# On melting and freezing for the 2D radial Stefan problem

### Mahir Hadžić

King's College London, UK### Pierre Raphaël

Université de Nice Sophia Antipolis, Nice, France

## Abstract

We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball ${r≤λ(t)}$. We revisit the pioneering analysis of [31] and prove the existence in the radial class of finite time *melting* regimes

which respectively correspond to the fundamental *stable* melting rate, and a sequence of codimension $k$ excited regimes. Our analysis fully revisits a related construction for the harmonic heat flow in [60] 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

which correspond respectively to the fundamental *stable* freezing rate, and excited regimes which are codimension $k$ stable.

## Cite this article

Mahir Hadžić, Pierre Raphaël, On melting and freezing for the 2D radial Stefan problem. J. Eur. Math. Soc. 21 (2019), no. 11, pp. 3259–3341

DOI 10.4171/JEMS/904