projector over divergence-free vector-fields.Consider the following equation

(4)

Multiplying the first sub-equation of Eq.(4) byuNand integrating by parts,we obtain

and

(5)

Therefore,uNis uniformly bounded inL∞(0,T;L2(R3)).Next,we will use Aubin-Lions lemma[12]to prove the strong convergence ofuN(or its subsequence) inL2(0,T;L2(Ω)) for any boundedΩ?R3.

In fact,for anyh∈L2(0,T;H3) andα<5/4,by the Gagliardo-Nirenberg inequality and the H?lder inequality,we obtain

(6)

Using the H?lder inequality and the Sobolev embeddingH3(R3)→Hα(R3),α<3/4<3,we deduce

(7)

and by the Gagliardo-Nirenberg inequality and the H?lder inequality,we get

(8)

Combining these estimates with the first sub-equation of Eq.(4),we obtain

?tuN∈L2(0,T;H-3(R3)).

SinceuNis bounded uniformly inL2(0，T;L2),there existsu∈L2(0，T;L2) and a subsequence of {uN} weakly converges touinL2(0，T;L2).By using the Aubin-Lions compactness theorem,we can get thatL2(0,T;L2(R3)) is compactly imbedded in the space

{u:u∈L2(0,T;Hα(R3));

ut∈L2(0,T;H-3(R3))}.

(9)

(10)

By the Holder inequality,we obtain

(11)

Noting thatβ>1,by the simple inequality

|ap-bP|≤p(ap-1+bp-1)|a-b|,
?a,b≥0,p∈R,

and employing the H?lder inequality,we have

for

Thus,usatisfies Eq.(3) and it is a weak solution to Eq.(1).

3 Proof of Theorem 2

In the following,we will use the classical Fourier splitting method to prove the theorem 2.

ProofTaking the Fourier transformation of the first sub-equation of Eq.(1),we get

(12)

where

(13)

Multiplying Eq.(12) by the integrating factor e|ξ|2αtgives

Then integrating in time from 0 tot,we obtain

(14)

Taking the divergence operator on the first sub-equation of Eq.(1),we yields

(15)

Because Fourier transform is a bounded map fromL1intoL∞,

(16)

and

(17)

By using the divergence free condition,we have

(18)

Combining inequalities (16)-(18),we have

(19)

Thus,solving the ordinary differential equation (14),and using inequality (19) and the boundedness of theL2norm of the solution,we obtain

(20)

Sinceu∈Lβ+1(0,T;Lβ+1(R3)) and

by using the interpolation inequality and the H?lder inequality,we have

(21)

(22)

From the energy inequality (5),it follows that

(23)

and applying the Plancherel’s theorem to inequality (23) yields

(24)

Let

(25)

whereγis a constant to be determined.Then using inequality (24),we have

(26)

(27)

By using Eq.(25),we haveH(t)=(t+1)γ,so

(28)

Applying the lemma 1,we can obtain

(29)

So plugging inequality (22) into inequality (28) and using inequality (29),we have

(30)

Integrating inequality (30) in time from 0 tot,we yields

Then

(31)

Hence inequality (31) is rewritten as

(32)

Choosingγsuitably large,we can deduce

Therefore the proof of the theorem 2 is complete.

4 Conclusions

In this paper,we proved that the three-dimensional generalized Navier-Stokes equations with damping have a weak solution for anyβ>1 and 0<α<5/4.TheL2decay of weak solutions for anyβ>2 and 0<α<3/4 was got.