Shifeng GENG Yanjuan TANG

School of Mathematics and Computational Science,Xiangtan University,Xiangtan 411105,China

E-mail:sfgeng@xtu.edu.cn;2393908820@qq.com

Abstract This article is involved with the asymptotic behavior of solutions for nonlinear hyperbolic system with external friction.The global existence of classical solutions is proven,and Lpconvergence rates are obtained.Compared with the results obtained by Hsiao and Liu,better convergence rates are obtained in this article.

Key words convergence rates;nonlinear di ff usion waves;nonlinear hyperbolic system;ex

1 Introduction

We consider the following nonlinear hyperbolic system

which describes the motion of isentropic gas fl ow with external friction acting on it,such as fl ow through porous media.Here,ρ is the density,u denotes velocity,and p(ρ)stands for the pressure with p′(ρ)>0,for ρ >0;m= ρu is the momentum and the friction constant α is a positive constant.Without loss of generality,α is normalized to be 1 throughout this article.

In Hsiao and Liu[5],it was proved that the solution of Cauchy problem(1.1)with

can be described time asymptotically by the self-similar solution of the following equations

or

and the authors obtained the L∞convergence rates as

In the Lagrange coordinate,system(1.1)can be transferred to the p?system with damping as follows

its corresponding nonlinear parabolic equation is

Hsiao and Liu[4]proved that the solution to the Cauchy problem(1.6)converges to that of(1.7)with a rate as k(v??v,u??u)(t)kL∞=O(t?1/2,t?1/2).Then,by taking more detailed but elegant energy estimates,Nishihara[13]succeeded in improving the convergence rates as k(v??v,u??u)(t)kL∞=O(t?3/4,t?5/4),when the initial perturbation is in H3.Furthermore,when the initial perturbation is in H3∩L1,by constructing an approximate Green function with the energy method together,Nishihara,Wang,and Yang[15]completely improved the rates as k(v??v,u??u)(t)kL∞=O(t?1,t?3/2),which are optimal in the sense comparing with the decay of the solution to the heat equation.Zhao[16]obtained the optimal Lpconvergence rates for strong di ff usive waves and large initial data,where only the initial oscillation is required to be small.By suitably choosing initial data of(1.7)and making heuristic analysis,Mei[11]realized that the best asymptotic pro fi le of the damped p-system(1.6)is a particular solution(?v,?u)(x,t)to the corresponding nonlinear parabolic equations(1.7),and the author further derived the convergence rates which are much better than the rates obtained in the previous works.He,Huang and Yong[3]obtained the stability of planar di ff usion wave for nonlinear evolution equation.For other related to p-system we refer to[1,2,6–8,14,17]and references therein.

For comparing the solution of(1.1)and(1.3),we use variables(ρ,m)instead of(ρ,u),where m=ρu,in which the problem(1.1),(1.2)and(1.3)become

or

As in[5],the following auxiliary functions are introduced as follows

where?ρ(x)is a smooth function with compact support and satis fi es

Under the above notations,if we let

where x0is uniquely determined by

then from(1.8),(1.9),(1.11)–(1.13),we can deduce that(y(x,t),z(x,t))solves the following Cauchy problem

Theorem 1.1Assume that(y0,z0) ∈ H3(R)× H2(R)and δ:=|ρ+? ρ?|? 1.Then,there exists a δ0>0 such that if ky0k3+kz0k2+ δ≤ δ0,the Cauchy problem(1.8)admits a unique global smooth solution(ρ,m)which satis fi es

Furthermore,under the additional assumption that(y0,z0)∈L1(R)×L1(R),the following Lp(2≤p≤∞)convergence rates are true

for any k≤2 if p=2 and k≤1 if p∈(2,∞].

This theorem is proved by energy method.In comparison with those arguments developed by Hsiao,Liu[5]and Nishihara[13],more careful a-priori estimates should be done in order to control the convection terms.

Remark 1.2Our result in(1.17)shows that the convergence rates are sharper than(1.5)given by Hsiao and Liu in[5].

For convenience,we only give the proof for the case when u+=u?=0 in which?ρ≡0,?m≡0 and(1.14)becomes

The general case can be treated in a similar way because the?ρ(x,t)and?m(x,t)decay to zero exponentially fast.

The rest of this article is organized as follows.In Section 2,we prove the convergence of the solution(ρ,m)for the Cauchy problem(1.8)to the solution of problem(1.9).The Lpconvergence rates of the solution of problem(1.8)are established in Section 3.

NotationsThroughout this article the symbol C or O(1)will be used to represent a generic constant which is independent of x and t and may vary from line to line.k·kLpand k·klstand for the Lp(R)-norm(1≤ p≤ ∞)and Hl(R)-norm.The L2-norm on R is simply denoted by k·k.Moreover,the domain R will be often abbreviated without confusions.

2 Global Existence and the Asymptotic of Solutions

In this section,we will obtain the existence of the global solution for problem(1.16).Consider the Cauchy proble m

where

Then we have the following theorem.

Theorem 2.1Under the assumptions in Theorem 1.1,there exists a unique time-global solution y(x,t)of the Cauchy problem(2.1)such that

Lemma 2.2(see[5]) For the self-similar solution of(1.10),it holds

The local existence of smooth solutions to the problem(2.1)can be obtained by the standard iteration method(cf.[9,12]).Thus,we only need to derive the a priori estimate under the following a priori assumption.Let T∈(0,+∞],we de fi ne

Let N(T)≤ ε,where ε is sufficiently small and will be determined later,then,it holds that

Hereafter the constant p?and p?are given by

Substituting(2.8)and(2.9)into(2.7),we have

Multiplying(2.1)by ytand integrating over R,we then have

Due to the fact

Integrating it over(0,t),we obtain(2.5).Similarly,using(2.5)and multiplying(2.15)by(1+t),we can have(2.6).Thus,the proof of Lemma 2.3 is completed. ?

Multiplying(2.18)by yxtand integrating it by parts over R,we get

Multiplying(2.28)by(1+t)and integrating it over(0,t)we then have(2.16).Moreover,by virtue of(2.16)and Lemma 2.3,multiplication of(2.27)by(1+t)2gives(2.17).This completes the proof of Lemma 2.4. ?

Similar to the proof of Lemma 2.4,we can have the following estimate.

In terms of Lemmas 2.3–2.5,applying the local existence result and the continuity,we can prove that for ky0k3+kz0k2+δ≤ δ0,there is the unique time-global solution of Cauchy problem(2.1).Furthermore,by considering((2.1)t)×ytt,((2.1))t×yt,((2.1)xt)×yxtt,((2.1))xt×yxt,((2.1))tt×yttt,respectively,integrating over R,multiplying with(1+t)kfor some integer k and integrating over[0,t],we can obtain estimates(2.3)and(2.4).Thus,the proof of Theorem 2.1 is completed.

3 Lp-Convergence Rates

In this section,we will obtain the Lpdecay estimates for the smooth solution of the problem(1.8).As in[15],one can rewrite(2.1)as

By employing the decay estimates(2.3)and(2.4)obtained in Theorem 2.1,we have by mimicing the arguments developed by Nishihara,Wang,and Yang in[15]that

for 0≤i≤5.Consequently

and(1.17)follows from(3.2)and Sobolev’s inequality.This completes the proof of Theorem 1.1.