On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity*

Hammadi ABIDI Ping ZHANG

(Dedicated to Professor Andrew J.Majda for the 70th birthday)

Abstract In this paper,the authors first consider the global well-posedness of 3-D Boussinesq system,which has variable kinematic viscosity yet without thermal conductivity and buoyancy force,provided that the viscosity coefficient is sufficiently close to some positive constant in L∞and the initial velocity is small enough in(R3).With some thermal conductivity in the temperature equation and with linear buoyancy force θe3 on the velocity equation in the Boussinesq system,the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L1(R3)and(R3)respectively.

Keywords Boussinesq systems,Littlewood-Paley theory,Variable viscosity,Maximal regularity of heat equation

1 Introduction

The purpose of this paper is to investigate the global well-posedness to the following threedimensional Boussinesq system with variable kinematic viscosity

Here θ,u = (u1,u2,u3)stand for the temperature and velocity of the fluid respectively,anddenotes the deformation tensor,Π is a scalar pressure function,and the kinematic viscous coefficient μ(θ)is a smooth,positive and non-decreasing function on [0,∞).The thermal conductivity coefficient ν ≥0,and e3= (0,0,1),ε ≥0,εθe3denotes buoyancy force.Furthermore,in all that follows,we shall always denote |D|sto be the Fourier multiplier with symbol |ξ|sfor s ≥0.

The Boussinesq system arises from a zeroth order approximation to the coupling between Navier-Stokes equations and the thermodynamic equations.It can be used as a model to describe many geophysical phenomena (see [28]).In the Boussinesq approximation of a large class of flow problems,thermodynamic coefficients such as kinematic viscosity,specific heat and thermal conductivity may be assumed to be constants,leading to a coupled system of parabolic equations with linear second order operators.

However,there are some fluids such as lubricants or some plasma flow for which this is not an accurate assumption (see [30]),and a quasilinear parabolic system as follows has to be considered:

One may check[17]and the references therein for more details about(1.2).Furthermore,under some technical assumptions,the global existence of weak solutions to (1.2)and in the case of constant viscosity,the uniqueness of such weak solutions in two space dimension was proved in[17].

Recently the System(1.2)has attracted a lot of attentions in the field of mathematical fluid dynamics.In particular,in two space dimension,with F(θ)= θe2for e2= (0,1)in (1.2),Wang and Zhang [32] proved the global existence of smooth solutions to (1.2).In this case,even with φ(θ)=0 and μ(θ)=μ＞0 in (1.2),Chae [11] and Hou,Li[23] independently proved the global existence of smooth solutions to (1.2),Hmidi and Keraani [20] proved the global existence of weak solutions to (1.2)with θ0,u0belonging to L2(R2)and the uniqueness of such solutions was proved for θ0,u0belonging to Hs(R2)for any s ＞0,the first author of this paper and Hmidi [3] established the global well-posedness of this system with initial data satisfyingWhen N ≥3,eN= (0,··· ,1),and F(θ)= θeN,φ(θ)= 0 and μ(θ)=μ＞0 in (1.2),which corresponds to ν =ε=0 and μ(θ)=μ＞0 in (1.1),Danchin and Paicu [14] proved the global well-posedness of this system withandfor p ∈[N,∞)provided that

for some sufficiently small constant c.

We should also mention that there are many studies on the so-called Boussinesq system with critical dissipation in two space dimension,which reads

When ν = 0 and μ ＞0,the above system is called Boussinesq-Navier-Stokes system with critical dissipation,Hmidi,Keraani and Rousset [21] proved the global well-posedness of such system.When ν ＞0 andμ=0,the System(1.3)is called Boussinesq-Euler system with critical dissipation,Hmidi,Keraani and Rousset [22] proved the global well-posedness of this system.Very recently even the logarithmically critical Boussinesq system was investigated by Hmidi in[19].There are also studies to the global well-posedness of the anisotropic Boussinesq system(with partial thermal conductivity and partial kinematic viscosity)in two space dimension (see[10,15] for instance).

On the other hand,Abidi[2]proved the global well-posedness of (1.2)in two space dimension under the assumptions that:φ(θ)= 0,F(θ)= 0,and the initial data satisfiesmoreover for some sufficiently small ε,there holds

Furthermore,the authors of[6]established the global well-posedness of a 2-D Boussinesq system,which has variable kinematic viscosity and with thermal conductivity of|D|θ,with general initial data provided that the viscosity coefficient is sufficiently close to some positive constant in L∞norm.

Motivated by [2,6] and the recent results of the authors of [4-5] concerning the global well-posedness of inhomogeneous Navier-Stokes system with variable density(see also[16,24]),we are going to investigate the global well-posedness to the following Boussinesq system with variable viscosity,which corresponds ν =ε=0 in (1.1):

In all that follows,we always make the convention that,for any α ＞0,α+means any constant greater than α,and

Theorem 1.1Letandbe a solenoidal vector filed for someThen there exists a sufficiently small constant ε0,and some small enough constant ε,which depends on,such that if

(1.4)has a unique global solution (θ,u,?Π)with

Remark 1.1Let us give the following remarks concerning this theorem:

(1)We point out that the exact dependence of ε onwill be given by(5.1)and(5.49).Furthermore,compared with the 2-D result in [2],here we do not require any smallness condition on the initial temperature in Theorem 1.1.

