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/r r 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 ∈ W1 q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1 q′(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W1 q (Ω) is a closed subspace of Ŵ1 q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq( Ω) that contains a space W1 q, Γ (Ω) = {p ∈ W1 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 (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 W1 q (Ω) = Ŵ1 q,Γ (Ω).

    Original languageEnglish
    Title of host publicationMathematical Fluid Dynamics, Present and Future
    PublisherSpringer New York LLC
    Pages203-285
    Number of pages83
    Volume183
    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

    Other

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

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    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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