Analysis and Mathematical Physics
On Singular Points in the Supercooled Stefan Problem
The supercooled Stefan problem models how water, cooled below freezing, forms into ice. Experiments have shown that this process can exhibit nucleation (pieces of ice forming spontaneously) and that fractal-like patterns can emerge in the ice. This stands in contrast with the melting problem, in which the ice is expected to be smooth at most points and times. This was recently demonstrated rigorously in groundbreaking work by Figalli, Ros Oton and Serra.
In joint work with Inwon Kim (UCLA) and Sebastian Munoz (UCLA), we analyze a subclass of solutions to the supercooled problem which are connected to a parabolic obstacle-type problem. For this subclass we show that some of the experimentally observed behavior can occur (such as nucleation) but that the singular set cannot be completely pathological (i.e. we give bounds on the size of the singular points).