Sensitivity to Initial Data for Physical and Minimal Solutions of the Supercooled Stefan Problem

TL;DR

This paper investigates how solutions to the supercooled Stefan problem depend on initial data, revealing conditions for continuity and the relationship between minimal and physical solutions' uniqueness.

Contribution

It establishes the conditions under which the solution map for physical and minimal solutions is continuous with respect to initial data, linking uniqueness to stability.

Findings

Solution map for physical solutions is continuous under initial data perturbations.

Solution map for minimal solutions is continuous when data shifts to the right.

Continuity of minimal solutions fails at initial data unless physical solutions are unique.

Abstract

We address the problem of well-posedness for physical and minimal solutions to a probabilistic reformulation of the supercooled Stefan problem by investigating the sensitivity of these solutions to changes in the initial data. We show that the solution map for physical solutions is continuous under perturbations to the initial condition $X_{0-}$ (in the weak sense of probability measures), provided that $X_{0-}$ admits a unique physical solution. Furthermore, we show continuous dependency of the solution map for minimal solutions when the data is shifted to the right; however, continuity of this solution map is shown to fail at $X_{0-}$ for shifts to the left unless uniqueness of physical solutions holds. As a result, we show that the question of whether the solution map for minimal solutions is continuous at $X_{0-}$ is equivalent to the question of whether the given data $X_{0-}$ admits a unique physical solution.