Hui Zhang

School of Mathematical Science,Anqing Teacher’s College,AnQing,Hunan 246011,P.R.China

Regularity Criteria for the Navier-Stokes Equations Containing Two Velocity Components

Hui Zhang∗

School of Mathematical Science,Anqing Teacher’s College,AnQing,Hunan 246011,P.R.China

.Based on the critical sobolev inequalities in the Besov spaces with the logarithmic form,the regularity criteria in terms of two velocity components for the 3D incompressible Navier-Stokes equations is improved.

AMS subject classifications:35Q30;76D03

Chinese Library Classifications:O175

Navier-Stokes equations,regularity criteria,Littlewood-Paley decomposition.

1 Introduction

In this paper,we consider the following incompressible Navier-Stokes equations in R3×(0,T)

Whereu=u(x,t)is the velocity field,p(x,t)is the scalar pressure andu0(x)with∇·u0=0 in the sense of distribution is the initial velocity field.

It is well known that foru0(x)∈L2(R3),(1.1)exist at least one weak solution that is called Leray-Hopf weak solution.Nevertheless,the fundamental problem of the uniqueness and regularity of such solutions is still open.

However,the solution regularity can been derived when certain growth conditions are satisfied.This is known as a regularity criterion problem introduced in the celebrated work of Serrin[1],and can be described as follows:

holds true.

There is a large literature on improvement of the condition(1.2).It may be superfluous to recall all results.To go directly to the main points of the present paper,we only review some known results which are closely related to our main result.Beirã da Veiga[2]improved the condition(1.2)in terms of two velocity components

A weak solution u is regular if the growth condition

Very recently,Zhang[3]extended the condition(1.3)into BMO space in the marginal caseq=∞

Another interesting contribution of this problem is due to Beirão da Veiga[4]on the regularity criterion with respect to the velocity gradient condition

Recently,based on the Littelewood-Paley decomposition to the equations(1.1),Dong and Zhang[5]extended the regularity criterion via two components of velocity field in homogeneous Besov space

Penel and Pokorny[6]obtained an improved regularity result inLpspaces

Dong and Chen[7]improved the condition(1.7)in Lorentz space,Morrey space and multiplier space.Actually,the weak solution remains regular if the single velocity component satisfies some conditions(see[8,9,16]).

The aim of this present paper is to extend the regularity criterion(1.6)and(1.7)in homogeneous Besovspace in the marginal case.More precisely,we will prove the following result.

Theorem 1.1.Suppose u0∈H3(R3)and∇·u0=0in the sense of distributions.Assume that u is a Leray-Hopf weak solutions of(1.1)on(0,T).If u satisfies the following condition

then the weak solution is regular on(0,T].

A very interesting consequence of(1.8)is the condition

which is a re fined improvement of the famous BKM criterion[11]to Navier-Stokes equations,and the same time is also an improvement of the condition(1.6).

Remark 1.1.Throughout the paper,Cstands for a constant,and changes from line to line;‖.‖pdenotes the norm of the Lebesgue spaceLp.

2 Preliminaries

We first introduce the Littlewood-Paley decomposition and the definition of Besov spaces,one may check[12,13]for more details.

Forf∈S,the Schwartz class of rapidly decreasing functions,de fine the Fourier transform

Choose two nonnegative radial functions χ,ϕ∈S(R3)supported respectively inB=such that

We de fine the pseudo-differential operators

and set

By telescoping the series,we thus have the following Littlewood-Paley decomposition

where BMO is the space of the Bounded Mean Oscillation.BMO is the space defined as a set for

It is deserved to point out that functionfsuch that

3 Proof of Theorem 1.1

Before going to the proof,we recall the following two inequalities established in[14,15]respectively

forf∈W1,rwith∇·f=0 and

forf∈Hs−1withs＞n2+1.

Remark 3.1.The idea of this proof is based on the method from[16,17].

Taking the inner product of thei−th equation of(1.1)with|ui|2ui,(i=1,2,3)and integrating by parts,we can show that

Here we have used the following equality,due to the free divergence condition

Applying the inequalities(3.1)and(3.2),we have

Here we have used the following relationship betweenpandu

Now,we take

then due to(3.4),we have

where∈is a small constant,such that

whereC∗is a positive constant depending onT∗.

Then,we do estimate for‖∇u‖2.

Multiplying the first equation of(1.1)by−Δu,after integration by parts,we have

where we have used the inequality

Integrating(3.5)on[T∗,t],we have Applying∇3to the first equation of(1.1),then takingL2inner product of the resulting equation with∇3,using integration by parts,we obtain

where we used the following Gagliardo-Nirenberg inequality:

and

It should be clear to readers that applying Gronwall’s inequality to(3.7)and taking(3.6)into account,we can obtainy(t)≤C.This completes the proof of Theorem 1.1.

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15 April 2014;Accepted 10 September 2014

∗Corresponding author.Email address:zhangaqtc@126.com(H.Zhang)