On melting and freezing for the 2d radial Stefan problem

Abstract

We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball $\{r\le \lambda (t)\}$. We revisit the pioneering analysis of [20] and prove the existence in the radial class of finite time melting regimes $$\lambda (t)=\{\begin{array}{l}(T-t{)}^{1/2}{e}^{-\frac{\sqrt{2}}{2}\sqrt{|\ln (T-t)|}+O(1)}\\ (c+o(1))\frac{(T-t{)}^{\frac{k+1}{2}}}{|\ln (T-t){|}^{\frac{k+1}{2k}}},k\in {\mathbb{N}}^{\ast }\end{array}\text{ as }t\to T$$ which respectively correspond to the fundamental stable melting rate, and a sequence of codimension $k\in {\mathbb{N}}^{\ast }$ excited regimes. Our analysis fully revisits a related construction for the harmonic heat flow in [42] by introducing a new and canonical functional framework for the study of type II (i.e. non self similar) blow up. We also show a deep duality between the construction of the melting regimes and the derivation of a discrete sequence of global-in-time freezing regimes $${\lambda }_{\infty }-\lambda (t)\sim \{\begin{array}{l}\frac{1}{\log t}\\ \frac{1}{t^k(\log t{)}^2},k\in {\mathbb{N}}^{\ast }\end{array}\text{ as }t\to +\infty$$ which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension $k$ stable.

70 pages, a few references added and typos corrected