  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.
