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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

一般初值的可压 NAVIER-STOKES 方程组

Cauchy 问题接触间断解的稳定性

郑婷婷

，赵俊宁

福建农林大学计算机与信息学院，福州 350002

厦门大学数学科学学院，厦门　 361005

摘要：本文研究了具有一般初值的一维可压 Navier-Stokes 方程 Cauchy 问题解的稳定性。在

某种条件下，证明了当 t → ∞ 时，解的渐进极限是粘性接触间断波。同时，证明了快速扩散

方程解的衰减速率。本文的证明方法基于基本的能量方法和快速反应扩散方程的渐进行为。

关键词：可压 Navier-Stokes 方程组，全局弱解，大时间渐进性

中图分类号： 0175

ON THE STABILITY OF CONTACT

DISCONTINUITY FOR CAUCHY PROBLEM

OF COMPRESS NAVIER-STOKES

EQUATIONS WITH GENERAL INITIAL

DATA

ZHENG Ting-Ting

, ZHAO Jun-Ning

School of Computer and Information Science, Fujian Agriculture and Forestry University,

Fuzhou 350002, P. R. China

School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China

Abstract: In this paper, we study the stability of solutions of the Cauchy problem for 1-D

compressible Narvier-Stokes equations with general initial data. The asymptotic limit of

solution is found, under some conditions. The results in this paper imply the case that the

limit function of solution as t → ∞ is a viscous contact wave, which approximates the contact

discontinuity on any ﬁnite-time interval as the heat conduction coeﬃcients toward zero. As a

by-product, the decay rates of solution for the fast diﬀusion equations are also obtained. The

proofs are based on the elementary energy method and the study of asymptotic behavior of

基金项目： This work was supported by a grant from the Ph.D. Programs Foundation of Ministry of Education of China

(200803840016)

作者简介： Correspondence author：ZHAO Jun-Ning（1945-），male，Professor，major research direction：Paritial dif-

ferential equations. ZHENG Ting-Ting（1980-），female，Assistant professor，major research direction：Paritial diﬀerential

equations.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

solution to the fast diﬀusion equation.

Key words: compressible Navier-Stokes equations, global weak solution, large-time

asymptotic behavior

0 Introduction

We study the asymptotic behavior of the solutions for the Cauchy problem to the one-

dimensional compressible Navier-Stokes equations in Lagrangian coordinates:











− u

= 0,



Rθ



= µ





γ−1

+ R

= κ





+ µ

(1)

with initial conditions

(v, u, θ)|

t=0

= (v

, u

, θ

) → (v

, 0, θ

) as x → ±∞. (2)

Here x ∈ R

, t > 0, v(x, t) > 0, u(x, t), θ(x, t) > 0 are the speciﬁc volume, velocity, temperature,

while µ > 0, κ > 0 denote the viscosity and heat conduction coeﬃcients respectively, v

, θ

are

positive constants.

In the ideal ﬂuids, i.e., κ = µ = 0, (1) become the well-known compressible Euler system:











− u

= 0,



Rθ



= 0 ,

γ−1

+ R

= 0 .

(3)

which is one of the most important nonlinear hyperbolic systems of conservation laws. It is well

known that the basic waves for the system (3) are dilation invariant solutions: shock waves,

rarefaction waves, contact discontinuities, and the linear combinations of these waves, called

Riemann solutions, which govern both local and large-time asymptotic behavior of general

solutions of Euler system (3)). Since the system (3) is an idealization when the dissipative

eﬀects are neglected, it is of great importance to study the large-time asymptotic behavior

of solutions of the corresponding viscous system, such as (1), toward the viscous versions

of these basic waves. Up to now, some deeper results have been obtained on the asymptotic

stability toward nonlinear waves, viscous shock proﬁles and viscous rarefaction, for quite general

perturbation of the Navier-Stokes system (1) and general systems of viscous strictly hyperbolic

conservation laws(see [1],[2]-[13]). However the problem of stability of contact discontinuities

is more subtle. To study the stability toward contact waves for systems of viscous conservation

- 2 -

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http://www.paper.edu.cn

laws, the concept of metastability of a weak contact discontinuity is introduced in [14], i.e.,

although a contact discontinuity is not an asymptotic attract for viscous system, there is a

viscous wave, which approximates the contact discontinuity on any ﬁnite time interval and is

the asymptotic limit of the solution of the viscous system. The problem of metastability of

contact discontinuity for the Navier-Stokes system (1) has already been well studied. For a

free-boundary value problem of (1) with a particle path as free boundary, the stability of a

viscous contact waves is proved in the supnorm (see [15, 16]). For the Cauchy problem of (1),

the stability of a viscous contact discontinuity is proved in [17], see also [18, 19].

The main purpose of this paper is to generalize the result in [18], we will give a class of

initial data, which ensures the contact discontinuity for the compressible Euler system (3) are

metastable waves patterns for the Navier-Stokes system (1), i.e., we will show that for a class

of initial value of the Cauchy problem (1), it is possible to construct a viscous contact wave

for the Navier-Sokes system, which is smooth, solves (1) asymptotically, and approximates the

given contact discontinuity on any ﬁnite-time interval as κ → 0.

We now construct the viscous contact wave for the Navier-Sokes system (1). First, consider

the corresponding Euler equations (3) with the Riemann initial data











(v, u, θ)(x, 0) = (v

−

, 0, θ

−

) if x < 0,

(v, u, θ)(x, 0) = (v

, 0, θ

) if x > 0.

(4)

Riemann problem (3)-(4) admits a contact discontinuity

(V , U, Θ) =



−

, 0, θ

−

), x < 0,

, 0, θ

), x > 0;

(5)

provided that

−

Rθ

−

= p

Rθ

In the next, as that in [15, 16, 17] we conjecture that the asymptotic limit (V, U, Θ) of (1) is as

follows

P (V, Θ) = R

= p

, U(x, t) =

κ(γ − 1)Θ

γRΘ

. (6)

and Θ is the solution of the following Cauchy problem







= a(ln Θ)

, a =

κp

(γ−1)

γR

> 0,

Θ(x, 0) = θ

(7)

- 3 -

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