Compressible-incompressible two phase flow of Korteweg type with phase transition: model problem

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 $W^1_q$ 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 $W^{2-1/q}_q$ regularity of $\rho_+$ on $\Gamma_t$, which means the regularity loss. Since the regularity of $\rho_+$ dominated by the Navier-Stokes-Korteweg equations is $W^3_q$ in space, we eliminate the problem by using the Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes equations.