### 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 L^{p}-norm for the smoothed rarefaction wave. We then employ the L^{2}-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity.

Original language | English |
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Pages (from-to) | 105-132 |

Number of pages | 28 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 194 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Aug 1 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Asymptotic stability of rarefaction wave for the navier-stokes equations for a compressible fluid in the half space

AU - Kawashima, Shuichi

AU - Zhu, Peicheng

PY - 2009/8/1

Y1 - 2009/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=70350774241&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350774241&partnerID=8YFLogxK

U2 - 10.1007/s00205-008-0191-8

DO - 10.1007/s00205-008-0191-8

M3 - Article

AN - SCOPUS:70350774241

VL - 194

SP - 105

EP - 132

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 1

ER -