Stefan problem with surface tension: global existence of physical solutions under radial symmetry

TL;DR

This paper introduces a novel probabilistic framework for the Stefan problem with surface tension under radial symmetry, proving global existence of solutions with sharp bounds on discontinuities in higher dimensions.

Contribution

It develops a new probabilistic notion of solution for the Stefan-Gibbs-Thomson problem, establishing global existence and sharp bounds on discontinuities in arbitrary dimensions.

Findings

Proves global existence of probabilistic solutions under radial symmetry.

Establishes sharp upper bounds on discontinuity sizes in dimensions d>2.

Uses stochastic representations to analyze discontinuities.

Abstract

We consider the Stefan problem with surface tension, also known as the Stefan-Gibbs-Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel "probabilistic" notion of solution, which can accommodate the discontinuities in time (of the radius) of the evolving aggregate. Our main result establishes the global existence of a probabilistic solution satisfying the natural upper bound on the sizes of the discontinuities. Moreover, we prove that the upper bound is sharp in dimensions d>2, in the sense that none of the discontinuities in the solution can be decreased in magnitude. The detailed analysis of the discontinuities, via appropriate stochastic representations, differentiates this work from the previous literature on weak solutions to the Stefan problem with surface tension.