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Asymptotic convergence of the Stefan problem to Hele-Shaw


Authors: Fernando Quirós and Juan Luis Vázquez
Journal: Trans. Amer. Math. Soc. 353 (2001), 609-634
MSC (2000): Primary 35B40, 35R35
DOI: https://doi.org/10.1090/S0002-9947-00-02739-2
Published electronically: October 23, 2000
MathSciNet review: 1804510
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Abstract:

We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as $t\to \infty $, and give the precise asymptotic growth rate for the radius.


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Additional Information

Fernando Quirós
Affiliation: Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain

Juan Luis Vázquez
Affiliation: Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain

DOI: https://doi.org/10.1090/S0002-9947-00-02739-2
Keywords: Stefan problem, Hele-Shaw, asymptotic behaviour
Received by editor(s): December 31, 1996
Received by editor(s) in revised form: October 28, 1997
Published electronically: October 23, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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