On the -boundedness of the solution operators in the study of the compressible viscous fluid flow with free boundary conditions

Dario Götz, Yoshihiro Shibata

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    Abstract

    In this paper, we consider a generalized resolvent problem for the linearization system of the Navier-Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the -bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal Lp-Lq regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the Lp in time and Lq space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55-89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2×2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3×3 matrix, so that the method presented here is slightly simpler.

    Original languageEnglish
    Pages (from-to)207-236
    Number of pages30
    JournalAsymptotic Analysis
    Volume90
    Issue number3-4
    DOIs
    Publication statusPublished - 2014

    Fingerprint

    Compressible Fluid
    Viscous Flow
    Free Boundary
    Viscous Fluid
    Fluid Flow
    Boundedness
    Boundary conditions
    Newtonian Fluid
    Free Boundary Problem
    Fluid Dynamics
    Operator
    Operator-valued Fourier multipliers
    Analytic Semigroup
    Invertibility
    Bounded Solutions
    Matrix Function
    Resolvent
    Surface Tension
    Existence Theorem
    Linearization

    Keywords

    • -boundedness
    • analytic semigroup
    • free boundary condition
    • general domain
    • maximal L-p-L-q regularity
    • Stokes equations

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

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