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

TL;DR

The paper addresses the uniqueness of the maximal weak solution to the 1D supercooled Stefan problem by embedding it into a free-target optimal transport framework and proving that the optimizer $\nu^*$ is independent of the cost function $u$ in one dimension. It shows that the optimal primal solution in 1D has an explicit, interval-based structure, yielding a unique maximal weak solution and a corresponding latent heat region characterized by $\nu^*$. The study also demonstrates the lack of monotonicity and $L^1$-Lipschitz stability in this setting, while establishing stability under weak convergence of measures. Altogether, the results clarify well-posedness and stability of the 1D supercooled Stefan problem within the subharmonic-ordered free-target transport framework, connecting to the broader GeneralDimensions and freetarget theory.

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.

Figures (2)

• Figure 2: Shape of $\mu_1$, $\mu_2$
• Figure 3: Shape of $\nu_1$, $\nu_2$

Theorems & Definitions (19)

• Definition 1.1: Maximal solution
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 3.1
• proof
• Remark 3.2
• Corollary 3.3
• proof
• Theorem 4.1