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.
Categories: math.AP