Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space

Shuichi Kawashima, Peicheng Zhu

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L2-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are "good" and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.

Original languageEnglish
Pages (from-to)3151-3179
Number of pages29
JournalJournal of Differential Equations
Volume244
Issue number12
DOIs
Publication statusPublished - 2008 Jun 15
Externally publishedYes

Fingerprint

Compressible Navier-Stokes Equations
Nonlinear Waves
Asymptotic stability
Asymptotic Stability
Half-space
Navier Stokes equations
Rarefaction Wave
Term
Large Time Behavior
Energy Method
Discrete Equations
A Priori Estimates
Stationary Solutions
Boltzmann Equation
Small Perturbations
Initial-boundary-value Problem
Superposition
Boltzmann equation
Decay
Boundary value problems

Keywords

  • Asymptotic stability
  • Compressible Navier-Stokes equations
  • Eulerian coordinate
  • Superposition of a rarefaction wave and a stationary solution

ASJC Scopus subject areas

  • Analysis

Cite this

Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space. / Kawashima, Shuichi; Zhu, Peicheng.

In: Journal of Differential Equations, Vol. 244, No. 12, 15.06.2008, p. 3151-3179.

Research output: Contribution to journalArticle

@article{dd458d5d77e54eb49ae9b27ab0c58e8f,
title = "Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space",
abstract = "In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L2-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are {"}good{"} and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.",
keywords = "Asymptotic stability, Compressible Navier-Stokes equations, Eulerian coordinate, Superposition of a rarefaction wave and a stationary solution",
author = "Shuichi Kawashima and Peicheng Zhu",
year = "2008",
month = "6",
day = "15",
doi = "10.1016/j.jde.2008.01.020",
language = "English",
volume = "244",
pages = "3151--3179",
journal = "Journal of Differential Equations",
issn = "0022-0396",
publisher = "Academic Press Inc.",
number = "12",

}

TY - JOUR

T1 - Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space

AU - Kawashima, Shuichi

AU - Zhu, Peicheng

PY - 2008/6/15

Y1 - 2008/6/15

N2 - In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L2-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are "good" and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.

AB - In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L2-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are "good" and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.

KW - Asymptotic stability

KW - Compressible Navier-Stokes equations

KW - Eulerian coordinate

KW - Superposition of a rarefaction wave and a stationary solution

UR - http://www.scopus.com/inward/record.url?scp=42649117067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42649117067&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2008.01.020

DO - 10.1016/j.jde.2008.01.020

M3 - Article

AN - SCOPUS:42649117067

VL - 244

SP - 3151

EP - 3179

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -