### Abstract

We discuss L^{p}-L^{q} type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed halfspace. Especially, we have the L^{p}-L^{q} type estimate of the gradient of the Stokes semigroup for any p and q with 1 < p ≤ q < ∞, while the same estimate holds only for the exponents p and q with 1 < p ≤ q ≤ n in the exterior domain case, where n denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case. Our proof of the L^{p}-L^{q} type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.

Original language | English |
---|---|

Pages (from-to) | 695-720 |

Number of pages | 26 |

Journal | Advances in Differential Equations |

Volume | 10 |

Issue number | 6 |

Publication status | Published - 2005 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Advances in Differential Equations*,

*10*(6), 695-720.

**On the stokes and Navier-Stokes equations in a perturbed half-space.** / Kubo, Takayuki; Shibata, Yoshihiro.

Research output: Contribution to journal › Article

*Advances in Differential Equations*, vol. 10, no. 6, pp. 695-720.

}

TY - JOUR

T1 - On the stokes and Navier-Stokes equations in a perturbed half-space

AU - Kubo, Takayuki

AU - Shibata, Yoshihiro

PY - 2005

Y1 - 2005

N2 - We discuss Lp-Lq type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed halfspace. Especially, we have the Lp-Lq type estimate of the gradient of the Stokes semigroup for any p and q with 1 < p ≤ q < ∞, while the same estimate holds only for the exponents p and q with 1 < p ≤ q ≤ n in the exterior domain case, where n denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case. Our proof of the Lp-Lq type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.

AB - We discuss Lp-Lq type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed halfspace. Especially, we have the Lp-Lq type estimate of the gradient of the Stokes semigroup for any p and q with 1 < p ≤ q < ∞, while the same estimate holds only for the exponents p and q with 1 < p ≤ q ≤ n in the exterior domain case, where n denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case. Our proof of the Lp-Lq type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.

UR - http://www.scopus.com/inward/record.url?scp=34248539352&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34248539352&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34248539352

VL - 10

SP - 695

EP - 720

JO - Advances in Differential Equations

JF - Advances in Differential Equations

SN - 1079-9389

IS - 6

ER -