On the ${\mathcal R}$-boundedness for the two phase problem with phase transition: compressible-incompressible model problem

TL;DR

This paper establishes maximal regularity results for a two-phase flow model with phase transition, using ${\mathcal R}$-bounded solution operators, advancing the mathematical understanding of such complex fluid dynamics problems.

Contribution

It proves the $L_p$-$L_q$ regularity for the compressible-incompressible two-phase flow with phase transition using ${\mathcal R}$-bounded operators, a novel approach in this context.

Findings

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

Established $\mathcal{R}$-boundedness of solution operators.

Facilitated proof of local well-posedness in related work.

Abstract

In this paper, we prove the maximal $L_p$-$L_q$ regularity of the compressible and incompressible two phase flow with phase transition in the model problem case with the help of ${\mathcal R}$-bounded solution operators corresponding to generalized resolvent problem. The problem arises from the mathematical study of the motion of two-phase flows having gaseous phase and liquid phase separated by a sharp interface with phase transition. Using the result obtained in this paper, in \cite{S0} we proved the local well-posedness of free boundary problem for the compressible and incompressible two phase flow separated by sharp interface with phase transition.