Local well-posedness of free surface problems for the navier-stokes equations in a general domain

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    Abstract

    In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain RN (N 2). The velocity field is obtained in the maximal regularity class W2;1 q;p (0 T)) = Lp((0; T);Wq ()N) W1 p ((0; T);Lq()N) (2<1 and N <q <1) for any initial data satisfying certain compatibility conditions. The assumption of the domain is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of. A bounded domain, a perturbed half space, and a perturbed layer satisfy the conditions for the domain, and therefore drop problems and ocean problems are treated in the uniform manner. Our method is based on the maximal Lp-Lq regularity theorem of a linearized problem in a general domain.

    Original languageEnglish
    Pages (from-to)315-342
    Number of pages28
    JournalDiscrete and Continuous Dynamical Systems - Series S
    Volume9
    Issue number1
    DOIs
    Publication statusPublished - 2016 Feb 1

    Keywords

    • Free boundary problems
    • Gravity force
    • Local well-posedness
    • Navier-Stokes equations
    • Surface tension

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics
    • Discrete Mathematics and Combinatorics

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