### 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 |
---|---|

Pages (from-to) | 1291-1330 |

Number of pages | 40 |

Journal | Indiana University Mathematics Journal |

Volume | 53 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2004 |

### Fingerprint

### Keywords

- Exterior domain
- Local energy decay
- Oseen equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Local energy decay of solutions to the oseen equation in the exterior domains.** / Enomoto, Yuko; Shibata, Yoshihiro.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 53, no. 5, pp. 1291-1330. https://doi.org/10.1512/iumj.2004.53.2463

}

TY - JOUR

T1 - Local energy decay of solutions to the oseen equation in the exterior domains

AU - Enomoto, Yuko

AU - Shibata, Yoshihiro

PY - 2004

Y1 - 2004

N2 - 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 Lp-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 Ln framework, and their properties of time decay.

AB - 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 Lp-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 Ln framework, and their properties of time decay.

KW - Exterior domain

KW - Local energy decay

KW - Oseen equation

UR - http://www.scopus.com/inward/record.url?scp=11144227016&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=11144227016&partnerID=8YFLogxK

U2 - 10.1512/iumj.2004.53.2463

DO - 10.1512/iumj.2004.53.2463

M3 - Article

AN - SCOPUS:11144227016

VL - 53

SP - 1291

EP - 1330

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 5

ER -