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 Jan 1

    Fingerprint

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this