YANG Xinguangand ZHANG Lingrui

College of Mathematics and Information Science,Henan Normal University,Xinxiang 453007,China.

College of Education and Teacher Development,Henan Normal University,Xinxiang 453007,China.

Received 24 May 2013;Accepted 2 July 2013

BKM’s Criterion of Weak Solutions for the 3D Boussinesq Equations

YANG Xinguang?and ZHANG Lingrui

College of Mathematics and Information Science,Henan Normal University,Xinxiang 453007,China.

College of Education and Teacher Development,Henan Normal University,Xinxiang 453007,China.

Received 24 May 2013;Accepted 2 July 2013

.In this present paper,we investigate the Cauchy problem for 3D incompressible Boussinesq equations and establish the Beale-Kato-Majdaregularity criterion of smooth solutions in terms of the velocity field in the homogeneous BMO space.

Incompressible Boussinesq equations;BKM’s criterion;smooth solutions.

1 Introduction

This paper is devoted to establish BKM’s criterion of smooth solutions for the Cauchy problem for 3D Boussinesq equations with viscosity inR3

here u is the velocity field,p is the pressure,θ is the small temperature deviations which depends on the density.η≥0 is the viscosity,ν≥0 is called the molecular diffusivity and e3=(0,0,1)T.The above systems describe the evolution of the velocity field u for a threedimensional incompressible fluid moving under the gravity and the earth rotation whichcome from atmospheric or oceanographic turbulence where rotation and stratification play an important role.When the initial density θ0is identically zero(or constant)and η=0,then(1.1)-(1.4)reduce to the classical incompressible Euler equation:

For the incompressible Euler equation and Navier-Stokes equation,a well-known criterion for the existence of global smooth solutions is the Beale-Kato-Majda criterion in[1] which states the control of the vorticity when ω=curl u in L1(0,T;L∞),this is sufficient to get the global well-posedness of solutions,i.e.,any solution u is smooth up to time T under the assumption that.Kozono and Taniuchi[2]improved the Beale-Kato-Majda criterion under the assumption.The regularity criteria for the Navier-Stokes equations,we can refer to Bahouri,Chemin and Danchin[3],Cao and Titi[4],Kato and Ponce[5],Kozono and Taniuchi[2],Zhou[6,7], Zhou and Lei[8],Zhang and Chen[9].

The global well-posedness for two-dimensional Boussinesq equations which has recently drawnalot ofattention.More precisely,the global well-posednesshas beenshown in various function spaces and for different viscosity,we can refer to[10-20].When η=ν=0,the Boussinesq system exhibits vorticity intensification and the global wellposednessissueremains anunsolvedchallenging openproblem(if θ0is aconstant)which may be formally compared to the similar problem for the three-dimensional axisymmetric Euler equations with swirl.

For the three-dimensional case,Hmidi and Rousset[16,17]proved the global wellposedness for the 3D Navier-Stokes-Boussinesq equations and Euler-Boussinesq equations with axisymmetric initial data without swirl respectively.Danchin and Paicu[12] obtained the global existence and uniqueness result in Lorentz space for the Boussinesq equations with small data.

Our purpose of this paper is to obtain logarithmically improved regularity(BKM’s) criterion of smooth solutions in terms of velocity field in BMO space.

Now we state our result as follows.

Theorem 1.1.Assume that(u0,θ0)∈Hm(R3)holds with divu0=0 and m≥3.If u satisfies the condition

then the solution(u,θ)for the Cauchy problem(1.1)-(1.4)can be extended smoothly beyond T.

The paper is organized as follows.We shall state some important inequalities in Section 2 and prove Theorem 1.1 in Section 3.

2 Preliminaries

Throughout this paper we use the following notations.Lp(R3)denotes the generic Lebegue space,Hm(R3)denotes the standard Sobolev space.BMO is the bounded mean oscillations space.is the homogeneous Besov space,where 0≤m,n≤+∞.S(Rn) be the Schwartz class of rapidly decreasing functions.

The Fourier transformation of f∈S(Rn)is defined as

and the inverse Fourier transformation of g∈S(Rn)is defined as

Next,we shall recall the Littlewood-Paley decomposition and define some functional spaces which can be found in[3,21,22].

