### Abstract

We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the L _{q} framework (1 < q < ∞) in a domain of R^{n} (n ≧ 2) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: v · u = 0 and au + β(T(u, p)v -(T(u, p)u, v)v) = h on the boundary of the domain with α, β ≧ 0 and α + β= 1, where u denotes a velocity vector, p a pressure, T(u, p) the stress tensor for the Stokes flow, and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and the non-slip one when α = 1, respectively.

Original language | English |
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Pages (from-to) | 469-519 |

Number of pages | 51 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 59 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 Apr |

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### Keywords

- Resolvent estimate
- Robin boundary condition
- Stokes system

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On a generalized resolvent estimate for the Stokes system with Robin boundary condition.** / Shibata, Yoshihiro; Shimada, Rieko.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 59, no. 2, pp. 469-519. https://doi.org/10.2969/jmsj/05920469

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TY - JOUR

T1 - On a generalized resolvent estimate for the Stokes system with Robin boundary condition

AU - Shibata, Yoshihiro

AU - Shimada, Rieko

PY - 2007/4

Y1 - 2007/4

N2 - We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the L q framework (1 < q < ∞) in a domain of Rn (n ≧ 2) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: v · u = 0 and au + β(T(u, p)v -(T(u, p)u, v)v) = h on the boundary of the domain with α, β ≧ 0 and α + β= 1, where u denotes a velocity vector, p a pressure, T(u, p) the stress tensor for the Stokes flow, and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and the non-slip one when α = 1, respectively.

AB - We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the L q framework (1 < q < ∞) in a domain of Rn (n ≧ 2) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: v · u = 0 and au + β(T(u, p)v -(T(u, p)u, v)v) = h on the boundary of the domain with α, β ≧ 0 and α + β= 1, where u denotes a velocity vector, p a pressure, T(u, p) the stress tensor for the Stokes flow, and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and the non-slip one when α = 1, respectively.

KW - Resolvent estimate

KW - Robin boundary condition

KW - Stokes system

UR - http://www.scopus.com/inward/record.url?scp=34547487815&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547487815&partnerID=8YFLogxK

U2 - 10.2969/jmsj/05920469

DO - 10.2969/jmsj/05920469

M3 - Article

AN - SCOPUS:34547487815

VL - 59

SP - 469

EP - 519

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 2

ER -