Local well-posedness of compressible-incompressible two-phase flows with phase transitions

This paper is concerned with the basic model for compressible and incompressible two phase flows with phase transitions The flows are separated by nearly flat interface represented as a graph over the $N-1$ dimensional Euclidean space ${\Bbb R}^{N-1}$ ($N \geq 2$). The local well-posedness is proved by the Banach fixed point theorem based on the maximal $L_p$-$L_q$ regularity theorm for the linearized problem.