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 $\{r\leq \lambda(t)\}$. We revisit the pioneering analysis of [20] and prove the existence in the radial class of finite time melting regimes $$ \lambda(t)=\left\{\begin{array}{ll} (T-t)^{1/2}e^{-\frac{\sqrt{2}}{2}\sqrt{|\ln(T-t)|}+O(1)}\\ (c+o(1))\frac{(T-t)^{\frac{k+1}{2}}}{|\ln (T-t)|^{\frac{k+1}{2k}}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to T $$ which respectively correspond to the fundamental stable melting rate, and a sequence of codimension $k\in \Bbb N^*$ 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 $$ \lambda_\infty - \lambda(t)\sim\left\{\begin{array}{ll} \frac{1}{\log t}\\ \frac{1}{t^{k}(\log t)^{2}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to +\infty $$ which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension $k$ stable.