Definition 2.1.DenoteCas the annulus of center on 0 with short radius 3/4 and long radius

8/3.Then there exist two positive functions ?∈C∞0(B(0,4/3))and χ∈C∞0(C)such that

Remark 2.1.The frequency localization operator is defined as

Definition 2.2.BMO denotes the homogeneous bounded mean oscillation space which equipped with the norm

Definition 2.3.(The Triebel-Lizorkin space˙Fsp,q)The homogeneous Triebel-Lizorkin space˙Fsp,qis defined as the set of tempered distributions u,i.e.,

Lemma 2.1.(The Bernstein inequality)LetCis a annulus of center on 0,B is a ball of center on zero,then there exists a constant C＞0such that for any integer k≥0 and function u∈Lα(Rn), b≥a≥1,we have

Proof.See,e.g.,[3].

Lemma 2.2.(The Special Bernstein inequality)For any integer k≥0,1≤p≤q≤+∞and function u∈Lp(Rn),we have

where c and C are positive constants independent of u and k. Proof.See,e.g.,[3].

Lemma 2.3.(The Gagliardo-Nirenberg inequality)

holds for all u∈L∞(Rn)∩Hm(Rn).

Lemma 2.4.(The Interpolation Inequalities)The following inequalities hold in the three dimensional Lebesgue space

Lemma 2.5.The following inequality holds:

Proof.See,e.g.,[3].

Lemma 2.6.There exists a uniform positive constant C such that holds for all u∈H3(R3)with?·u=0.Proof.Usingthesimilar techniqueasin[8],wecanderiveourresult.FromtheLittlewood-Paley decomposition,we have

Denote Rj=(?/?xj)(?Δ)?1/2(R=(R1,R2,···,Rn))be the Riesz transformation,from the Biot-Savard law,we see uXj=Rj(R×?u)(j=1,2,···,n),Since R is a bounded operator in BMO,we derive that‖?u‖BMO≤C‖?×u‖BMO.Combining(2.20)and(2.21),we complete the proof of lemma.

3 Proof of main theorem

Proof of Theorem 1.1:Multiplying(1.1)by u,using(1.3)and integrating by parts inR3, we derive

Multiplying(1.2)by θ,using(1.3)and integrating inR3,we obtain

Combining(3.1)and(3.2),integrating with respect to t,using the Gronwall inequality, we conclude that

Applying?to the both sides of(1.1),taking a L2inner product of the resulting equation with?u,integrating by parts,we derive

The similar steps to(1.2),we obtain

From(3.6),(3.7)and?·u=0,it follows that

By Lemmas 2.5 and 2.6,using the incompressible condition?·u=0,we give the estimate of Ii(i=1,2,3).

It follows from(3.8)-(3.11)that

By the Gronwall inequality,we arrive at

From(1.8),there exist an arbitrary small constant ε＞0 and T?＜T such that

Hence,combining(3.13)and(3.3)-(3.5),we conclude

where A(t)=supT?≤s≤t(‖?3u(s)‖2L2+‖?3θ(s)‖2L2),t∈[T?,T],C0dependson‖?u(T?)‖2L2+‖?θ(T?)‖2L2,C1＞0 is a uniform constant.

Applying?mto(1.1)and(1.2),then taking L2inner product of the resulting equation with?mu and?mθ respectively,integrating by parts,we get

It follows from(3.16),(3.17)and?·u=0 that

Since the proof for the case m＞3 is similar to m=3,here we only need to prove the case m=3.By the H¨older inequality,the Cauchy inequality and Lemma 2.6,we get

Using Lemma 2.4 and‖θ‖L∞≤‖θ0‖L∞,we obtain

Noting that‖θ‖L∞≤‖θ0‖L∞and using the Cauchy inequality,we deduce

From direct computation,we have

Combining(3.18)-(3.24)and(3.4),we conclude

Thus,it follows from(3.18)-(3.25)that

holds for all T?≤t＜T.

Integrating(3.26)over[T?,s]with respect to t,using Lemma 2.5,we arrive at

which implies

For all T?≤t＜T,then using the Gronwall inequality and(3.28),we deduce that e+A(t) is bounded,i.e.,

where C is dependent on‖?3u(T?)‖2L2+‖?3θ(T?)‖2L2.Thus,we complete the proof of Theorem 1.1.?

Acknowledgments

Xinguang Yang was in part supportedby the NSFC(No.11326154),the Key Scientific and Technological Project of Henan Province(No.142102210448),the Innovational Scientists and Technicians Troop Construction Projects of Henan Province(No.114200510011)and the Young Teacher Research Fund of Henan Normal University(qd12104).

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?Corresponding author.Email addresses:yangxinguang@hotmail.com(X.Yang),zhanglingrui@126.com(L. Zhang)

AMS Subject Classifications:52B10,65D18,68U05,68U07

Chinese Library Classifications:O175.27