On the r-boundedness of solution operators for the stokes equations with free boundary condition

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    Abstract

    In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform Wr 2-11/r domain of N-dimensional Euclidean space ℝN (N ≥ 2, N < r < ∞). We prove the R-boundedness of solution operators with spectral parameter λ varying in a sector Σε,λ0 = {λ ε C: |arg λ| ≤ π - ε, |λ| ≥ λ0}, from which we can deduce the Lp-Lq maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. 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 ∈ Wq 1(Ω) to the variational problem:(p, φ)Ω = (f, φ)n for any φ ∈ W q 1 (Ω) with 1 < q < ∞ and q′ = q/(q - 1), where Wq 1(Ω) is a closed subspace of Ŵ 1(Ω) = {p ∈ L q,loc(Ω): p ∈ Lq(Ω) N, p|λ = 0} with respect to gradient norm || · || Lq(Ω) that contains a space Wq 1(Ω) = {p ∈ Wq 1(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 [28]. Our assumption is satisfied for any q ∈ (l, ∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with Wq 1(Ω) = Ŵq 1(Ω), and by exterior domains with W q 1(Ω) = the closure of W 1(Ω) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform Wr 2-1r domain.

    Original languageEnglish
    Pages (from-to)313-368
    Number of pages56
    JournalDifferential and Integral Equations
    Volume27
    Issue number3-4
    Publication statusPublished - 2014

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    Boundedness of Solutions
    Stokes Equations
    Free Boundary
    Mathematical operators
    Neumann Problem
    Boundary conditions
    Dirichlet Problem
    Unique Solvability
    Operator
    Half-space
    Navier Stokes equations
    Boundary value problems
    Surface tension
    Flow of fluids
    Existence of Solutions
    R-boundedness
    Viscosity
    Gradient
    Stokes Operator
    Variable Viscosity

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

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    title = "On the r-boundedness of solution operators for the stokes equations with free boundary condition",
    abstract = "In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform Wr 2-11/r domain of N-dimensional Euclidean space ℝN (N ≥ 2, N < r < ∞). We prove the R-boundedness of solution operators with spectral parameter λ varying in a sector Σε,λ0 = {λ ε C: |arg λ| ≤ π - ε, |λ| ≥ λ0}, from which we can deduce the Lp-Lq maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. 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 ∈ Wq 1(Ω) to the variational problem:(p, φ)Ω = (f, φ)n for any φ ∈ W q 1 (Ω) with 1 < q < ∞ and q′ = q/(q - 1), where Wq 1(Ω) is a closed subspace of ŴqΓ 1(Ω) = {p ∈ L q,loc(Ω): p ∈ Lq(Ω) N, p|λ = 0} with respect to gradient norm || · || Lq(Ω) that contains a space Wq 1(Ω) = {p ∈ Wq 1(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 [28]. Our assumption is satisfied for any q ∈ (l, ∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with Wq 1(Ω) = Ŵq 1(Ω), and by exterior domains with W q 1(Ω) = the closure of WqΓ 1(Ω) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform Wr 2-1r domain.",
    author = "Yoshihiro Shibata",
    year = "2014",
    language = "English",
    volume = "27",
    pages = "313--368",
    journal = "Differential and Integral Equations",
    issn = "0893-4983",
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    TY - JOUR

    T1 - On the r-boundedness of solution operators for the stokes equations with free boundary condition

    AU - Shibata, Yoshihiro

    PY - 2014

    Y1 - 2014

    N2 - In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform Wr 2-11/r domain of N-dimensional Euclidean space ℝN (N ≥ 2, N < r < ∞). We prove the R-boundedness of solution operators with spectral parameter λ varying in a sector Σε,λ0 = {λ ε C: |arg λ| ≤ π - ε, |λ| ≥ λ0}, from which we can deduce the Lp-Lq maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. 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 ∈ Wq 1(Ω) to the variational problem:(p, φ)Ω = (f, φ)n for any φ ∈ W q 1 (Ω) with 1 < q < ∞ and q′ = q/(q - 1), where Wq 1(Ω) is a closed subspace of ŴqΓ 1(Ω) = {p ∈ L q,loc(Ω): p ∈ Lq(Ω) N, p|λ = 0} with respect to gradient norm || · || Lq(Ω) that contains a space Wq 1(Ω) = {p ∈ Wq 1(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 [28]. Our assumption is satisfied for any q ∈ (l, ∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with Wq 1(Ω) = Ŵq 1(Ω), and by exterior domains with W q 1(Ω) = the closure of WqΓ 1(Ω) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform Wr 2-1r domain.

    AB - In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform Wr 2-11/r domain of N-dimensional Euclidean space ℝN (N ≥ 2, N < r < ∞). We prove the R-boundedness of solution operators with spectral parameter λ varying in a sector Σε,λ0 = {λ ε C: |arg λ| ≤ π - ε, |λ| ≥ λ0}, from which we can deduce the Lp-Lq maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. 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 ∈ Wq 1(Ω) to the variational problem:(p, φ)Ω = (f, φ)n for any φ ∈ W q 1 (Ω) with 1 < q < ∞ and q′ = q/(q - 1), where Wq 1(Ω) is a closed subspace of ŴqΓ 1(Ω) = {p ∈ L q,loc(Ω): p ∈ Lq(Ω) N, p|λ = 0} with respect to gradient norm || · || Lq(Ω) that contains a space Wq 1(Ω) = {p ∈ Wq 1(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 [28]. Our assumption is satisfied for any q ∈ (l, ∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with Wq 1(Ω) = Ŵq 1(Ω), and by exterior domains with W q 1(Ω) = the closure of WqΓ 1(Ω) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform Wr 2-1r domain.

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    JO - Differential and Integral Equations

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