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

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Herrero, Miguel A. and Medina Reus, Elena and Velázquez, J.J. L. (2000) The birth of a cusp in the two-dimensional, undercooled Stefan problem. Quarterly of Applied Mathematics, 58 (3). pp. 473-494. ISSN 0033-569X

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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 crisps in finite time, when they exhibit an undercooling Delta = 1 - O(c) < 1, where 0 < c << 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.


Item Type:Article
Uncontrolled Keywords:Stefan problem; undercooling; interfaces; asymptotic behaviour; matched asymptotic expansions
Subjects:Sciences > Mathematics > Differential equations
ID Code:17887
Deposited On:23 Jan 2013 11:15
Last Modified:12 Dec 2018 15:07

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