On the Radius of Spatial Analyticity for the Inviscid Boussinesq Equations

CHENG Fengand XU Chao-Jiang

1 Hubei Key Laboratory of Applied Mathematics,Faculty of Mathematics and Statistics,Hubei University,Wuhan 430062,China.

2 Department of Mathematics,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China.

3 Université de Rouen,CNRS UMR 6085,Laboratoire de Mathématiques,76801 Saint-Etienne du Rouvray,France.

Abstract. In this paper,we study the problem of analyticity of smooth solutions of the inviscid Boussinesq equations. If the initial datum is real-analytic,the solution remains real-analytic on the existence interval. By an inductive method we can obtain a lower bound on the radius of spatial analyticity of the smooth solution.

Key Words: Boussinesq equations;analyticity;radius of analyticity.

1 Introduction

In this paper,we consider the following multi-dimensional invisicd Boussinesq equation on the torus Td,

with divu0=0. Here,u=(u1,...,ud)is the velocity field,pthe scalar pressure,andθthe scalar density.eddenotes the vertical unit vector(0,...,0,1).The Boussinesq systems play an important role in geophysical fluids such as atmospheric fronts and oceanic circulation(see, e.g., [1–3]). Moreover, the Boussinesq systems are important for the study of the Rayleigh-Benard convection,see[4,5].

Besides the physical importance,the invisicd Boussinesq equations can also be viewed as simplified model compared with the Euler equation. In the cased=2,the 2D inviscid Boussinesq equations share some key features with the 3D Euler equations such as the vortex stretching mechanism. It was also pointed out in[6]that the 2D invisicid Boussinesq equations are identical to the Euler equations for the 3D axisymmetric swirling flows outside the symmetric axis.

The inviscid Boussinesq equations have been studied by many authors through the years, for instance [7–16]. Specially, Chae and Nam [8] studied local existence and uniqueness of the inviscid Boussinesq equation and some blow-up criterion in the Sobolev space,Yuan[11]and Liu et al.[14]in the Besov space,Chae and Kim[7]and Cui et al.[15]in the H?lder spaces, Xiang and Yan[16] in the Triebel-Lizorkin-Lorentz spaces. It was remarked that the global regularity for the inviscid Boussinesq equations even in two dimensions is a challenging open problem in mathematical fluid mechanics.

In this paper,we are concerned with the analyticity of smooth solutions of the inviscid Boussinesq equations(1.1). The analyticity of the solution for Euler equations in the space variables,for analytic initial data is an important issue,studied in[17–23]. In particular,Kukavica and Vicol[21] studied the analyticity of solutions for the Euler equations and obtained that the radius of analyticityτ(t) of any analytic solutionu(t,x) has a lower bound

for a constantC0>0 depending on the dimension andC>0 depending on the norm of the initial datum in some finite order Sobolev space. The same authors in[22] obtained a better lower bound forτ(t) for the Euler equations in a half space replacing (1+t)?2by (1+t)?1in (1.2). In [24], we have investigated the Gevrey analyticity of the smooth solution for the ideal MHD equations following the method of[21]. The approach used in[21–24]relies on the energy method in infinite order Gevrey-Sobolev spaces.Recently,Cappiello and Nicola [25] developed a new inductive method to simplify the proof of[21,22]. In this paper, we shall apply this inductive method to study the analyticity of smooth solution for the inviscid Boussinesq equations. The main additional difficulty arises from the estimate of the weak coupling termu·?θ.

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

2 Notations and Main Theorem

In this section we will give some notations and function spaces which will be used throughout the following arguments. Throughout the paper,Cdenotes a generic constant which may vary from line to line. Since we work on the torus Tdthroughout the paper,we shall write the function spaceL2orHkto represnt the functions that are squre integrable or squre integrable up tok-th derivative without mentioning the domain Td.

Letv=(v1,...,vd)be a vector function,we say thatv∈L2which meansvi∈L2for each 1≤i≤d. We denote theL2norm ofvbyLetρbe a scalar function,we say the pare(v,ρ)∈L2ifv,ρ∈L2. We denote theL2norm of the pare(v,ρ)to be

Denote〈·,·〉to be the inner product inL2either for vector function or scalar function.

