Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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The birth of a cusp in the two-dimensional, undercooled Stefan problem


Authors: M. A. Herrero, E. Medina and J. J. L. Velázquez
Journal: Quart. Appl. Math. 58 (2000), 473-494
MSC: Primary 35R35; Secondary 35B40, 35C20, 35K05
DOI: https://doi.org/10.1090/qam/1770650
MathSciNet review: MR1770650
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Abstract: This paper deals with the one-phase, undercooled Stefan problem, in space dimension $ N = 2$. We show herein that planar, one-dimensional blow-up behaviours corresponding to the undercooling parameter $ \Delta = 1$ are unstable with respect to small, transversal perturbations. The solutions thus produced are shown to generically generate cusps in finite time, when they exhibit an undercooling $ \Delta = 1 - O\left( \epsilon \right) < 1$, where $ 0 < \epsilon << 1$, and $ \epsilon $ is a parameter that measures the strength of the perturbation. The asymptotic behaviour of solutions and interfaces near their cusps is also obtained. All results are derived by means of matched asymptotic expansions techniques.


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DOI: https://doi.org/10.1090/qam/1770650
Article copyright: © Copyright 2000 American Mathematical Society


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