(2)The assumption that u0∈(R3)is to make sure that the solution decay to 0 as time t goes to ∞,which will be essential to obtain the a priori estimate of the velocity u in the space L1(R+,Lip(R3)).In fact,the exact decay rate of u(t)will be given by (5.31)and (5.37).The assumption thatis due to the variable viscosity,one may check (3)of Remark 1.2 for details.

(3)Compared with the results in[4-5]and[24]for the inhomogeneous Navier-Stokes system with variable viscosity,here ε0is a uniform small positive constant,which does not depend on θ0.While in [4-5]and [24],the smallness condition for μ(ρ0)-1 is in some sense formulated as

for some δ ＞0.In general,under the assumption that

for some ε0sufficiently small,Desjardins[16]only proved the global existence of strong solutions for 2-D inhomogeneous Navier-Stokes system.Yet the uniqueness and regularities of such strong solutions are still open.Moreover,here we do not require the initial velocity u0∈H1as was assumed in [4-5,16,24].

However,with linear buoyancy force θe3on the right-hand side of the velocity equation in(1.4),we still do not know how to prove such global well-posedness result as Theorem 1.1 for the corresponding system.Yet with some dissipation on the temperature equation,more precisely,for the system below,

we have the following global well-posedness result.

Theorem 1.2Let sand αandbe a solenoidal vector field.Then there exist sufficiently small constants η and η0,which depend on,such that if

(1.9)has a unique global solution (θ,u)with

Remark 1.2Let us mention the following facts about Theorem 1.2.

(1)We remark that the exact smallness conditions for η and η0will be given by (7.1),(7.2),(7.12)and(7.20).This result in some sense extends the global well-posedness result in[14]with constant viscosity to the case of variable kinematic viscosity.As a matter of fact,the method of the proof to Theorem 1.2 is also motivated a lot from that in [14,27],namely,we need first to control the L∞(R+;L3,∞(R3))norm for the velocity field u before we deal with the evolution of the Besov norm for u.

(2)We also remark that the global well-posedness of the System (1.9)is easier for larger s.However,we just choose s ≤1 for simplicity.And the reason why we choose s ＞3-is due to the facts that when we manipulate the globalfor the velocity in Subsection 7.2,we need θ(t1)belongs towhich is obtained by using the smooth effect of |D|sin the θ equation of (1.9),and psgiven by (5.39)satisfies s ＞,which is crucial for us to work the paraproduct estimate in (7.28).We do not claim the assumption thatin Theorem 1.2 is optimal in any sense.

(3)We emphasize that the main difficulty in the proof of Theorems 1.1 and 1.2 is due to the variable viscosity.In this case,when we apply Littlewood-Paley theory and smoothing effect of heat semigroup to prove the global L1in time of the space Lipschize estimate for the velocity,we need some positive space derivative estimate of θ.Yet to propagate the positive space derivative estimate for θ,we require the the global L1in time of the space Lipschize estimate for the convection velocity u.Especially for the transport equation of (1.4),one has

which makes impossible to close the a priori estimates.

Let us complete this section with the notations we are going to use in this context.

NotationsLet A,B be two operators,we denote [A;B] = AB -BA,the commutator between A and B.For ab,we mean that there is a uniform constant C,which may be different on different lines,such that a ≤Cb.We shall denote by (a | b)(or (a | b)L2)the L2(R3)inner product of a and b,and denote by (dj)j∈Z(resp.(cj)j∈Z)a generic element of?1(Z)(resp.?2(Z))so that ‖(dj)j∈Z‖?1(Z)=1 (resp.‖(cj)j∈Z‖?2(Z)=1).

For X a Banach space and I an interval of R,we denote by C(I; X)the set of continuous functions on I with values in X,and by Cb(I; X)the subset of bounded functions of C(I; X).For q ∈[1,+∞],the notation Lq(I; X)stands for the set of measurable functions on I with values in X,such that t‖f(t)‖Xbelongs to Lq(I).Finally for any vector field v = (v1,v2,v3),we denoteand the Leray projection operatordiv.

2 Strategies to the Proof of Theorems 1.1 and 1.2

As the existence part of both Theorems 1.1 and 1.2 basically follows from the a priori estimates for smooth enough solutions of (1.4)and (1.9).We shall only outline the main steps in the derivation of the estimates.

2.1 Strategy to the proof of Theorem 1.1

By applying maximal regularity estimates for heat semi-group,we prove that under the assumption of (1.6),for smooth enough solution (θ,u)of (1.4)on [0,T*[,there holds

for any p ∈[6,8].If one assumes moreover the smallness condition (5.1),we get,by using Littlewood-Paley analysis that T*≥1 and

With (2.2),we can prove the propagation of regularities of (θ0,u0)for (θ,u)on [0,1],namely the estimate (5.14).This in particular ensures some t0∈]0,1[ so that there holds

Due to (2.3),we can prove the following Desjardins type (see [16])energy estimates for t ∈]t0,T*[,

for C given by (5.31).Based on (2.5),we deduce that for p ∈[6,8] and satisfying (5.39),

By virtue of (2.4)and (2.6),we can prove the following key estimate by applying the smoothing effect of heat semi-group and Littlewood-Paley theory

for any t ∈]0,T*[,where ‖u(t0)‖H1is determined by (2.3)andby (5.31).

