Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension

Gelification and mass transport in a static non-isothermal waxy solution

“…on x = L. In particular, the fact that Z > 0 close to the free boundary, so that phase is shrinking, makes it similar to a supercooled Stefan problem, for which blow-up can be a typical feature (for a review on non-standard Stefan problems see [7]). A rather comprehensive analysis of the phenomenon of blow-up for parabolic free boundary problems with data of Cauchy type (like (67)) or of supercooled (superheated) Stefan type (like (68)) has been performed in a series of papers [9,13,14], but only in case of constant coefficients. We note for instance that in (68) the source is not constant and even has variable sign, and the 'latent heat', though of constant sign, is non-constant (although, by assumption, it is bounded away from zero in the time interval considered).…”

“…Therefore our strategy to exclude blow-up for (FBP3) is to exclude it for (70). In order to make use of the theory developed in [9,13,14], we introduce the variables…”

Mushy Regions in Negative Squeeze Films

“…Furthermore, if u + additionally satisfies the constraint u + ≥ 0, which is here enforced by our cavitation condition, then even if the corresponding Hele-Shaw problem has a receding free boundary (as it does when the mush is expanding), it does not suffer terminal blow-up via cusps or other singularities in its free boundary. Instead, if necessary, it avoids this fate by nucleating new components of its free boundary [12].…”

The Supercooled One-Phase Stefan Problem in Spherical Symmetry

“…Problems of this kind have been studied by other authors in connection with the freezing of a supercooled liquid. Several different boundary conditions were analysed in [3], [5], [6], [7], [10], [11], [13], for the one-dimensional case, in [1], [2] for cylindrical symmetry and in [9] for spherical symmetry.…”