In[26], it is stated that a smooth functionfis uniformly analytic in Tdif there existM,τ>0 such that

Letn≥0 be an integer andα,β∈Ndbe multi-indices,then the sequence

satisfies

for all multi-indexα,β∈Ndand some universal constantC,for proof please refer to[17].With these notations,we can state our main results.

Theorem 2.1.Letbe analytic inTd,satisfyingdivu0=0and

for someand A≥1. Letbe the corresponding Hk max-imal solution of the inviscid Boussinesq equations(1.1), with the initial datum(u0,θ0)0,C1>0, depending only on k and d,such that the radius of analyticity satisfying

Remark 2.1.In the caseθ=0,Theorem 2.1 recovers the result of Kukavica and Vicol[21]and Cappiello and Nicola[25]for the incompressible Euler equation.

Remark 2.2.When the dimensiond=2,the blow-up criterion proved by Chae and Nam in[8]stated that the solution remains smooth up toTas long asSo,it will be very interesting if the quantitydsin the lower bound of the radius of analytic solution in(2.4)can be replaced by

3 The estimate of the Sobolev norm

In order to prove the main Theorem 2.1, we recall the following results about the local existence and uniqueness ofHk-solution of the inviscid Boussinesq equations(1.1)which is a proposition in[27]ford=3.

Theorem 3.1.(Wang-Xie[27,Proposition 1.2])Ifand u0(x)satisfiesthe divergence-free condition,then there exists T2>0such that the inviscid problem(1.1)admits a unique solution

The domain considered in [27] is a bounded domain with smooth boundary conditions and this case can be naturally extended to periodic domain with periodic boundary conditions. The proof is due to the argument in [28] and [29]. Since it is standard, we omit the details here. When the dimensiond=2,Chae and Nam[8]also proved the local existence and blow-up criterion.

In order to prove the main Theorem,we will need the following Lemma.

Lemma 3.1.Let d=2,3,andbe fixed. Let(u,θ)∈C(0,T1;Hk)be the correspondingmaximal Hk-solution of(1.1)with intial data(u0,θ0)∈Hk and u0satisfies the divergence-free condition and periodic boundary condition,then?0≤t

where the constant C0depending on the dimension and k.

Proof.Since the initial data (u0,θ0)∈Hksatisfies ?·u0=0 and the periodic boundary conditions, the local existence of theHk-solution(u,θ) is already known. We here only need to show theHkenergy estimate(3.1).

Letα∈Ndsatisfies 0 ≤|α|=α1+α2+...+αd≤k. We first apply the?αon both sides of the first equation of(1.1)and then take theL2-inner product with?αuwith both sides,which gives

where the pressure term 〈??αp,?u〉=0 is due to the factuis divergence free and the domain considered here is a periodic domain.

We then apply the?αon both sides of the second equation of (1.1) and take theL2inner product with?αθon both sides,which gives

Now adding(3.2)with(3.3)and taking summation over 0≤|α|≤k,we can obtain

Notice thatuis divergence free, by use of the Sobolev inequality which can be found in[6]we can obtain

where the constantCdepends onkand the dimensiond. In the same way,we can obtain

where the constantCalso depends onkand the space dimensiond. Substituting (3.5),(3.6)and(3.7)into(3.4),we obtain

whereCis some constant depending onk,d. By the Gronwall inequality, (3.1) is then proved.

Lemma 3.1 tells us that the solution for the inviscid Boussinesq equation has the same Sobolev regularity as the initial data. In the following Lemma we will show that if the initial data (u0,θ0)∈Hkfor arbitraryk≥3, there exists an interval [0,T] uniformly with respect toksuch that the unique smooth solution(u,θ)∈L∞([0,T];Hk).

Lemma 3.2.Let(u,θ)be the H3-solution of the inviscid Boussinesq equation(1.1)on the time interval[0,T],with initial data(u0,θ0)∈H3.Then for all k≥3,if(u0,θ0)∈Hk,the corresponding solution(u,θ)satisfies

Proof.We claim that for every 3≤m≤kthere exist a constantCmsuch that

Form=3 the statement follows from our assumption. Take 4≤m≤kand suppose that the statement is true form?1,i. e.

