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In this paper, we prove the maximal $L_p$-$L_q$ regularity of the compressible and incompressible two phase flow with phase transition in the model problem case with the help of ${\mathcal R}$-bounded solution operators corresponding to generalized resolvent problem. The problem arises from the mathematical study of the motion of two-phase flows having gaseous phase and liquid phase separated by a sharp interface with phase transition. Using the result obtained in this paper, in \cite{S0} we proved the local well-posedness of free boundary problem for the compressible and incompressible two phase flow separated by sharp interface with phase transition.

On the $\mathcal R$-Boundedness for the Two Phase Problem with Phase Transition: Compressible-Incompressible Model Problem