The supercooled Stefan problem in one dimension

Abstract

We study the 1D contracting Stefan problem with two moving boundaries that describes the <em>freezing</em> of a supercooled liquid. The problem is borderline ill--posed with a density in excess of unity indicative of the dividing line. We show that if the initial density, $\rho_0(x)$ does not exceed one and is not too close to one in the vicinity of the boundaries, then there is a unique solution for all times which is smooth for all positive times. Conversely if the initial density is too large, singularities may occur. Here the situation is more complex: the solution may suddenly freeze without any hope of continuation or may continue to evolve after a local instant freezing but, sometimes, with the loss of uniqueness.