Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

TL;DR

This paper investigates the qualitative behavior of solutions to a thermodynamically consistent two-phase Stefan problem with surface tension, analyzing stability, global existence, and the dynamics of equilibria.

Contribution

It establishes the generation of local semiflows, proves global existence under certain conditions, and analyzes stability and instability of equilibria, including multiple spheres.

Findings

Solutions generate local semiflows in well-defined state manifolds.

Solutions without singularities exist globally and are relatively compact.

Multiple spheres of the same radius are shown to be unstable.

Abstract

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.