We take theHminner product of the first equation of(1.1)withuand take theHminner product of the second equation of(1.1)withθ,which gives

For 4≤m≤k,by(3.8)we obtain

whereCis a constant depending onm,d. Then the Gronwall inequality yields

By Sobolev embedding inequality,we have

for some constantC′. Then from(3.9)and the assumption,we have

So it easily follows that

which proves the Lemma by induction.

Remark 3.1.In Lemma 3.2,the uniform lifespan[0,T]of the Sobolev solution is independent of the Sobolev orderkwhich allows us to takek→∞. In other words,if the initial datum(u0,θ0)isC∞,then solution(u(t,x),θ(t,x))is alsoC∞for almost everyt∈[0,T].

4 Proof of Theorem 2.1

In this Section,we will give the proof of the main theorem.Proof of Theorem 2.1.By Lemma 3.2, we know that the solution (u,θ) is smooth, since(u0,θ0)is. Now we claim that for all|α|=N>2 we have

whereC0,C1are positive constants depending only onkandd. We set

To prove the claim, we proceed by induction onN. The result is true forN=2 by (3.1)with notice thatk+2<2k+1 andHence,letN≥3 and assume(4.1)holds for multi-indicesαof length 2≤|α|≤N?1 and prove it for|α|=N.

For|α|=N,|γ|≤k, we first apply?α+γon both sides of the first and the second equation of(1.1)and then take theL2-inner product with?α+γuand?α+γθrespectivily,which gives

DenoteLw=Luw:=u·?wand

Taking summation with 0≤|γ|≤kin(4.2),we have by the Cauchy-Schwartz inequality

where we used the the standard argument in[pp.47-48,[30]]and the constant C depends onkandd. It remains to estimate I1and I2.Note that by the Leibniz rule,we can expand the expression of I1as follows

where the restriction|β|+|δ|<|α|+|γ| is due to the fact that〈u·??α+γu,?α+γu〉=0 becauseuis divergence free. By[25],we have

We then follow the ideal of[25]to estimate I2. Similarly,we can expand I2as

Then we divide the summation of the right of(4.4)into three parts

where

and

Estimation of I21: With the fact thatHkis an algbra ifand|γ|≤k,we have

Noting that 2≤|α?β|≤N?1 and 2≤|β|+1≤N?1,the hypothesis(4.1)for 2≤|α|≤N?1 indicates that

and

Substituting(4.6)and(4.7)into(4.5)and employing(2.2),we obtain

Estimation of I22: In a similar way,we rewrite I22as

For R21,we obtain

for some constantCdepending onk.

For R22,we have the following estimate

Then for R23,we have

Summing up the estimates of(4.8)–(4.10),we obtain

Estimate of I23: We divide I23as follows

For R31,we have

whereCdepends onkanddand we used the fact|γ?δ|+k≤2kin(4.11),the inductive hypothesis(4.1),and the fact thatB>‖(u0,θ0)‖H2k+1. For R32,note that the mult-indexαis fixed,we have

where the the constantCd,kdepends ondandk. For R33,in a similar way we have

Summing up(4.11),(4.12)and(4.13),we obtain

Combining the estimates I1and I2,we have from(4.3)

where we used the Cauchy-Schwartz inequality and the factifN≥3 and the constantC>0 depending only on the dimensiondandk.

Now we integrate(4.14)from 0 totand take the supremum on|α|=N. We obtain

We can takeC0≥Cin(4.15),so that Grownwall inequality[25,Lemma 2.1]gives

Note that we have EN[(u,θ)(0)]≤BAN?1by the assumption(2.3).If we chooseC1=3C,so thatthen usingA≥1 andN≥3 gives

and we obtain exactly(4.1)for|α|=N. Then the theorem is proved.

Acknowledgement

The research of the second author is supported partially by“The Fundamental Research Funds for Central Universities of China”.