On the evolution of compressible and incompressible viscous fluids with a sharp interface

Takayuki Kubo*, Yoshihiro Shibata

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq 2-1/q domain in RN (N ≥ 2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2 < p < ∞ and N < q < ∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.

Original languageEnglish
Article number621
JournalMathematics
Volume9
Issue number6
DOIs
Publication statusPublished - 2021 Mar 2

Keywords

  • Local in time unique existence theorem
  • Navier-Stokes equations
  • R-bounded operator
  • Two phase problem

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint

Dive into the research topics of 'On the evolution of compressible and incompressible viscous fluids with a sharp interface'. Together they form a unique fingerprint.

Cite this