On well-posedness of incompressible two-phase flows with phase transitions: the case of equal densities

TL;DR

This paper establishes the well-posedness and thermodynamic consistency of a model for incompressible two-phase flows with phase transitions, focusing on the case of equal densities, and analyzes the system's long-term behavior.

Contribution

It proves local well-posedness using maximal Lp-regularity and identifies the system's equilibria and stability properties in the equal densities case.

Findings

The model conserves total energy and non-decreases total entropy.

Existence of a local semiflow on a nonlinear state manifold.

Global existence and stability of solutions under certain conditions.

Abstract

The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal $L_p$-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.