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

Authors: Emmanuele DiBenedetto and Avner Friedman
Journal: Trans. Amer. Math. Soc. 282 (1984), 183-204
MSC: Primary 35R35; Secondary 35K05, 80A20
MathSciNet review: 728709
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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$. 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.

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