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

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

### Cite this

**On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain.** / Enomoto, Yuko; Shibata, Yoshihiro.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 252, no. 12, pp. 6214-6249. https://doi.org/10.1016/j.jde.2012.03.003

}

TY - JOUR

T1 - On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain

AU - Enomoto, Yuko

AU - Shibata, Yoshihiro

PY - 2012/6/15

Y1 - 2012/6/15

N2 - 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.

AB - 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.

KW - 2-dimensional exterior domain

KW - Compressible viscous fluid

KW - L -L decay

KW - Local energy decay

KW - Stokes semigroup

UR - http://www.scopus.com/inward/record.url?scp=84859716070&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859716070&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2012.03.003

DO - 10.1016/j.jde.2012.03.003

M3 - Article

VL - 252

SP - 6214

EP - 6249

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -