Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface

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    2 Citations (Scopus)

    Abstract

    In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the Lp in time Lq in space maximal regularity class, (2<p<∞, N<q<∞, and 2/p+N/q<1), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the Lp-Lq framework for the linearized equations with the help of maximal Lp-Lq regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

    Original languageEnglish
    Pages (from-to)117-152
    Number of pages36
    JournalEvolution Equations and Control Theory
    Volume7
    Issue number1
    DOIs
    Publication statusPublished - 2018 Jan 1

    Fingerprint

    Stokes Operator
    Regularity Theory
    Maximal Regularity
    Laplace-Beltrami Operator
    Global Well-posedness
    Stokes Equations
    Free Boundary Problem
    Exponential Decay
    Spectral Analysis
    Free Boundary
    Surface Tension
    Free Surface
    Spectrum analysis
    Velocity Field
    Navier Stokes equations
    Surface tension
    Mathematical operators
    Bounded Domain
    Navier-Stokes Equations
    Ball

    Keywords

    • Free boundary problems
    • Global well-posedness
    • Navier-stokes equations
    • Surface tension

    ASJC Scopus subject areas

    • Modelling and Simulation
    • Control and Optimization
    • Applied Mathematics

    Cite this

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    abstract = "In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the Lp in time Lq in space maximal regularity class, (2p-Lq framework for the linearized equations with the help of maximal Lp-Lq regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.",
    keywords = "Free boundary problems, Global well-posedness, Navier-stokes equations, Surface tension",
    author = "Yoshihiro Shibata",
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    AB - In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the Lp in time Lq in space maximal regularity class, (2p-Lq framework for the linearized equations with the help of maximal Lp-Lq regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

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