Asymptotic stability of rarefaction wave for the navier-stokes equations for a compressible fluid in the half space

Shuichi Kawashima, Peicheng Zhu

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers' equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates in Lp-norm for the smoothed rarefaction wave. We then employ the L2-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity.

Original languageEnglish
Pages (from-to)105-132
Number of pages28
JournalArchive for Rational Mechanics and Analysis
Volume194
Issue number1
DOIs
Publication statusPublished - 2009 Aug 1
Externally publishedYes

Fingerprint

Rarefaction Wave
Compressible Fluid
Asymptotic stability
Asymptotic Stability
Half-space
Navier Stokes equations
Navier-Stokes Equations
Fluids
Smooth Approximation
Decay Estimates
Lp-norm
Energy Method
Burgers Equation
Small Perturbations
Initial-boundary-value Problem
Boundary value problems
State Space
Classify
Infinity
Boundary conditions

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

Cite this

Asymptotic stability of rarefaction wave for the navier-stokes equations for a compressible fluid in the half space. / Kawashima, Shuichi; Zhu, Peicheng.

In: Archive for Rational Mechanics and Analysis, Vol. 194, No. 1, 01.08.2009, p. 105-132.

Research output: Contribution to journalArticle

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