## Abstract

In this paper, we proved the generalized resolvent estimate and the maximal L _{p}-L _{q} regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector Σ _{∈,γ0} = {λ ∈ C \ {0} | | arg λ| ≤ π - ∈, |λ| > γ _{o}} with 0 < ∈ < π/2 and γ _{o} ≥0. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal L _{p}-L _{q} regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal L _{p} regularity implies that the resolvent estimate of A in λ ∈ Σ _{∈,γ0}, but the opposite direction is not true in general (cf. Kalton and Lancien[19]). However, in this paper using the R boundedness of the operator family in the sector Σ _{∈ λ0}, we derive a systematic way to prove the resolvent estimate and the maximal L _{p} regularity at the same time.

Original language | English |
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Pages (from-to) | 561-626 |

Number of pages | 66 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 64 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 |

Externally published | Yes |

## Keywords

- Gravity force
- Half space problem
- Maximal regularity
- Resolvent estimate
- Stokes equation
- Surface tension

## ASJC Scopus subject areas

- Mathematics(all)

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