(董玉) (黃耀芳) (李莉) (盧青)

School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China

E-mail: 2011071009@nbu.edu.cn; 2011071017@nbu.edu.cn; lili2@nbu.edu.cn; 2111071026@nbu.edu.cn

Abstract In this paper,we obtain new regularity criteria for the weak solutions to the three dimensional axisymmetric incompressible Boussinesq equations.To be more precise,under some conditions on the swirling component of vorticity,we can conclude that the weak solutions are regular.

Key words Boussinesq equations;regularity criteria;axisymmetry

1 Introduction

In this paper we consider the regularity of the 3D axisymmetric Boussinesq system.The incompressible viscous Boussinesq equations in R3have the following form:

Hereu=(u1,u2,u3) is the velocity field,Prepresents the scalar pressure,ρis the temperature fluctuation,νandκ>0 denote the kinematic viscosity and thermal diffusion,respectively,ande3=(0,0,1) is the unit vector in a vertical direction.

The Boussinesq equations can be used to model numerous geophysical phenomena such as atmospheric fronts,the dynamics of oceans,katabatic winds,and dense gas dispersion,etc.Global regularity of the weak solution to the Boussinesq equations in three dimensional space is a challenging open problem.Many scholars have studied related fluid system with additional assumptions such as the axisymmetric or no-swirl conditions.

There are many relevant results with axisymmetric structures for Boussinesq equations.Hanachi,Houamed and Zerguine in [8]showed that if the initial data (v0,ρ0) is axisymmetric and (ω0,ρ0) belongs to the critical spaceL1(?)×L1(R3),withω0being the initial vorticity associated tov0and ?={(r,z)∈R2:r>0},then the viscous Boussinesq system has a unique global solution.Hmidi and Rousset in[10]indicated that if the initial datav0∈Hs,s>,divv0=0,ρ0∈Hs-2∩Lm,m>6 andr2ρ0∈L2,then there is a unique global solution for the three-dimensional Euler-Boussinesq system with axisymmetric initial data without a swirl.For the global well-posedness to the three dimensional Boussinesq equations with horizontal dissipation,Miao and Zheng in [13]established a relationship betweenby taking full advantage of the structure of the axisymmetric fluid without a swirl and some tricks of harmonic analysis.Subsequently,in [14],they assumed that the support of the axisymmetric initial dataρ(r,z) does not intersect thez-axis,and they proved the global well-posedness of the tridimensional anisotropic Boussinesq equations.Sulaiman explored the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial datawithp>6;see [16].Jin,Xiao and Yu proved in [11]the global well-posedness of the two dimensional Boussinesq equations with three types of partial dissipation,under the assumption that the initial data and partial derivatives of initial data is square integrable.For other results regarding Boussinesq equations,we refer to[1,3,5,9,12,15,18].

In [17],Wang,Wang and Liu assumed thatν>0,initial datau0∈H2(R),∈L3(R3)∩L∞(R3) andifκ>0,ρ0∈H1(R3) ifκ=0.Then they established six new regularity criteria of the weak solutions to the incompressible axisymmetric Boussinesq equations,which are independent of the temperature.Guo,Wang and Li in[7]studied the regularity criteria of axisymmetric weak solutions to the three-dimensional incompressible magnetohydrodynamic equations with a non-zero swirl.Inspired by their work,the main purpose of this paper is to extend the results of the MHD system to the Boussinesq system.In order to do this,we need some estimates onρand its derivatives.

Any vectorucan be represented in cylindrical coordinates asu=urer+uθeθ+uzez,whereer=(cosθ,sinθ,0),eθ=(-sinθ,cosθ,0),ez=(0,0,1).We say a vectoruis axisymmetric ifur,uθ,uzare independent ofθ.We callur,uθ,uzthe radial,swirling andz-components of velocity,respectively.In cylindrical coordinates,the gradient and Laplacian operator on scalar functions have the expressionrespectively.

In order to consider Boussinesq equations in the axisymmetric scheme,we rewrite (1.1) in cylindrical coordinates,and obtain that

The main results of this paper can be given as follows:

Theorem 1.1Let(u,ρ)be an axisymmetric divergence-free weak solution for the Boussinesq equation (1.1) in [0,T],with initial datau0∈H1(R3).Suppose that the swirling component of vorticityωθsatisfies that

Remark 1.2An important feature of (1.1) is that it has a scaling invariance property;namely,if (u(t,x),ρ(t,x),P(t,x)) is a solution of (1.1),then

is also a solution.This property has inspired people to consider the regularity of solutions in the scaling invariant functions space,such asu ∈L3(R3),withIn this sense,the present paper establishes sufficient conditions in scaling invariant spaces;This guarantees the regularity of solutions.

The rest of this paper is organized as follows: In Section 2,we present some preliminaries which will be used in the ensuing content.The proof of main results will be completed in Section 3.

2 Preliminaries

In order to explain the definition of homogeneous Besov spaces,we first present some notations.Let?be a smooth function satisfying that

(i) supp? ?

