Feng CHENG

Hubei Key Laboratory of Applied Mathematics;School of Mathematics and Statistics,Hubei University,Wuhan 430062,China

E-mail:fengcheng@hubu.edu.cn

Chaojiang XU

School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China;Universit′e de Rouen,CNRS UMR 6085,Laboratoire de Math′ematiques,76801 Saint-Etienne du Rouvray,France

E-mail:Chao-Jiang.Xu@univ-rouen.fr

Abstract In this article,we study the analytical smoothing e ff ect of Cauchy problem for the incompressible Boussinesq equations.Precisely,we use the Fourier method to prove that the Sobolev H1-solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time.So the incompressible Boussinesq equations admit exactly same smoothing e ff ect properties of incompressible Navier-Stokes equations.

Key words analyticity;smoothing e ff ect of solutions;Boussinesq equation

1 Introduction

In this article,we consider the following incompressible Boussinesq equations on the torus TN=[0,2π]Nwith N=2 or 3,

where u(t,x)=(u1,···,uN)(t,x),(t,x) ∈ R+× TN,is the velocity vector fi eld,p=p(t,x)is the scalar pressure,θ= θ(t,x)is the scalar temperature in the content of thermal convection and the density in the modeling of geophysical fl uids,ν >0 is the viscosity,and κ >0 is the thermal di ff usivity,and eN=(0,···,1)is the unit vector in the xN-direction.In addition to(1.1),we assume that u,θ,p are periodic for the spatial variable and the average value of u,θ,p on TNvanishes:

In physics,the Boussinesq system(1.1)is commonly used to model large scale atmospheric and oceanic fl ows,for example,tornadoes,cyclones,and hurricanes.It describes the dynamics of fl uid in fl uenced by gravitational force,which plays an very important role in the study of Raleigh-Bernard convection,see[14,18–20].

In mathematics,the Boussinesq system is one of the most commonly used simplifed models to understand some key features of the 3-D incompressible Navier Stokes equations and Euler equations.Actually,if we set θ ≡ 0,the Boussinesq system(1.1)reduces to the incompressible Navier Stokes equations.

The well-posedness of the Boussinesq system(1.1)was studied extensively in recent years.The global well-posedness of weak solutions,or strong solutions in the case of small data for the Boussinesq equations was considered by many authors,see[1,3,9–12,24].The global existence and uniqueness of smooth solutions to the 2D Boussinesq system with partial viscosity was also studied,see[7,8,22,23].However,similar to three-dimensional Navier Stokes equation,the global existence or fi nite time blow-up of smooth solutions for the 3-D incompressible Boussinesq equations is still an open problem.

In this article,we study the analytical regularity of the strong solution to the incompressible Boussinesq equation(1.1),which is similar to the analytical regularity of Navier Stokes equation.Since Foias and Temam[13]studied the Gevrey class regularity of Navier Stokes equations in 1989,there were many works on the Gevrey class regularity of solutions for many kinds of equations,see[2,6,21].In two dimensions,the authors of[4]studied Gevrey class regularity for the two-dimensional Newton-Boussinesq equations.They considered the vorticity instead of the velocity to eliminate the pressure by transforming the velocity equation into the vorticity equation.In this article,we use the Leray projection operator to deal with the pressure and the method can be applied to higher dimensions,which improves the previous work.

The article is organized as follows.In Section 2,we will give some notations and state our main results.In Section 3,we fi rst recall some known results and then give some lemmas which are needed to prove Theorem 2.1.In Section 4,we give the proof of Theorem 2.1.

2 Notations and Main Theorems

In this section,we will give some notations and function spaces which will be used throughout the following arguments.Throughout the article,C denotes a generic constant which may vary from line to line.

Denote

We denote L2(TN)Nthe CN-valued vector functions space of each vector component in L2(TN),and it is a Hilbert space for the inner product,

where f=(f1,···,fN),g=(g1,···,gN) ∈ L2(TN)N.Let m>0 be an integer,we denote Hm(TN)Nthe CN-valued vector function space of each vector component in Sobolev space Hm(TN),and the corresponding norm k·kmis

where?u0=0 meets condition(1.2).Let r>0 be a real number,we can then identify the Sobolev space(TN)as

Recall that we say a smooth function f belongs to Gevery class Gs(TN)for some s>0,if there exists C,τ>0 such that

The parameter τ is called the radius of Gevrey class s.In particular,when s=1,it is the analytical functions.Let us de fi ne the Zygmund operator Λ=with

The same de fi nition of Λθ for the scalar function θ∈ H1(TN).

We de fi ne now the function space D(ΛreτΛ1/s)for r,s,τ>0,u ∈ D(ΛreτΛ1/s),if u ∈and

For the scalar function θ,we also write θ∈D(ΛreτΛ1/s),which means θ∈L2(TN),?θ0=0 and

It was proved in[17]that D(ΛreτΛ1/s)? Gs(TN).

