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

研究成果: Conference contribution

6 被引用数 (Scopus)

抄録

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).

本文言語English
ホスト出版物のタイトルMathematical Fluid Dynamics, Present and Future
編集者Yoshihiro Shibata, Yukihito Suzuki
出版社Springer New York LLC
ページ203-285
ページ数83
ISBN(印刷版)9784431564553
DOI
出版ステータスPublished - 2016
イベント8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 - Tokyo, Japan
継続期間: 2014 11 112014 11 14

出版物シリーズ

名前Springer Proceedings in Mathematics and Statistics
183
ISSN(印刷版)2194-1009
ISSN(電子版)2194-1017

Other

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

ASJC Scopus subject areas

  • Mathematics(all)

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