The $\mathcal{R}-$bounded operator families arising from the study of the barotropic compressible flows with free surface

TL;DR

This paper investigates the mathematical properties of free boundary problems in barotropic compressible flows, establishing $ ext{R}$-bounded operator families to analyze regularity in fluid dynamics models.

Contribution

It introduces the existence of $ ext{R}$-bounded operator families for the resolvent problem, advancing the understanding of maximal regularity in free boundary compressible flow models.

Findings

Established $ ext{R}$-boundedness of operator families

Proved maximal $L_p-L_q$ regularity for the model problem

Applied Weis' theory to free boundary Navier-Stokes equations

Abstract

In this paper, we study some model problem associated to the free boundary value problem of the barotropic compressible Navier-Stokes equations in general smooth domain with taking surface tension into account. To obtain the maximal $L_p-L_q$ regularity property of the model problem, we prove the existence of $\mathcal{R}-$bounded operator families of the resolvent problem via Weis' theory on operator valued Fourier multipliers.