### Abstract

In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform W^{3−1/r} _{r} domain of N-dimensional Euclidean space ℝ^{N} (N ⩾ 2, N < r < ∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σ_{ε,λ0} = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ_{0}} (0 < ε < π/2), and the maximal L_{p}-L_{q} regularity with the help of the R-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution p ∈ W^{1} _{q} (Ω) to the variational problem: (∇p,∇ϕ)_{Ω} = (f,∇ϕ)_{Ω} for any ϕ ∈ W^{1} _{q′}(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W^{1} _{q} (Ω) is a closed subspace of Ŵ^{1} _{q,Γ} (Ω) = {p ∈ L_{q,loc}(Ω) | ∇p ∈ L_{q}(Ω)^{N}, p|_{Γ} = 0} with respect to gradient norm ∥∇ · ∥_{Lq(} _{Ω)} that contains a space W^{1} _{q,} _{Γ} (Ω) = {p ∈ W^{1} _{q} (Ω) | p|_{Γ} = 0}, and Γ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (λ_{0},∞), which was proved in Shibata [13]. Our assumption is satisfied for any q ∈ (1,∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W^{1} _{q} (Ω) = Ŵ^{1} _{q,Γ} (Ω).

Original language | English |
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Title of host publication | Mathematical Fluid Dynamics, Present and Future |

Publisher | Springer New York LLC |

Pages | 203-285 |

Number of pages | 83 |

Volume | 183 |

ISBN (Print) | 9784431564553 |

DOIs | |

Publication status | Published - 2016 |

Event | 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 - Tokyo, Japan Duration: 2014 Nov 11 → 2014 Nov 14 |

### Other

Other | 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 |
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Country | Japan |

City | Tokyo |

Period | 14/11/11 → 14/11/14 |

### Keywords

- Analytic semigroup
- Free boundary condition
- Maximal L-L regularity
- R-Boundedness
- Stokes equations
- Surface tension
- UniformW domain

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

_{p}-l

_{q}regularity of the stokes equations with free boundary condition. In

*Mathematical Fluid Dynamics, Present and Future*(Vol. 183, pp. 203-285). Springer New York LLC. https://doi.org/10.1007/978-4-431-56457-7_9