This is basically the contents of Section 5.With (2.7),we shall complete the proof to the existence part of Theorems 1.1 in Subsection 6.1 by constructing appropriate approximate solutions and passing to the limit.And finally the uniqueness part of Theorems 1.1 will be proved in Subsection 6.2 by applying Osgood lemma.

2.2 Strategy to the proof of Theorem 1.2

As in the proof of Theorem 1.1,under the smallness conditions (7.1)and (7.2),we shall prove that the corresponding local smooth solution (θ,u)of (1.9)on [0,T*[ satisfies T*≥1,and

Thanks to (2.8),we get,by applying the smoothing effect of etΔand et|D|s,that there exists some t1∈],1[ so that

Starting from t1,we write the velocity equation of (1.9)as

from which,Lemma 4.2 and under the additional smallness assumption (7.20),we infer the following.

Lemma 2.1Under the assumptions of Proposition 7.3 and for η4given by (7.20),we have

Lemma 2.2Under the assumptions of Proposition 7.3,for 0 ＜s ＜1,one has for any t ∈]t1,T*[,

where psis given by (7.21).

In view of (2.10)and the above lemmas,we obtain the a priori estimate forfor any t ＜T*.The detail will be presented in Proposition 7.3.

Finally the existence part of Theorem 1.2 will be proved by constructing appropriate approximate solutions and passing to the limit in Subsection 8.1.Whereas the uniqueness part of Theorems 1.2 will be proved in Subsection 8.2 through Osgood lemma type argument.

3 Littlewood-Paley Analysis and Lorentz Spaces

The proofs of Theorems 1.1 and 1.2 require Littlewood-Paley decomposition.Let us briefly explain how it may be built in the case x ∈RN(see e.g.[7]).Let φ be a smooth function supported in the annulusand χ(ξ)be a smooth function supporte d in the ballsuch that

Then for u ∈S′h(RN)(see [7,Definition 1.26]),which means u ∈S′(RN)and

we set

we have the formal decomposition

Moreover,the Littlewood-Paley decomposition satisfies the property of almost orthogonality:

We recall now the definition of homogeneous Besov spaces and Bernstein type inequalities from [7].Similar definitions in the inhomogeneous context can be found in [7].

Definition 3.1(see [7,Definition 2.15])Let (p,r)∈[1,+∞]2,s ∈R and u ∈S′h(RN),we set

Lemma 3.1Let B be a ball and C an annulus of RN.A constant C exists so that for any positive real number δ,any non-negative integer k,any smooth homogeneous function σ of degree m,and any couple of real numbers (a,b)with b ≥a ≥1,there hold

Lemma 3.2(see [7,Lemma 2.4])Let C be an annulus.A positive constant C exists so that for any p ∈]1,∞[ and any couple (t,λ)of positive real numbers,we have

We also recall Bony’s decomposition from [9]:

where

In order to obtain a better description of the regularizing effect of the transport-diffusion equation,we need to use Chemin-Lerner type spacesfrom [7].

Definition 3.2Let (r,λ,p)∈[1,+∞]3and T ∈]0,+∞].We defineas the completion of C([0,T]; S(RN))by the norm

with the usual change if r =∞.For short,we just denote this space by

To prove Theorem 1.2,we also need to use Lorentz space Lp,q(R3).For the convenience of the readers,we recall some basic facts on Lp,q(RN)from [18,25].

Definition 3.3(see [18,Definition 1.4.6])For a measurable function f on RN,we define its non-increasing rearrangement by

where μ denotes the usual Lebesgue measure.For (p,q)∈[1,+∞]2,the Lorentz space Lp,q(RN)is the set of functions f such that ‖f‖Lp,q＜∞,with

We remark that Lorentz spaces can also be defined by real interpolation from Lebesgue spaces (see for instance [25,Definition 2.3]):

where 1 ≤p0＜p ＜p1≤∞,β satisfiesand 1 ≤q ≤∞.

Lemma 3.3(see [25,pages 18-20])Let 1 ＜p ＜∞and 1 ≤q ≤∞,we have

· For 1 ≤p ≤∞and 1 ≤q1≤q2≤∞,we have

Lemma 3.4(see [14,Lemma 3.9])For 1 ＜p ＜q ≤∞,one has

As an application of the above basic facts on Littlewood-Paley theory and Lorentz spaces,we prove the following estimates.

Lemma 3.5Let p ≥,s ∈]-1,∞[,α ∈[0,1[ and a,b ∈S(R3).We have

ProofThe proof of the first inequality of (3.6)basically follows from that of (26)in [14].For completeness,we present its proof here.Using Bony’s decomposition (3.5),we write

It follows from Lemma 3.1 that

for some (cj,r)j∈Z∈?r(Z)so that ‖(cj,r)j∈Z‖?r(Z)=1.Similar estimate holds for Tba.

Whereas applying Lemma 3.1 and Lemma 3.3 yields

Hence by virtue of Lemma 3.4,we obtain the first inequality of (3.6).

Along the same line,by using Bony decomposition (3.5),we write

It follows from the classical commutator’s estimate (see [7])that

While it is easy to observe that

4 Some Technical Lemmas

In this section,we shall collect some technical lemmas which will be used throughout this paper.The first one is concerning the definition of Besov spaces with negative indices through heat semi-group.

