On the ℛ-boundedness for the two phase problem: compressible-incompressible model problem

Takayuki Kubo,Kohei Soga,Yoshihiro Shibata

doi:10.1186/s13661-014-0141-3

Abstract

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal L p - L q regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).MSC: 35Q35, 76T10.

Highlights

1 Introduction This paper is concerned with the evolution of compressible and incompressible viscous fluids separated by a sharp interface
Typical examples of the physical interpretation of our problem are the evolution of a bubble in an incompressible fluid flow, or a drop in a volume of gas
We assume that no phase transitions occur and we do not consider the surface tension at the interface t and the free boundary St+ for mathematical simplicity

Summary

Introduction

This paper is concerned with the evolution of compressible and incompressible viscous fluids separated by a sharp interface. Div K is an N -vector with components stands for the identity matrix, nt the unit normal to t pointed from t– to t+, nt– the unit outward normal to St–, and μ± and ν+ are first and second viscosity coefficients, respectively, which are assumed to be constant and satisfy the condition μ± > , ν+ > , The case (C ) is used to prove the existence of R-bounded solution operator to

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