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The Stefan problem with surface tension in the three dimensional case with spherical symmetry: nonexistence of the classical solution

Published online by Cambridge University Press:  26 September 2008

A. M. Meirmanov
A. M. Meirmanov
Affiliation:
Institut für Angewandte Mathematik, Univeristät Bonn, Bonn, Germany
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Abstract

The Stefan problem with surface tension in the three-dimensional case with spherical symmetry is considered. We first establish the existence and uniqueness of the classical solution with surface tension and kinetic undercooling effects for all time, and then pass to the limit as the kinetic undercooling tends to zero. The limiting solution is the global-in-time weak solution and the local-in-time classical solution for the Stefan problem with surface tension. This solution cannot be the global-in-time classical solution. If S(t) is the radius of a solid ball in a supercooled liquid, then (1) there exists at least one point t* of discontinuity of the function S(t):

or (2) the continuous function S(t) cannot be absolutely continuous, and it maps some zero-measure set of (t*, T*) onto some set of Ω with a strictly positive measure.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 1 , March 1994 , pp. 1 - 19
Copyright
Copyright © Cambridge University Press 1994

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References

[1]Visintin, A. 1985 Models for supercooling/superheating effects. Volume 120 of Research Notes in Mathematics. Pitman, 200–207.Google Scholar
[2]Luckhaus, S. 1990 Solutions for the two-phase Stefan problems with the Gibbs-Thomson Law for the melting temperature. Euro. Jnl Appl. Math. 1, 101–111.Google Scholar
[3]Chen, X. & Reitich, F. 1990 Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling. IMA Preprint Series, 715, November.Google Scholar
[4]Radkevitch, E. 1991 Gibbs–Thomson law and existence of the classical solution of the modified Stefan problem. Soviet Dokl. Acad. Sci. 316 (6), 1311–1315.Google Scholar
[5]Chadam, J., Howison, S. & Ortoleva, P. 1987 Existence and Stability for Spherical Crystal Growing in a Supersaturated Solution. IMA J. Appl. Math. 39, 1–15.CrossRefGoogle Scholar
[6]Ladyzenskaya, O. A., Solonnikov, V. A. & Ural'ceva, N. N. 1968 Linear and Quasilinear Equations of Parabolic Type. Transl. American Math. Society, Providence, RI.CrossRefGoogle Scholar
[7]Meirmanov, A. 1992 The Stefan Problem. Walter De Gruyter, Berlin-New York.Google Scholar
[8]Xie, W. 1990 The Stefan Problem with a kinetic condition at the free boundary. SIAM J. Math. Anal. 21 (2), 362–373.Google Scholar