On the r-bounded solution operator and the maximal lp-lq regularity of the stokes equations with free boundary condition

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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 W3−1/rr domain of N-dimensional Euclidean space ℝN (N ⩾ 2, N < r < ∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σε,λ0 = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ0} (0 < ε < π/2), and the maximal Lp-Lq 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 ∈ W1q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1q′(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W1q (Ω) is a closed subspace of Ŵ1q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq(Ω) that contains a space W1q,Γ (Ω) = {p ∈ W1q (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 (λ0,∞), 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 W1q (Ω) = Ŵ1q,Γ (Ω).

Original languageEnglish
Title of host publicationMathematical Fluid Dynamics, Present and Future
EditorsYoshihiro Shibata, Yukihito Suzuki
PublisherSpringer New York LLC
Pages203-285
Number of pages83
ISBN (Print)9784431564553
DOIs
Publication statusPublished - 2016
Event8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 - Tokyo, Japan
Duration: 2014 Nov 112014 Nov 14

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume183
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

Other8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014
CountryJapan
CityTokyo
Period14/11/1114/11/14

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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