Sharp global-existence condition for the supercooled Stefan problem

Let $\eta_0$ be the initial data, with initial domain ${\eta_0>0}$. Condition $(C)$ means that $\eta_0$ is strictly subharmonically ordered with $\nu$ with respect to ${\eta_0>0}$ for some $\nu$ satisfying $\nu\leq\chi_{\{\eta_0>0\}}$; strict subharmonic order means that $\eta_0\leq_{SH}\mu$ with strictly positive stopping time. Sharp global-existence conjecture. There exists a global-time weak solution of $(St)$ with initial data $\eta_0$ and initial domain ${\eta_0>0}$ if and only if $\eta_0$ satisfies condition $(C)$. The condition is necessary by prior analysis, and the conjecture asserts its sufficiency; the additional hypotheses used in the paper's existence theorem are expected not to be essential.