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

TL;DR

This paper proves the global existence and stability of solutions for a complex free boundary problem involving compressible and incompressible two-phase flows with phase transitions, under small initial data conditions.

Contribution

It establishes a unique global-in-time solution in maximal regularity class for the two-phase flow problem with phase transitions in bounded domains.

Findings

Existence of a unique global solution under small initial data.

Solution exhibits exponential stability over infinite time.

Solution constructed in maximal $L_p-L_q$-regularity class.

Abstract

Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains $\Omega_{t +}, \Omega_{t -} \subset \mathbb{R}^N$, $N \ge 2$, 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 sufficiently small and the initial domain of the incompressible fluid is close to a ball. 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.