The ill-posed Hele-Shaw model and the Stefan problem for supercooled water

Abstract

The Hele-Shaw flow of a slow viscous fluid between slightly separated plates is analyzed in the ill-posed case when the fluid recedes due to absorption through a core G G . Necessary and sufficient conditions are given on the initial domain occupied by the fluid to ensure the existence of a solution. Regularity of the free boundary is established in certain rather general cases. Similar results are obtained for the analogous parabolic version, which models the one-phase Stefan problem for supercooled water.