Well-posedness of the supercooled Stefan problem with oscillatory initial conditions

Abstract

We study the one-phase one-dimensional supercooled Stefan problem with oscillatory initial conditions. In this context, the global existence of so-called physical solutions has been shown recently in [3], despite the presence of blow-ups in the freezing rate. On the other hand, for regular initial conditions, the uniqueness of physical solutions has been established in [6]. Here, we prove the uniqueness of physical solutions for oscillatory initial conditions by a new contraction argument that replaces the local monotonicity condition of [6] by an averaging condition. We verify this weaker condition for fairly general oscillating probability densities, such as the ones given by an almost sure trajectory of ${(1+W_x-\sqrt{2x|log|logx||})}_{+}\wedge 1$ near the origin, where W is a standard Brownian motion. We also permit typical deterministically constructed oscillating densities, including those of the form $(1+sin1∕x)∕2$ near the origin. Finally, we provide an example of oscillating densities for which it is possible to go beyond our main assumption via further complementary arguments.