### Abstract

In this paper, we prove a local energy decay of the Oseen semigroup in the n-dimensional exterior domain (n ≥ 3). In the three dimensional exterior domain case, Kobayashi and Shibata [16] already proved the local energy decay. Our theorem is not only an extension of the results due to Kobayashi and Shibata to the n-dimensional case but also the complete study of the local energy decay theorem for the Oseen equation with optimal time decay rates. The local energy decay gives us a crucial step to obtain the L_{p}-L _{q} estimates of the Oseen semigroup, which enable us to prove the unique existence of globally in time solutions to the Navier-Stokes equation in an exterior domain with small initial data in the L_{n} framework, and their properties of time decay.

Original language | English |
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Pages (from-to) | 1291-1330 |

Number of pages | 40 |

Journal | Indiana University Mathematics Journal |

Volume | 53 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2004 |

Externally published | Yes |

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### Keywords

- Exterior domain
- Local energy decay
- Oseen equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*53*(5), 1291-1330. https://doi.org/10.1512/iumj.2004.53.2463