Proposition 4.1(see [7,Theorem 2.34])Let s be a negative real number and (p,r)∈[1,∞]2.A constant C exists such that

The other key ingredient used in this paper is the maximal Lp(Lq)regularity for the heat kernel.

Lemma 4.1(see [25,Lemma 7.3])The operator A defined by

is bounded from Lp(]0,T[;Lq(R3))to Lp(]0,T[;Lq(R3))for every T ∈(0,∞] and 1 ＜p,q ＜∞.Moreover,there holds

Lemma 4.2Let 3 ＜p ＜∞.The operator B defined byis bounded fromfor every T ∈(0,∞],and there holds

If moreover,p ∈]3,6],one has

ProofNote that

Applying Young’s inequality in the space variables yields

where 1[0,t](t′)denotes the characteristic function on [0,t],from which and Hardy-Littlewood-Sobolev inequality,we conclude the proof of (4.1).

It remains to prove the limiting case,i.e.,(4.2).Indeed it follows by a similar derivation of(4.4)that

Note that for 3 ＜p ≤6,we have

as a result,it comes out (4.2).This completes the proof of Lemma 4.2.

Lemma 4.3Let s ∈]0,1] and u be a smooth solenoidal vector filed on [0,T].Let θ be a smooth enough solution of

Then one has for any t ≤T,

(1)for all p ∈[1,∞],‖θ(t)‖Lp≤‖θ0‖Lp;

(3)for all p ∈]1,∞[ and r ∈[1,∞],

(4)for any α ∈]0,1[,

ProofPart (1)follows directly from [12].While by taking the L2inner product of (4.5)with θ and using div u=0,we get

integrating the above inequality over [0,t] yields part (2)of the lemma.

To deal with part(3),we first apply the dyadic operatorto(4.5)and then use a standard commutator’s process to write

Taking L2inner product of (4.6)withand using Hlder inequality,we have

Recalling from [19,33] the following generalized Bernstein inequality that

for some p independent constant c.We thus obtain

from which,we infer

and hence

However recalling from [22,Lemma 4.3] that

Resuming the above inequality into (4.8)and using part (1)of the lemma gives rise to part (3)of the lemma.

Finally we deduce from Lemma 3.5 and (4.6)that for any α ∈]0,1[,

Applying Gronwall’s inequality gives rise to

Whereas it follows from (4.6)and the proof of part (1)in [12] that

which ensures

Applying Gronwall’s lemma gives

Substituting the above inequality into (4.9)leads to part (4)of the lemma,and we complete the proof of Lemma 4.3.

Let us complete this section by recalling the following proposition from [1].

Proposition 4.2(see [1,Proposition 3.3],see also [7,Theorem 3.37])Let p ∈]2,∞[ andLet u,v be two solenoidal vector field which satisfy u ∈C([0,T];and

Then there exists a constant C such that

Moreover,there holds

5 The a Priori Estimates Related to the System (1.4)

In this section,we shall establish the a priori estimates which will be used to prove the global existence part of Theorem 1.1.

5.1 The short time estimates for smooth enough solutions of (1.4)

Proposition 5.1Let (θ,u)be a smooth enough solution of (1.4)on [0,T*[.Then under the assumption of (1.6),for all t ∈[0,T*[,we have (2.1)for any p ∈[6,8].If moreover we assume that

for some ε1sufficiently small,one has (2.2).

ProofLetdiv be the Leray projection operator to the divergence free vector space.In order to prove (2.1),we first apply the operator P to the u equation of (1.4)to get

or equivalently,

Note that 3 ＜p ＜∞,we infer from Proposition 4.1 that

from which and Lemma 4.2,we deduce from (5.3)that for any p ∈[6,8],

Along the same line,since p ≥6,we deduce from Proposition 4.1 and Lemma 3.1 that

from which,(5.3)and Lemma 4.1,we infer for any p ∈[6,8] that

By summing up (5.4)and (5.5),for ε0being sufficiently small in (1.6),we write

Then for ε sufficiently small in (1.6),we infer

While we deduce from (4.2)for p=6 and (5.3)that

Note that a similar proof of (4.1)also yields

so that we deduce from (5.3)that

On the other hand,it follows from Lemma 4.1 that

which together with the fact (see [13])that

implies

Then we deduce from (5.3)that

Thanks to the fact that

and ‖1-μ(θ0)‖L∞?1,we get,by summing up (5.8)and (5.9)that

On the other hand,it follows from Proposition 4.1 that

Yet by virtue of the real interpolation with the pairs(see [8,Theorem 6.4.5,p.160]),we get

As a result,it comes out

and

Then for ε sufficiently small in (1.6),we deduce from (5.10)that

Inserting the above inequality into (5.7)yields

This together with (5.6)concludes the proof of (2.1).

On the other hand,by virtue of Lemma 3.2,we deduce from (5.3)that

Yet applying Bony’s decomposition (3.5)and standard paraproduct estimates (see [7])leads to

Along the same line,we write

Applying Lemma 3.1 gives

The same estimate holds for Td(μ(θ)-1).

Resuming the above two inequalities into (5.11)and summing up the resulting inequality over j in Z,we arrive at

Without loss of generality,we may assume that T*＞1.We define

We claim that under the assumptions of Proposition 5.1,t*=1.Otherwise,it follows from [2,Lemma 3.1] that

and (2.1)ensures that

Then taking t=t*in(5.12),for ε0,ε being sufficiently small in(1.6),we deduce from(2.1)and(5.12)that

which together with (5.1)ensures

In particular,if we take ε,ε1so small thatwe obtain

which contradicts with(5.13)if t*＜1,and this in turn shows that t*=1 and there holds(2.2).Hence we complete the proof of Proposition 5.1.

Proposition 5.2Under the assumptions of Proposition 5.1,one has

ProofWe first apply the dyadic operatorto(5.2)and then taking the L2inner product of the resulting equation withu that

from which and Lemma 3.1,we infer

Whereas using Bony’s decomposition (3.5)and div u=0,one has

Applying commutator’s estimate (see [7,29])gives

The same estimate holds for div.We thus obtain

where we used (2.2)in the last step.

Along the same line,we get,by using paraproduct estimates in [7] that

However applying Lemma 3.1 gives

And it follows from [7,Theorem 3.14] and (1.6),(2.2)that

Therefore by virtue of (2.1)for p=12,we obtain

Resuming the above estimate and (5.16)into (5.15),we write

Taking ε0,ε small enough in (1.6)and ε1small enough in (5.1),we obtain the second line of(5.14).

Finally similar to (5.17),we can prove

This concludes the proof of the proposition.

Corollary 5.1Under the assumptions of Proposition 5.1,we can find some t0∈]0,1[ such thatu(t0)∈H1(R3),moreover,there holds (2.3).

ProofNote that since δ ∈]0,12[,we have

from which and (5.14),we deduce that there exists some t0∈]0,1[ so that

This together with (5.14)concludes the proof of (2.3).

5.2 The propagation of H1 regularity for u

As a convention in the remaining of this section,we shall always denote t0to be the positive time determined by Corollary 5.1.

Proposition 5.3Let (θ,u)be a smooth enough solution of (1.4)on [0,T*[.Then under the assumptions of Proposition 5.1,we have (2.4)for t ∈]t0,T*[.

ProofThe proof of this proposition is motivated by that of [16,Theorem 1] for 2-D inhomogeneous Navier-Stokes system and that of [5,Proposition 2.1] for 3-D inhomogeneous Navier-Stokes system.In fact,since div u=0,we get,by taking L2inner product of the velocity equation of (1.4)with u,that

integrating the above inequality over [t0,t] and using (1.5)and div u = 0,we obtain the first line of (2.4).

Whereas by taking the L2inner product of (5.2)with ?tu,we write

Motivated by the derivation of (29)in [16],we get,by using integration by parts,that

Using the θ equation of (1.4)and then integration by parts,we get

Notice that

Hence due to div u=0,we obtain

which together with the velocity equation of (1.4)implies that

Integrating the above inequality over [t0,t] and using again div u = 0 and ‖u‖L6≤C‖?u‖L2,we infer

To deal with the pressure function Π,we get,by taking space divergence to the velocity equation of (1.4),that

and

from which,we deduce

We thus deduce from (5.20)that

On the other hand,it is easy to observe that

from which and

we infer

Taking ε0sufficiently small in (1.6),we obtain for p ∈[3,6],

Substituting the above inequality for p=4 into (5.22),we obtain

Applying Gronwall’s lemma and using (2.1)for p=6,we infer

This concludes the proof of (2.4).

An immediate consequence of Proposition 5.3 is the following corollary concerning the estimate offor 3 ≤p ≤8.

Corollary 5.2Under the assumptions of Proposition 5.1,for any p ∈[3,8] and any t ∈]t0,T*[,we have

ProofFor p ∈[3,6],by virtue of (5.25),(2.1)and (2.4),we have for any t ∈]t0,T*[,

which together with (1.6)yields the first inequality of (5.28).For p ∈[6,8],applying H?lder’s inequality gives rise to

from which,(2.1)and the first inequality of(5.28),we conclude the proof of the second inequality of (5.28).

Corollary 5.3LetThen under the assumption of Proposition 5.1,one has

ProofThe proof of this proposition basically follows from that of Proposition 5.3.We first get,by multiplying (5.19)by 〈t〉 and then by using a similar derivation of (5.26),that

By (2.1)and Gronwall’s lemma,we thus obtain

which together with the second inequality of (1.6)leads to (5.30).

5.3 Large time decay estimate for u

Lemma 5.1Under the assumptions of Proposition 5.1,we have

ProofWe first deduce from (5.18)that

Applying Schonbek’s strategy in [31],we split the frequency space R3into two time-dependent domains:R3= S(t)∪S(t)c,wherefor somewhich will be chosen hereafter.Then we deduce from (5.32)that

To deal with the term on the right hand side of (5.33),we write the velocity equation of (1.4)as

Taking Fourier transform with respect to x variables gives rise to

so that

We now estimate term by term above.It is easy to observe that

and

and

Resuming the above estimates into (5.34)and using (2.4)yields

Inserting the resulting inequality into (5.33),we get forgiven by (5.31).

from which,we infer

Hence a similar derivation of (5.35)leads to

where in the last step,we used once again that.This ensures (5.31)forand we completes the proof of Lemma 5.1.

