Global Strong Solution to the 3D Incompressible Navierv-Stokes Equations with General Initial Data

Tingting Zheng and Peixin Zhang

1Computer and Message Science College,Fujian Agriculture and Forest University,Fuzhou 350002,Fujian Province,P.R.China.

2School of Mathematical Sciences,Huaqiao University,Quanzhou 362021,Fujian Province,P.R.China.

Global Strong Solution to the 3D Incompressible Navierv-Stokes Equations with General Initial Data

Tingting Zheng1and Peixin Zhang2,∗

1Computer and Message Science College,Fujian Agriculture and Forest University,Fuzhou 350002,Fujian Province,P.R.China.

2School of Mathematical Sciences,Huaqiao University,Quanzhou 362021,Fujian Province,P.R.China.

.We study the existence ofglobalstrong solution to an initial–boundary value(or initial value)problem for the 3D nonhomogeneous incompressible Navier-Stokes equations.In this study,the initial density is suitably small(or the viscosity coefficient suitably large)and the initial vacuumis allowed.Results show thatthe unique solution of the Navier-Stokes equations can be found.

AMS subject classifications:35B65,35Q35,76N10

Incompressible Navier-Stokes equations,strong solutions,vacuum.

1 Introduction

The motion of a nonhomogeneous incompressible viscous fluid in a domain Ω ofR3is governed by the Navier-Stokes equations

the initial and boundary conditions(1.1)with the following conditions:

Here we denote the unknown density,velocity and pressure fields of the fluid byρ,uandP,respectively.fis a given external force driving the motion.Ω is either a bounded domain inR3with smooth boundary or the whole spaceR3.

It is interesting to studing the regularity criterion for strong solution of(1.1).Many people devote to researching these kind of results.In particular,Kim[1]proved that ifT∗was the blowup time of a local strong solution,then

whereLrwdenoted the weakLr−space.In[1],Kim also proved that the unique strong solution existed globally when ‖∇u0‖L2was small enough.

For the case the initial density is away from zero,the nonhomogeneous equations(1.1)have been studied by many people,see[2–4]and their references therein.In these papers,the authors proved the existence and uniqueness of the local strong solution for general initial data and they also got global well-posedness results for small solutions in 3D(or higher dimensional)space,while for 2D space they established the existence of large strong solutions.In[5–7],the authors obtained the global well-posedness results for initial data belonging to certain scale invariant space.

In this paper,base on Kim’s work,we are interested in the existence of global strong solution with general initial data.The main result of this paper can be stated as follows:

Theorem 1.1.Assume that(ρ0,u0,f)satisfies

and the compatibility condition

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Throughout this paper,we denote

1＜r＜∞,kis a positive constant,the standard Sobolev space is described as follows:

We will give the proof of Theorem 1.1 in Section 2.

2 Proof of Theorem 1.1

Before the proof,we recall the local existence result.In[10],Choe and Kim gave the following local strong solution existence theorem.

Theorem 2.1.Under the conditions of(1.3)and(1.4),there exists a time T＞0and a unique strong solution(ρ,u,P)to the initial boundary problem(1.1)–(1.2)satisfying

To extend the local classical solution guaranteed by Theorem 2.1,we prove it by contradiction.

Now,we establish priori estimates for smooth solutions to the initial boundary problems(1.1)-(1.2).LetT＞0 be the fixed time and(ρ,u,P)be the smooth solution to(1.1)-(1.2)on Ω×(0,T]in the class(2.1)with smooth initial data(ρ0,u0,P0)satisfying with(1.3),(1.4).

Lemma 2.1.Let(ρ,u,P)be a smooth solution of(1.1)-(1.2).Then

where,the letter C denotes a generic positive constant depending on the constants in some Sobolev inequalities.

Remark 2.1.If Ω is a bounded domain,the constantCmust depend on Ω comparing to the unbounded domain.

Proof.Multiplying(1.1)1bypρp−1(p≥2)then integratingxover Ω,one gets

Integrating(2.4)on[0,T]and takingp→∞,we obtain(2.2).Multiplying(1.1)2byu,integratingxover Ω and using Sobolev inequalities,we have

By applying the H¨older and Sobolev inequalities,we have

whereCis dependent of the constants in the Sobolev inequalities.From this and(2.5),using Young’s inequality,we have

then by integrating(2.6)on[0,T],we have(2.3).

We define

Lemma 2.2.Let(ρ,u,P)be a smooth solution of(1.1)-(1.2).If¯ρ is suitably small orµis suff iciently large,then

provided A(T)≤2M.

Proof.Multiplying(1.1)2byutand integrating over Ω,one gets

With the H¨older and Sobolev inequalities,one has

for someδ∈(0,1)and for any(r,s)satisfying2s+3r=1,3＜r＜∞.Takingv=|u|,w=|∇u|ands=4,r=6 in(2.9),with Sobolev inequality,one has

On the other hand,since(u,P)is a solution of the stationary Stokes equations

whereF=ρf−ρut−ρu·∇u,it follows from the classical regularity theory that

where we assumeµ≥1.Then from(2.10)and(2.11),one deduces

By integrating the last inequlity on[0,T],it yields that

Proof of Theorem 1.1.To prove the global existence,we argue by contradiction.Assume that(ρ,u)blows up at some finite timeT∗,0 ＜T∗＜ ∞.Since(ρ,u)satisfies the regularity(2.1)for anyT＜T∗,in view of Sobolev embedding again,we conclude that

which contradicts Theorem 1.3 in[1].This completes the proof of Theorem 1.1.

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15 June,2014;Accepted 23 March,2015

∗Corresponding author.Email addresses:nljj2011@126.com(T.Zheng),zhpx@hqu.edu.cn(P.Zhang).