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Some Uniform Estimates and Large-Time Behavior for One-Dimensional Compressible Navier-Stokes System in Unbounded Domains with Large Data

Jing Li, Zhilei Liang

Abstract

This paper is concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above independently of both time and space. Moreover, the global solution is showed to be asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.

Some Uniform Estimates and Large-Time Behavior for One-Dimensional Compressible Navier-Stokes System in Unbounded Domains with Large Data

Abstract

This paper is concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above independently of both time and space. Moreover, the global solution is showed to be asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.

Paper Structure

This paper contains 1 section, 3 theorems, 14 equations.

Table of Contents

  1. Introduction

Key Result

Lemma 1.1

Assume that the initial data $(v_0,u_0,\theta_0)$ satisfy and are compatible with 1.6, 1.7. Then there exists a unique global (large) generalized solution $(v ,u ,\theta )$ with positive $v(x, t)$ and $\theta(x, t)$ to 1.1-1.5, or 1.1-1.41.6, or 1.1-1.41.7 satisfying that for any $T>0,$

Theorems & Definitions (3)

  • Lemma 1.1
  • Lemma 1.2: ji1ji2
  • Theorem \oldthetheorem