Proposition 5.4Under the assumptions of Proposition 5.1,we have

for any t ∈[t0,T*[ andgiven by (5.31).

ProofMultiplying (5.32)by〈t〉2δ-and then integrating the resulting inequality over[t0,t],we get,by applying (5.31),that

On the other hand,by multiplying (5.19)by,we deduce,by a similar derivation of(5.26),that

Applying Gronwall’s lemma and using (2.1)gives rise to

This together with(2.3)and(5.38)implies(5.37),and we completes the proof of the proposition.

We now present the key estimate of this section.

Proposition 5.5Under the assumptions of Theorem 1.1,for anysatisfying

and C given by (5.31),we have (2.6).

ProofWe first deduce from (5.25)that

from which,(2.1)and H?lder’s inequality,we get

On the other hand,since p satisfies (5.39),we get,by applying H?lder’s inequality,that

This together with (2.1)concludes the proof of the proposition.

5.4 The L1(R+; )estimate for the velocity field

The goal of the this section is to present the a priori L1(R+;(R3))estimate for the velocity field,which is the most important ingredient used in the proof of Theorem 1.1.

Lemma 5.2Let ∈∈]0,[,let (θ,u,?Π)be a smooth enough solution of (1.4)on [0,T*[.Then under the assumptions of Theorem 1.1,we have

for any t ∈]t0,T*[ andgiven by (5.31).

ProofApplyingto (5.2)and using a standard commutator’s process,we write

Throughout this paper,we always denoteand abbreviate d(u)as d.

Taking L2inner product of (5.41)withand using Lemma 3.1,we obtain

Using integration by parts and [14,Lemma A.5],one has

for some uniform positive constant c.

Whereas it follows from Lemma 3.1 that

Thus,by taking ε0sufficiently small in (1.6),we deduce from (5.42)that

This gives rise to

Then by virtue of Definition 3.2,we infer

In what follows,we shall handle term by term the right-hand side of (5.43).Firstly applying Bony’s decomposition (3.5)yields

Applying [29,Lemma 1] gives

And by applying Lemma 3.1,one has

The same estimate holds foru.Hence,thanks to the interpolation inequality

and Corollary 5.2 implies

The same process along with (2.6)for p=6 ensures

Substituting (5.44)and (5.45)into (5.43)leads to (5.40),and we complete the proof of Lemma 5.2.

With Lemma 5.2,we can prove the a priori L1(R+;(R3))estimate for u.

Proposition 5.6Under the assumptions of Theorem 1.1,one has (2.7)for any t ∈[0,T*[.

ProofBy virtue of (2.2),we only need to prove the estimate offor any t ＜T*.As a matter of fact,for any integer N and p ∈]6,8] satisfying (5.39),we deduce from Lemma 3.1 and Lemma 5.2 that

However as

we get,by taking

in (5.46),that

Therefore,in view of (2.3)and (2.6),wheneveris so small that

we infer from (5.48)that

which together with (2.2)leads to (2.7).This completes the proof of the proposition.

6 The Proof of Theorem 1.1

6.1 Existence part of Theorem 1.1

The proof to the existence part of Theorem 1.1 basically follows the following strategy.We begin by solving an appropriate approximate problem,and then we provide uniform estimates for such approximate solutions,and finally we prove the convergence of such approximate solution sequence to a solution of (1.4)through a standard compactness argument.

Lemma 6.1Let (θ,u,?Π)be a smooth enough solution of (1.4)on [0,T*[.Then under the assumption that

for ε0sufficiently small,one has for any t ＜T*,

ProofWe first notice that u solves

from which,we get,by using Proposition 4.2,that

whereas it follows from product laws in Besov spaces that

which together with (6.1)ensures that

On the other hand,we get,by first applying the operatorto the θ equation of (1.4)and then taking the L2inner product of the resulting equation withthat

As div u=0,by applying (2)of (3.6)for α=we write

which together with

and Definition 3.2 implies that

Summing up (6.3)and (6.4),we obtain

Yet due to (2.1)for p=6,we have

Then applying Gronwall’s lemma to (6.5)and using the fact that

we conclude the proof of (6.2).

Now let us recall the following lemma from [1].

Lemma 6.2(see [1,Lemma 4.2])Let s ∈R and (p,r)∈[1,∞[2.Let G ∈(R3).Then there exists Gn∈H∞(R3),such that for all ε ＞0,there exists some n0∈N such that

If moreover G = (G1,G2,G3)and div G = 0,we can choose Gn= (Gn1,Gn2,Gn3)so that div Gn=0.

Proof(The Existence Part of Theorem 1.1)By virtue of Lemma 6.2,we can find θn0,un0∈H∞(R3)for n ∈N so that

Then according to [1,Theorem 1.1],we deduce that the System (1.4)with the initial data(θn0,un0)admits a unique local in time solution (θn,un,?Πn)on [0,T*n[ verifying

Moreover,whenever ε0,ε are small enough in (6.6),we deduce from Proposition 5.6 that there exists a positive constant C0,which depends onso that

from which and (5.47),Lemma 6.1,we infer that (θn,un,?Πn)is uniformly bounded infor any fixed t ＜T*n.This implies that T*n=∞.To prove that there is a subsequence of {(θn,un,?Πn)}n∈N,which converges to a solution(θ,u,?Π)of (1.4),which satisfies(1.7),we only need to use a standard compactness argument of Lions-Aubin’s lemma.Since this argument is rather classical,we shall not present the details here.One may check similar argument from page 582 to page 583 of [1] for details.

