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.
Categories: math.AP