### Abstract

In this paper, we proved the local energy decay and some L _{p}-L _{q} decay properties of solutions to the initial-boundary value problem for the Stokes equations of compressible viscous fluid flow in a 2-dimensional exterior domain. Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] treated the same problem in a 3-dimensional exterior domain, and our results obtained in this paper are an extension of results in Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] to the 2-dimensional case. The fundamental solution to the resolvent equations for the Stokes equations in ℝ ^{2} has a logarithmical singularity at λ=0, λ being a resolvent parameter, while it is continuous up to λ=0 in ℝ ^{3}. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required L _{p}-L _{q} decay estimates in the exterior domain are obtained by combining the local energy estimate and the L _{p}-L _{q} estimates in ℝ ^{2} by a cut-off technique.

Original language | English |
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Pages (from-to) | 6214-6249 |

Number of pages | 36 |

Journal | Journal of Differential Equations |

Volume | 252 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2012 Jun 15 |

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### Keywords

- 2-dimensional exterior domain
- Compressible viscous fluid
- L -L decay
- Local energy decay
- Stokes semigroup

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics