On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

Takayuki Kubo, Yoshihiro Shibata, Kohei Soga

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    Abstract

    In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the Lp in time and the Lq in space framework with 2 <p <1 and N <q <∞ under the assumption that the initial domain is a uniform Wq 2-1/q domain in ℝN(N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal Lp-Lq regularity theorem.

    Original languageEnglish
    Pages (from-to)3741-3774
    Number of pages34
    JournalDiscrete and Continuous Dynamical Systems- Series A
    Volume36
    Issue number7
    DOIs
    Publication statusPublished - 2016 Jul 1

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    Keywords

    • Compressible viscous fluid
    • Free boundary problem
    • Local well-posedness theorem
    • Maximal L-L regularity
    • R-bounded solution operator
    • Two phase problem
    • Uniform W domain

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Applied Mathematics
    • Analysis

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