(ii) 0≤? ≤1;

The homogeneous dyadic blocksare defined for allj ∈Z through

Now we are ready to give the definition of homogeneous Besov spaces.

Definition 2.1([2]) Lets ∈R and (p,r)∈[1,∞]2.The homogeneous Besov spaceconsists of the distributionsuinsuch that

Lemma 2.2Let=urer+uzezbe an axisymmetric vector field.Then we have the equalities

ProofThe above estimates can be obtained by direct calculation of all terms for|?u|2,and then chooseu=urer+uzezandu=uθeθ,respectively.

Lemma 2.3([2,4]) Letube an axisymmetric vector fields with divu=0 andω=curlu.Suppose thatωvanishes sufficiently quickly near infinity in R3.Then?(uθeθ) can be represented as singular integral forms

It should be noted that Calderon-Zygmund operators are bounded onandLrfor anyp,q ∈[1,∞],s ∈R,1

Lemma 2.4([2]) Let 1≤p1≤p2≤∞and 1≤r1≤r2≤∞.Then,for anys ∈R,we have that

ifp ∈(3,∞).

In order to obtain our regularity criteria,we need a trilinear estimate similar to Lemma 2.5 on the termdx.To control this term,a direct application of Lemma 2.5 will lead to the control of‖?2uθ‖2,which is difficult,since we do not have a representation of?2uθin the form of a Calderon-Zygmund type convolution.However,a more delicate proof of Lemma 2.5 will lead to the following corollary:

Proof(i) The case ofp ∈(3,∞).

We first prove the casep ∈(3,∞).Through the Littlewood-Paley decomposition,we know that

By the homogeneous Bony decomposition,we have that

We now deal with these estimates separately.

Since the homogeneous paraproduct operator ˙Tis continuous;see [2],we have forp>3 that

In a similarly way,we get that

Since the remainder operator is continuous,by continuously embeddingforq ∈[1,∞],we know that

Combining the previous inequalities,forp>3,we obtain that

(ii) The case ofp ∈[,3].

For the casep ∈[,3],we have that

whereηandη′are conjugate indices.

By the homogeneous Bony decomposition,we get that

wheret,p,rsatisfyt>0,1≤pi,rj ≤∞,i=1,2,3,j=2,3,and

Therefore,by (2.1),the embedding theorem,Lemma 2.3 and the interpolation theorem,we obtain that

Using a similar way,we find that

Combining this with the above estimates,we get that,forp ∈[,3],

The proof of corollary is complete.

3 Proof of Theorem 1.1

ApplyingL2estimates to equation (1.3),we get that

Similarly,through theL2estimate forρin the fourth equation of (1.2),we obtain that

Next,let us apply?r,?zto (1.2)4to obtain that

Applying energy estimates to the above two equalities,integrating by parts and using the divergence-free condition,we get that

Combining (3.1),(3.2),(3.3) and (3.4),we infer that

Now we deal with the terms in (3.5),successively.

(1) Estimates ofI1andI2.

It is easy to see that

Forα ∈(3,∞),by Lemma 2.5,we get that

Forα ∈[,3],one has that

Combining (3.6) and (3.7),forα ∈[,∞),we have that

since Calderon-Zygmund operators are bounded on.

It should be noted that,ifα=,we obtain from (3.8) that

(2) Estimate ofI3.

Using a similar calculation as that forI1andI2,we have that

By Lemma 2.5,it is evident that,forα ∈(3,∞),

Combining (3.9),(3.10),Lemma 2.3 and Young’s inequality,we know forα ∈(,∞) that

On the other hand,we know from (3.10) that,ifα=,then

(3) Estimate ofI4.

It is clear that

By Corollary 2.6,we immediately get that,forα ∈(3,∞),

Combining (3.11),(3.12) and Young’s inequality,forα ∈(,∞),we infer that

Forα=,we get from (3.12) that

(4) Estimate ofI5.

The estimates ofI5,I6,I7involve the control of derivatives ofρ.

Using H?lder’s and Young’s inequalities,for anyα ∈[,∞),we have that

(5) Estimate ofI6andI7.

It is obvious that

We use Lemma 2.5 and Young’s inequality to obtain that,forα ∈(3,∞),

Forα ∈[,3],through Lemma 2.5,we have that

Using Young’s inequality,we infer that,forα ∈(,3],

Combining (3.13) and (3.15),and using Lemma 2.3,we deduce that,forα ∈(,∞),

Ifα=,from (3.14),Young’s inequality and Lemma 2.3,we have that

Summing up the above estimates,forα ∈(,∞),we deduce that

Then by Gr?nwall’s inequality,we arrive at

which means thatω ∈L∞（0,T,L2）,sinceis finite.According to the Biot-Savart law,we have that

The Sobolev inequality suggests thatu ∈L∞（0,T,L6）.Through one of the regularity criteria in [17],we can conclude thatubelongs to the regular class.

For the case ofα=,it follows that

Therefore,the smallness assumption onωθimplies the regularity ofu.This completes the proof of Theorem 1.1.

Conflict of InterestThe authors declare no conflict of interest.