Compressible-incompressible two-phase flows with phase transition: model problem

Abstract

We study the compressible and incompressible two-phase flows separated by a sharp interface with a phase transition and a surface tension. In particular, we consider the problem in ${\mathbb{R}}^N$, and the Navier-Stokes-Korteweg equations is used in the upper domain and the Navier-Stokes equations is used in the lower domain. We prove the existence of $\mathcal{R}$-bounded solution operator families for a resolvent problem arising from its model problem. According to Shibata \cite{GS2014}, the regularity of ${\rho }_{+}$ is $W_q^1$ in space, but to solve the kinetic equation: ${\mathbf{u}}_{\Gamma }\cdot {\mathbf{n}}_t=[[\rho \mathbf{u}]]\cdot {\mathbf{n}}_t/[[\rho ]]$ on ${\Gamma }_t$ we need $W_q^{2-1/q}$ regularity of ${\rho }_{+}$ on ${\Gamma }_t$, which means the regularity loss. Since the regularity of ${\rho }_{+}$ dominated by the Navier-Stokes-Korteweg equations is $W_q^3$ in space, we eliminate the problem by using the Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes equations.

Typos corrected