### Abstract

The main concern of this paper is to study large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half-space. We consider an outflow problem and obtain a convergence rate of solutions toward a corresponding stationary solution. Here the existence of the stationary solution is proved under a smallness condition on the boundary data with the aid of center manifold theory. We also show the time asymptotic stability of the stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. The proof is based on deriving a priori estimates by using a time and space weighted energy method.

Original language | English |
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Pages (from-to) | 2201-2235 |

Number of pages | 35 |

Journal | Mathematical Models and Methods in Applied Sciences |

Volume | 20 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2010 Dec 1 |

Externally published | Yes |

### Keywords

- Compressible Navier-Stokes equation
- Eulerian coordinate
- boundary layer solution
- ideal polytropic model
- outflow problem
- weighted energy method

### ASJC Scopus subject areas

- Modelling and Simulation
- Applied Mathematics

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## Cite this

*Mathematical Models and Methods in Applied Sciences*,

*20*(12), 2201-2235. https://doi.org/10.1142/S0218202510004908