$\mathcal R$-bounded operator families arising from a compressible fluid model of Korteweg type with surface tension in the half-space

TL;DR

This paper establishes $ ext{R}$-boundedness of solution operators for a resolvent problem related to the Navier-Stokes-Korteweg system with surface tension, enabling maximal regularity results for liquid-vapor flow models.

Contribution

It proves $ ext{R}$-boundedness of solution operator families for the resolvent problem in the Korteweg-type fluid model, advancing the analysis of free boundary problems with surface tension.

Findings

$ ext{R}$-boundedness of solution operators established

Maximal regularity in $L_p$-$L_q$ setting achieved

Application to free boundary problems with surface tension

Abstract

In this paper, we consider a resolvent problem arising from the free boundary value problem for the compressible fluid model of Korteweg type, which is called as the Navier-Stokes-Korteweg system, with surface tension in the half-space. The Navier-Stokes-Korteweg system is known as a diffuse interface model for liquid-vapor two-phase flows. Our purpose is to show the $\mathcal R$-boundedness for the solution operator families of the resolvent problem, which gives us the maximal regularity estimates in the $L_p$-in-time and $L_q$-in-space setting by applying the Weis's operator valued Fourier multiplier theorem.