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

TL;DR

This paper proves the existence and uniqueness of strong solutions for a complex two-phase flow model involving compressible and incompressible fluids with phase transitions, surface tension, and free boundary conditions.

Contribution

It establishes the first rigorous mathematical result for strong solutions to a general compressible-incompressible two-phase flow with phase transitions and free boundary in arbitrary domains.

Findings

Unique strong solution exists locally in time for small initial data.

Solutions are in the maximal $L_p - L_q$ regularity class.

The model includes surface tension and phase transition effects.

Abstract

We consider a free boundary problem of compressible-incompressible two-phase flows with phase transitions in general domains of $N$-dimensional Euclidean space (e.g. whole space; half-spaces; bounded domains; exterior domains). The compressible fluid and the incompressible fluid are separated by either compact or non-compact sharp moving interface, and the surface tension is taken into account. In our model, the compressible fluid and incompressible fluid are occupied by the Navier-Stokes-Korteweg equations and the Navier-Stokes equations, respectively. This paper shows 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.