Strong solutions for compressible-incompressible two-phase flows with phase transitions

We consider the compressible-incompressible two-phase flows with phase transitions in a general domain of $N$-dimensional Euclidean space. The compressible fluid and the incompressible fluid are separated by a sharp interface, and the surface tension is taken into account. To describe the model, the Navier-Stokes-Korteweg equations are used for the compressible fluid and the Navier-Stokes equations are used for the incompressible fluid. The aim of this paper is to show that for given $T > 0$ the problem admits a unique strong solution on $(0, T)$ in the maximal $L_p$-$L_q$ regularity class provided the initial data are small in their natural norms.