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

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 R-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. of the solution operators following system of equations:

The half-Space Model Problem for Compressible Fluid Flow

“…Substituting (21), ( 22), (23), ( 24) and ( 25) to (7), we obtain the formula of � = � = 〈 � 1 , � 2 , … , � −1 〉 and � in equation system of (5) which completes the proof of Theorem 2…”

“…In this paper, we put the different approach of the general solution of velocity as in [7]. This research focus on considering the solution formula of the compressible Stokes equation system (2) without surface tension by using Fourier transform.…”

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

“…In this section, we shall show the proof of Theorem 2.8. In order to prove Theorem 2.8, we use the following lemmas which is proven by Götz and Shibata [7].…”

Compressible–Incompressible Two-Phase Flows with Phase Transition: Model Problem

“…He proved the existence of R bounded solution operators for the model problem that derives the maximal L p -L q regularity of solutions to the linearized equations automatically with the help of Weis's operator valued Fourier multiplier theorem [35]. According to Shibata [13], the regularity of ρ + is W 1 q in space, but to solve the kinetic equation: u Γ · n t = [[ρu]] · n t /[[ρ]] on Γ t we need W 2−1/q q regularity of ρ + on Γ t , which means the regularity loss. On the other hand, the regularity of ρ + dominated by the Navier-Stokes-Korteweg equations is W 3 q in Ω t+ (cf.…”