Global solvability of compressible-incompressible two-phase flows with phase transitions in bounded domains

Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains $\Omega_{t+}, \Omega_{t-} \subset \mathbb{R}^N$ where the domains are separated by a sharp compact interface $\Gamma_t \subset \mathbb{R}^{N - 1}$. We prove a global in time unique existence theorem for such free boundary problem under the assumption that the initial data are near the equilibrium and initial domains are bounded. In particular, we obtain the solution in the maximal $L_p$-$L_q$ regularity class with $2 < p <\infty$ and $N < q < \infty$ and exponential stability of the corresponding analytic semigroup on the infinite time interval.