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Nonlinear Stability for the Superposition of Viscous Contact Wave and Rarefaction Waves to Non-isentropic Compressible Navier-Stokes System with General Initial Perturbations

Yi Peng, Xiaoding Shi, Yuhang Wu

Abstract

In this paper, the large time behavior of the solutions for the Cauchy problem to the one-dimensional compressible Navier-Stokes system with the motion of a viscous heat-conducting perfect polytropic gas is investigated.Our result shows that the combination of a viscous contact wave with rarefaction waves is asymptotically stable, when the large initial disturbance of the density, velocity and temperature belong to $H^{1}(\mathbb{R})$, $L^{2}(\mathbb{R})\cap L^{4}(\mathbb{R})$ and $L^{2}(\mathbb{R})$, provided the strength of the combination waves is suitably small. In addition, the initial disturbance on the derivation of velocity and temperature belong to $L^{2}(\mathbb{R})$ can be arbitrarily large.

Nonlinear Stability for the Superposition of Viscous Contact Wave and Rarefaction Waves to Non-isentropic Compressible Navier-Stokes System with General Initial Perturbations

Abstract

In this paper, the large time behavior of the solutions for the Cauchy problem to the one-dimensional compressible Navier-Stokes system with the motion of a viscous heat-conducting perfect polytropic gas is investigated.Our result shows that the combination of a viscous contact wave with rarefaction waves is asymptotically stable, when the large initial disturbance of the density, velocity and temperature belong to , and , provided the strength of the combination waves is suitably small. In addition, the initial disturbance on the derivation of velocity and temperature belong to can be arbitrarily large.
Paper Structure (4 sections, 15 theorems, 199 equations)

This paper contains 4 sections, 15 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem 1.1
  4. Proof of Theorem 1.2

Key Result

Theorem 1.1

(Viscous contact wave). Under the assumption of the initial condition 98, for any given left end state $(v_{-},u_{-},\theta_{-})$, suppose that the right end state $(v_{+},u_{+},\theta_{+})$ satisfies 71. Let $(V,U,\Theta)$ be the viscous contact wave defined in 18 with strength $\delta=\mid \theta_ such that if $\delta<\delta_{0}$, the Cauchy problem 01--03, 23 admitting a unique global solution

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 9 more