Yingshu ZHANGLang LIShaomei FANG

Department of Mathematics,South China Agricultural University,Guangzhou 510642,China

E-mail:yinshu728@sina.com;lilang@scau.edu.cn;dz90@scau.edu.cn

Abstract The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(n ≥ 3)is considered.Based on Green’s function method,some ingenious energy estimates are given.Then the global existence and pointwise convergence rates of the classical solutions are established.Furthermore,the Lpconvergence rate of the solution is obtained.

Key words Kuramoto-Sivashinsky equation;global existence;pointwise estimates;Green’s function method;energy method

1 Introduction

In this paper,we concern the global existence and pointwise estimates of the solutions to the following Cauchy problem for the generalized Kuramoto-Sivashinsky system in the multidimensions

was derived by Kuramoto[1]and Sivashinsky[2]independently.By setting a new variable u(x,t)= ?Φ(x,t)=(Φx1,Φx2,···,Φxn),we get the conservative form of(1.2)

At fi rst,it was obtained in the context of plasma ion mode instabilities[3].Since then,the KS equation was used to study the turbulence in reactive systems,di ff usive instabilities in laminar fl ame fronts,Rayleigh-Benard convection and fl ow stability of thin liquid fi lms on inclined planes[4],and the population growth and di ff usion[5].It also arises in the context of viscous fi lm layer fl ow and bifurcating solutions of the Navier-Stokes equations[6].

The KS equation(1.3)is one of a few nonlinear systems that were extensively explored,for example,the well-posedness problem(see[7–10]),the long-time behavior(see[11]),the control problem(see[12,13]).Guo[14]studied the initial value problems for the multidimensional generalized KS equation

this paper established the existence and uniqueness theorems of the global smooth solutions to this problem.Furthermore,Zhang[15]investigated the decay of the solution in 1≤n≤3 with any initial data.Allouba[16]gave an explicit solution to a linearized KS equation in all spatial dimensions,and Zhao[17]studied the decay rate of the multidimensional generalized KS system and the nonlinear term was equaassumed as f(u)=O(|u|2).Based on these previous works,we set the nonlinear term as f(u)=O(|u|ρ)(ρ ≥ 2).

The main purpose of this paper is to establish the global existence and pointwise estimates of solutions to the generalized KS equation(1.1).Up to now,there are few works on the pointwise estimates of solutions of the Cauchy problem for multidimensional KS equation.The difficulty is that KS equation consists of higher order derivatives.To overcome this difficulty,we mainly use the Green’s function method(see e.g.[18–20,22]).In this paper,we fi rst express the solution to(1.1)by Green’s function and obtain the pointwise estimates of the Green’s function.Then,we show the pointwise estimates of the solutions to(1.1)by using the Green’s function method.Finally,the Lp(1≤p≤∞)estimate of the solution can be obtained easily.

Now we give the main results in this paper.

The rest of the paper is organized as follows.In Section 2,we take the Fourier transform to derive the solution formula of the Cauchy problem.Then we show some estimates to the low-frequency part,middle-frequency part and high-frequency part respectively.In Section 3,we will construct a global classical solution.In Section 4,we give the pointwise estimate of Green’s function of problem(1.1)obtained in Section 2.Finally in Section 5,we give the pointwise eatimate to the solution of problem(1.1)which completes the proof of Theorem 1.1.