6.2 Uniqueness part of Theorem 1.1

Let (θi,ui,?Πi),for i=1,2 be two solutions of (1.4),which satisfy (1.7).We denote

Then due to (1.4),the system for (δθ,δu,δ?Π)reads

where δF is determined by

We first deduce from the transport equation of (6.8)that

Yet notice that

and

we infer

with

On the other hand,it follows from (6.8)and Proposition 4.2 that

While we deduce from (6.9)and standard product laws in Besov spaces (see [7])that

Observing that

Resuming the above inequality into (6.12),we obtain for t ≤t1,

As for α ≥0 and x ∈(0,1],there holds

therefore,by virtue of (6.11),we eventually find for t ≤t1,

Due to

applying Osgood lemma to (6.14)leads to

This proves the uniqueness part of Theorem 1.1 for t ≤t1.The global in time uniqueness of solutions to (1.4),which satisfies (1.7),can be obtained by a boot-strap argument.This completes the proof of Theorem 1.1.

7 The a Priori Estimates Related to the System (1.9)

In this section,we shall present the a priori estimates,which we need to prove Theorem 1.2,for the System (1.9).As in Section 5,we shall first present the short time a priori estimates.

7.1 The short time estimates for smooth enough solutions of (1.9)

Proposition 7.1Let (θ,u)be a smooth enough solution of (1.9)on [0,T*[.We assume that ‖are so small that

and that

for some sufficiently small constants η1,η2.Then one has (2.8).

ProofWe first rewrite the velocity equation of (1.9)as

Yet it follows from [19] that

And it is easy to observe from the proof of[2,Lemma 3.1]that this lemma still holds for smooth enough solutions of (4.5),so that we have

Whereas according to Definition 3.1,for any fixed integer N,one has

where we used the fact thatin the last step.Taking N in the above inequality so that

we obtain

Along the same line,we have

Hence by virtue of (7.1),we deduce from (7.5)that

To estimate ‖u‖L∞1(L3,∞),we need the following lemma,which we admit for the time being.

Lemma 7.1Under the assumptions of Proposition 7.1,one has

Thanks to (7.2)and (7.8),we infer from (7.7)that

which leads to (2.8),and this completes the proof of Proposition 7.1.

Proposition 7.1 is proved provided that we present the proof of Lemma 7.1,which we give as follows.

Proof of Lemma 7.1In fact,it follows from (7.3)and [26,Lemma 23] that

Applying Lemma 4.2 for p=6 yields

Whereas due to (7.3),we get,by a similar proof of (5.8)and (5.9),that

and

so that thanks to (7.1)and (7.6),one has

Since for p ≥q,there holds (see [18])

we have

As a result,it comes out

In particular,if

we infer

Finally it is easy to observe that

Inserting the above two inequalities into (7.9),we arrive at

Note from (7.2)that

from which and (7.11),we conclude the proof of (7.8).

Proposition 7.2Let (θ,u)be a smooth enough solution of (1.9)on [0,T*[.Then under the assumptions of Proposition 7.1 and

for some sufficiently small η3,we have

ProofLet us denoteThen we get,by a similar derivation of (6.3),that

from which,(7.2)and (2.8),we infer

However it follows from part (4)of Lemma 4.3 and Proposition 7.1 that

So that we obtain

Yet by using interpolation and Young’s inequalities,one has for any σ ＞0,there exists some positive constant Cσ＞0 such that

and

Resuming the above two inequalities into (7.14),we conclude the proof of (7.13)if there holds(7.12)and the assumptions of Proposition 7.1.

Corollary 7.1Let (θ,u)be a smooth enough solution of (1.9)on [0,T*[.Let η2and η3be given by (7.2)and (7.12)respectively.Then under the assumptions of Proposition 7.2,we can find some t1∈],1[ so that there holds (2.9).

ProofWe first deduce from (7.13)that

which ensures that there exists someso that

Next we claim that

Indeed we first infer from (7.4)that

Note that by using (1)of (3.6)and (7.8),we have

and

Hence by virtue of (7.1),we obtain

from which,and the para-product estimates (see e.g.[7]),we infer

However,it follows from (2.8)that

While part (4)of Lemma 4.3 and (7.13)ensures that

Resuming the above estimates into (7.18)and using (7.13)leads to (7.17).

On the other hand,it follows from(7.19)that there exists some t1∈],1[so that there holds the second inequality of (2.9).And (7.17)ensures the first inequality of (2.9).This completes the proof of Corollary 7.1.

7.2 The L1(R+; )estimate for the velocity field

As a convention in the remaining of this subsection,we always denote t1to be the positive time determined by Corollary 7.1.

Proposition 7.3Let 3-＜s ≤1,and (θ,u)be a smooth enough solution of the System(1.9)on [0,T*[.Then under the assumptions of Proposition 7.2 and

for some small enough constant η4,and where

we have

which is bounded according to (2.9).

Proof of Proposition 7.3We admit Lemmas 2.1 and 2.2 for the time being and continue the proof of Proposition 7.3.

