### Abstract

We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier-Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L^{2}-energy method with the aid of the Poincaré type inequality.

Original language | English |
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Pages (from-to) | 483-500 |

Number of pages | 18 |

Journal | Communications in Mathematical Physics |

Volume | 240 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2003 Jan 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*240*(3), 483-500. https://doi.org/10.1007/s00220-003-0909-2

**Asymptotic Stability of the Stationary Solution to the Compressible Navier-Stokes Equations in the Half Space.** / Kawashima, Shuichi; Nishibata, Shinya; Zhu, Peicheng.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 240, no. 3, pp. 483-500. https://doi.org/10.1007/s00220-003-0909-2

}

TY - JOUR

T1 - Asymptotic Stability of the Stationary Solution to the Compressible Navier-Stokes Equations in the Half Space

AU - Kawashima, Shuichi

AU - Nishibata, Shinya

AU - Zhu, Peicheng

PY - 2003/1/1

Y1 - 2003/1/1

N2 - We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier-Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L2-energy method with the aid of the Poincaré type inequality.

AB - We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier-Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L2-energy method with the aid of the Poincaré type inequality.

UR - http://www.scopus.com/inward/record.url?scp=0141792302&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141792302&partnerID=8YFLogxK

U2 - 10.1007/s00220-003-0909-2

DO - 10.1007/s00220-003-0909-2

M3 - Article

VL - 240

SP - 483

EP - 500

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -