Uniqueness of the maximal solution of the supercooled Stefan problem in 1D

TL;DR

This paper proves the uniqueness of the maximal weak solutions to the 1D supercooled Stefan problem by linking it to an optimal transport problem and analyzing its stability properties.

Contribution

It demonstrates that in 1D, the supercooled Stefan problem's maximal solution is unique and independent of the cost function, and establishes weak measure stability despite lacking monotonicity.

Findings

Uniqueness of maximal weak solutions in 1D

Independence from cost function in the optimal transport formulation

Stability in weak convergence of measures

Abstract

We prove uniqueness of the maximal weak solutions to the supercooled Stefan problem in 1 dimension. This follows by showing that in 1 dimension, the optimal solution of the corresponding free target optimal transport problem given in \cite{GeneralDimensions}, is independent of the choice of the cost function. Moreover, we show that the supercooled Stefan problem lacks monotonicity and $L^1$-Lipschitz stability, which are available in a similar problem considered in a previous paper \cite{freetarget}. However, in $1$ dimension, it has stability in the weak convergence of measures.