With these preparations,we can state our main results.

Theorem 2.1If the initial data(u0,θ0)∈(TN)×H1(TN).Then the following results hold.

(1)For two dimensions case,N=2,the Boussinesq equations(1.1)admit an unique global solution(u,θ)satisfying

(2)For three dimensions case,N=3,the Boussinesq equations(1.1)admit a unique local solution(u,θ)satisfying

where T1>0 depends on the initial data(u0,θ0).

Remark 2.21.Following the arguments of Foias and Temam[13],one can improve the regularity of time t for the solution(u,θ)by extending to the complex plane and showing that the solution is actually analytic in time variable.Since the computations are standard,we omit these details.

2.The analytical smoothing e ff ect of Theorem 2.1 imply the Gevery smoothing e ff ect in D(ΛreτΛ1/s)? Gs(TN)for any s ≥ 1.

3 Premilinary Results

In this Section,we recall some known results on the incompressible Boussinesq system(1.1).Let us fi rst recall the de fi nition of weak solutions and strong solutions for the Boussinesq system(1.1).

De fi nition 3.1(weak solutions) For any T>0,one call(u,θ)the weak solution of the Boussinesq system(1.1),if

and u satis fi es the following weak formulation

The pressure term p in the velocity equation of(1.1)is the Lagrange multiplier,which is eliminated by taking the L2-inner product with the divergence-free vector fi eld.However,it can be determined by

with p satisfying periodic boundary conditions and(1.2).

Analogous to the incompressible Navier Stokes equation,the global existence of weak solutions to the Boussinesq equations is standard,see[5,12,15,16]and references therein.

Theorem 3.2(weak solutions) Let(u0,θ0) ∈ L2σ(TN)× L2(TN).There exists a global weak solution(u,θ)of the Boussinesq equations(1.1)such that

Such solutions satis fi es,for all t∈[0,T],the energy inequalities

and

for any T>0,and some constant C>0.

Remark 3.3For two dimensions case,N=2,the weak solution obtained in Theorem 3.2 is unique.Whether the uniqueness of weak solution for three dimensions case N=3 is still unknown.

In the following,we study a class of more regular solutions to the Boussinesq equations,which is called strong solutions analogous to the Navier-Stokes equations and introduced by Temam in[25].

De fi nition 3.4(strong solutions)Let(u0,θ0)∈H1σ(TN)×H1(TN).Usually,one call a strong solution(u,θ)of the Boussinesq equations(1.1)on[0,T]if

and(u,θ)satis fi es De fi nition 3.1.

The existence and uniqueness of strong solutions for the incompressible Boussinesq equations(1.1)is analougous to the results of Navier Stokes equations,which we state as follows.

Theorem 3.5Let(u0,θ0)∈H1σ(TN)×H1(TN).Then the following results hold.

(1)For two dimensions case N=2,for any T>0,the Boussinesq equations(1.1)admit an unique strong solution(u,θ)on[0,T]satisfying

(2)For three dimensions N=3,there exists T1>0 such that the Boussinesq equations(1.1)admit a unique strong solution(u,θ)on[0,T1]satisfying

where T1>0 depends on the initial data(u0,θ0).

ProofFor the self-contenant of the article,we give the`a priori estimate of the solutions,then the construction of approximate solutions follows from the standard Galerkin method which we refer readers to[25].

Let us assume that(u,θ)is the smooth solution of the incompressible Boussinesq equations.We multiply the velocity equation of(1.1)by Λ2u and integrate over TN.After integrating by parts,we have

where??p,Λ2u?L2(TN)vanishes because u is divergence-free and Λ2= ?? does not violate this property on the torus.Then we multiply the thermal equation of(1.1)by Λ2θ and integrate over TN.Integrating by parts,we obtain

We fi rst recall the following Sobolev inequality(see[25],now usually called the Gagliardo-Nirenberg inequality),

Using the Cauchy-Schwartz inequality and the Sobolev inequality,we have

Case ofN=3 We apply the Young’s inequality on(3.6)and(3.8)to bound the right hand side of(3.1)and(3.2),

This proved the local existence of strong solution for three-dimensional space.

The proof of the existence for both two dimensions and three dimensions can be made rigorous by considering the Galerkin approximation procedure,which is standard in the book[25]. ?

Remark 3.6The di ff erence in the Sobolev inequality(3.3)and(3.4)caused by the spatial dimension N makes huge in fl uence on the lifespan of the solution,which is showed in the above proof.

4 Proof of Theorem 2.1

In this section,we give the proof of Theorem 2.1.To prove Theorem 2.1,we recall the following lemmas in[13]concerning the estimates of nonlinear terms.

Lemma 4.1Let u,v,w be given in D(eτΛ1/sΛ2)for τ>0,s>0.Then the following estimates hold,for N=2 or 3,

and

where C>0 is a constant.