NotationsIn this paper, α =(α1,α2,···,αn)and β =(β1,β2,···,βn)are multiindexes.C denotes a positive generic constant.Lp,Wm,pare the usual Lebesgue and Sobolev space on Rnand Hm=Wm,2,with the norm||·||Lp,||·||Wm,p.In particular,we use the notation Hm(Rn)=Wm,2(Rn)and de fi ne the homogeneous Sobolev space˙Hm(Rn)and the corresponding norm to be

2 Green’s Function and Energy Estimate

As usual,suppose that f(x,t)∈L1(Rn),we de fi ne its Fourier transform with respect to the spatial x∈Rnas

and the inverse Fourier transform to be

Now we study Green’s function of(1.1),i.e.,we consider the solution to the following initial value problem

By Fourier transform with respect to the variable x,we deduce that

In the rest section,we denote μ(ξ)=a|ξ|2? b|ξ|4? c.As a result,one has

By the Duhamel’s principle,we can get the representation of the solution for the nonlinear problem(1.1)be two smooth cut-o fffunctions,where r,R are constants and 2r

ProofBecause of a2<4bc,we know μ(ξ)=a|ξ|2? b|ξ|4? c is negative and ξ lies in a bounded interval away from ξ=0 and ξ= ∞.Thus,there exist constant b0>0 such that≤ Ce?b0t.Then this proposition can be obtained by the similar method as above.?

By the de fi nition of high frequency part,we have the following lemma.

Lemma 2.3(Poincar′e-like inequality) Assume u(x) ∈ Hm,m ≥ 0 is an integer.Then for high frequency part,there exist constant C such that

holds for any integer s∈[0,m].

Proposition 2.4Assume that u(x,t)is the solution of problem(1.1)and u3(x,t)is the high frequency part of u(x,t),when c>,there exists C>0 is constant such that

ProofWithout loss of generality,we only consider|α|=0.We rewrite problem(1.1)in the high frequency form

Multiply(2.15)by u3(x,t),then integrating by parts with respect to x,and using Cauchy’s inequality,we have

Recall Lemma 2.3,we have

Thus,there exists some constant C>0 such that

By integrating on(0,t)with respect to t,by virtue of Lemma 2.3,it gives

The estimates for high order derivatives can be obtained similarly. ?

3 Global Existe nce

We will consider the problem in the space Xl,E,which will be introduced as follows.For a given integer l≥+5 and some constant E>0,we de fi ne

where Dl(u)is de fi ned to be

It is easy to show that Xl,E,equipped with norm Dl(·)is a non-empty Banach space.To obtain the global solution,we construct a convergent sequence{um(x,t)}by the following linearized scheme

Finally,we have Dl(vm)≤ EDl(vm?1).Since E= ε13?1,{um}is a Cauchy sequence in Banach space Xl,E,and then its limit function u is a global solution of Cauchy problem(1.1).Therefore,the proof of this theorem is fi nished. ?

4 Pointwise Estimates on Green’s Function

From[20],we know the decay of the solution of the linearized equation is mainly related to the properties ofin low frequency part.We use the cut-o fffunctions to divide the frequency space into low frequency part,middle frequency part and high frequency part.

Lemma 4.1Suppose t>0,ξ∈ Rn,there exists constant C=C(β,b)such that

ProofIf ξ∈ Rn,the result can be obtained by direct calculation.Suppose ξ=(ξ1,ξ2,···,ξn)∈ Rn,thus

It holds for|β|=0 obviously.Suppose that it holds for|β|≤ l?1,we will prove that it is true for|β|=l.Applying χ2(ξ)andto(2.2),we get

Thus(4.15)following from(4.14)and(4.23).?

On the other hand

Using(4.14)again and by the same discussion as above,we get(4.24)from(4.28)and(4.29).?

5 Pointwise Estimates of u(x,t)

Now we will give a pointwise estimate for the solution u of(1.1).Taking Dαxon(2.4)and applying Duhamel’s principle,we obtain

where n3=min{n1,n2}.

ProofThe proof of this lemma can be found in Lemma 2.4 of[21],and it is the the case of θ=1.We omit it here. ?

According to Proposition 4.3,

we now employ Lemma 5.2 to have

Combining(5.3),(5.11),(5.12)and(5.13),we discover

Using the de fi nition of M(t),we conclude

Since

it is easy to imply

Since ε? 1,using continuity of M(t)and induction,we assert

Thus we complete the proof of Corollary 1.1.?