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Singularity formation in the one-dimensional supercooled Stefan problem

Published online by Cambridge University Press:  26 September 2008

Miguel A. Herrero
Affiliation:
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Juan J. L. Velázquez
Affiliation:
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain

Abstract

It is well-known that solutions to the one-dimensional supercooled Stefan problem (SSP) may exhibit blow-up in finite time. If we consider (SSP) in a half-line with zero flux conditions at t = 0, blow-up occurs if there exists T < ∞ such that limtTs(t) > 0 and lim inftT⋅(t) = – ∞,s(t) being the interface of the problem under consideration. In this paper, we derive the asymptotics of solutions and interfaces near blow-up. We shall also use these results to discuss the possible continuation of solutions beyond blow-up.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

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