Remark 4.2In Lemma 4.1,u is CN-valued vector function while v,w can be either CN-valued vector function or C-valued scalar function.

Then we are able to prove Theorem 2.1 following the idea of Foias and Temam[13].

Proof of Theorem 2.1For the sake of simplicity,we shall only consider the time variable in the real case and we remark that the results can be extended to the complex case following the same arguments of[13].For this part,we set the radius of Gevrey class τ(t)=t.

We recall that the usual strategy to approximate the Navier Stokes equation is to project the velocity fi eld onto the divergence-free fi eld.Here we also need to write the velocity equation of the Boussinesq system(1.1)into the following form

where P is the Helmholtz-Leray orthogonal projection,that is,projection onto divergence-free vector fi elds.In this way,we eliminate the pressure term as Foias and Temam did in[13].Denote the operator A= ?P?,the eigenvectors{Ej}of A constitutes the othonormal basis of(TN).We denote the corresponding eigenvalues bywith 0< λ1< ···≤λj≤ λj+1≤ ···,then AEj= λjEj.We note that P commutes with ?? on the torus,and A=??when acting on the divergence-free vector fi eld.

The thermal equation of the Boussinesq system(1.1)is the second order parabolic equation and the eigenvectors of?? also constitute the orthonormal basis of L2(TN).Letbe the orthonormal basis of L2(TN)andbe the corresponding eigenvalues,which satis fi es

We will then approximate equation(4.1)and the thermal equation of(1.1)by the Galerkin method.For fi xed integer m ∈ N,let Pmbe the projection from(TN)onto the subspace Wmspanned by{Ej,1≤j≤m,j∈Z}and Pmbe the projection from L2(TN)onto the subspace Vmof L2(TN)spanned by{eα,1≤α≤m,α∈Z}.

We are looking for solution(um,θm)to the following approximate equations

where

The function um,θmare analytical for x variables,in fact the sequence of Ej’s and eα’s is the linear combinations of the sequence of functions Wk,?and wn,

where k=(k1,···,kN)∈ ZN{0},n=(n1,···,nN) ∈ ZN{0},e1,···,eNis the canonical basis of RN.ak,?is the coefficient that make the sequence Wk,?to be orthonormal in(TN).

After taking L2-inner product of(4.2)with Ejand taking L2-inner product of(4.3)with eα,for 1 ≤ j ≤ m and 1 ≤ α ≤ m.The Cauchy problem(4.2)–(4.4)is equivalent to the following Cauchy problem for ordinary di ff erential system

where

The standard theory of ordinary di ff erential equations indicates that the ODE system(4.5)–(4.7)admit a unique local solution?ξj,m(t),ηα,m(t)?1≤j,α≤mon some interval[0,Tm].

We prove now the solution of the ODE system(4.5)–(4.7),?ξj,m(t),ηα,m(t)?can be extended to a global in time solution for any fi xed m.To show this,we note that each Ejand eαare analytical.Then we can perform integration by parts to obtain

where we used the fact?·Ek=0.Integrating by parts,one can also obtain

So if we multiply(4.5)by ξj,m(t)and take sum over{1 ≤ j ≤ m,j∈ Z},we obtain

where we infer from(4.8)the following fact

If we multiply(4.6)by ηα,m(t)and take sum over{1 ≤ α ≤ m,α ∈ Z},we obtain

where we infer from(4.9)the fact

Equation(4.11)implies that,for any m∈N,

With this fact,one can infer from(4.10)that

The above priori estimates show that the solution(ξj,m(t),ηα,m(t))is bounded only by the initial data,then we can repeat the arguments above to extend the local solution to arbitrary time interval[0,T].Thus for fi xed integer m,the Cauchy problem of ordinary di ff erential equations(4.5)–(4.7)possess a unique solution(ξj,m(t),ηα,m(t)) ∈ C1([0,T])for all T>0.Equivalently,we have the solution(um,θm)to the approximate equation(4.2)–(4.4)satis fi es

for fi xed m∈N.

Once we obtained the approximating solution,we then perform the Gevrey norm estimates on(um,θm).For fi xed integer m,as a fi nite summation of Ejand eα,the solution?um,θm

?belongs to Gevrey class D(ΛreτΛ)for arbitrary r>0.Let T>0 be fi exed.For 0

where we used the fact that umis divergence free and P is symmetric.Taking the L2-inner product of the thermal equation of(4.3)with Λ2e2tΛθm,we obtain,similarly,

By Plancherel’s theorem and summing over(4.13)and(4.14),we can write

where we use the compact embedding theorem[25].To show that E(u)=etΛu and F(θ)=etΛθ,one only need to recall that

Then we have,by the uniqueness of the strong solution,

For now we prove the local solution with values in Gevrey class functions.Now if we know that

for some large positive number depending on T and the initial data,then we can repeat the argument above at any time 0

But estimate(4.26)is not true for three dimensions N=3,because we do not have the uniform bound(3.11)for all t in three dimensions. ?