As a matter of fact,due to(2.8),we only need to prove(7.22)and(7.23)forIndeed by virtue of (2.10)and Lemma 3.2,we deduce that

Choosing N in the above inequality so that

we obtain

Whereas applying (1)of (3.6)and para-product estimates (see [7] for instance)leads to

and

where in the last step,we used (1.5)so that

Resuming the above estimates into (7.24)and using (7.1)and (2.11)gives rise to

It remain to handle the last term in (7.26).In the case when s = 1,p1given by (7.21)equals 6,and then we get,by using para-product estimate,that

Yet it follows from part (3)of Lemma 4.3 and (7.31)below that

Whereas similar to (7.24),we infer from (2.10)that

Note that para-product estimates (see [7])along with (7.27)and (7.31)below ensures that

Hence by virtue of (7.1)and (7.31),we infer

We thus obtain

Resuming the above estimate into (7.26)leads to (7.22).

Yet it follows from part (3)of Lemma 4.3 that

And according to Definition 3.1 and Lemma 3.3,for any integer L,we write

Taking L so that

in the above inequality,we obtain

from which,we infer

Resuming the estimates (7.29),(7.30)into (7.28)and using Lemmas 2.1 and 2.2,we arrive at

Substituting the above estimate into (7.26)and using (2.8)gives rise to (7.23).This completes the proof of Proposition 7.3.

We now turn to the proof of Lemmas 2.1 and 2.2.

Proof of Lemma 2.1Note that for s ∈]0,1],psgiven by (7.21)belongs to ]3,6].Then we get,by a similar derivation of (5.4)and (5.5)that

and

By virtue of Part (1)and Part (2)of Lemma 4.3,we have

and

Therefore,whenever there hold (7.1)and (7.20),we have

On the other hand,since psgiven by (5.39)is greater than,we get,by applying Lemma 4.2,that

Then by virtute of (7.9),(7.10)and (7.31),we conclude

We thus infer from (7.20)that

which leads to (2.11).

Proof of Lemma 2.2Thanks to (7.3),we get,by applying Proposition 4.1 and Lemma 4.2,that

Note that for q ∈]3,∞[,we have

Applying the above inequality for q =gives

Resuming the above estimate into (7.32)and using (2.11)and the fact that

we obtain the first inequality of (2.12).

Similar to the proof of (7.32),one has

By using interpolation inequality that

and (7.31),we obtain

While applying Lemma 4.2 gives rise to

Resuming the above two estimates into(7.33)and using the assumption(7.1),we get the second inequality of (2.12).This completes the proof of Lemma 2.2.

8 Proof of Theorem 1.2

8.1 Existence part of Theorem 1.2

As in Subsection 6.1,we first present the following a priori estimate.

Lemma 8.1Let (θ,u)be a sufficiently smooth solution of (1.9)on [0,T*[.We assume that there holds (7.1).Then for any α ∈]0,1[,one has

ProofWe first write the u equation of (1.9)as

from which and Proposition 4.2,we deduce

Yet by using Bony’s decomposition,we have

and it is easy to observe that

Hence by virtue of (7.1)and (7.25),we obtain

which together with part (4)of Lemma 4.3 implies that

However,it follows from interpolation and Young’s inequalities that for any σ ＞0,there exists some Cσ＞0 so that

We thus infer from (8.2)that

Applying Gronwall’s lemma to the above inequality leads to (8.1).

We now turn to the existence part of Theorem 1.2.

The existence part of Theorem 1.2We basically follow the same line as that in Subsection 6.1.Firstly by virtue of Lemma 6.2,we can findfor n ∈N so that

Then according to [1,Theorem 1.1],we deduce that the System (1.9)with the initial data(θn0,un0)admits a unique local in time solution (θn,un,?Πn)on [0,T*n[ verifying

and

Moreover,whenever η0,η are small enough in(1.10),we deduce from Proposition 7.3 that there exists a positive constant C1,which depends onso that

from which and Lemma 8.1,we infer that (θn,un,?Πn)is uniformly bounded infor any fixed t ＜T*n.This implies that

To prove that there is a subsequence of {(θn,un,?Πn)}n∈N,which converges to a solution(θ,u,?Π)of (1.9),which satisfies (1.11),we need to use a standard compactness argument of Lions-Aubin’s lemma,which we shall not present the details here.One may check similar argument from page 582 to page 583 of [1] for details.

8.2 Uniqueness part of Theorem 1.2

Let (θi,ui,?Πi),for i=1,2 be two solutions of (1.9)which satisfy (1.11).We denote

Then thanks to (1.9),the system for (δθ,δu,δ?Π)reads

where δG is determined by

for δF given by (6.9).

Then similar to (6.10),we first deduce from the transport diffusion equation of (8.5)and part (1)of Lemma 4.3 that

and

While a similar derivation of (6.13)yields for some small enough positive time t2and for t ≤t2,

Resuming the Estimate(8.6)into(8.7)ensures that for some small enough positive time t2and t ≤t2,

With (8.8),we can follow the same line as that in Subsection 6.2 to complete the uniqueness part of Theorem 1.2.

AcknowledgementPart of this work was done when we were visiting Morningside Center of Mathematics,CAS,in the summer of 2013.We appreciate the hospitality and the financial support from the Center.