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The inverse Stefan problem for the heat equation in two space variables

Part of: Parabolic equations and systems Thermodynamics and heat transfer

Published online by Cambridge University Press:  26 February 2010

David Colton
David Colton
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47401.
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Extract

The Stefan problem is a particular free boundary problem for the heat equation which arises in the investigation of the melting of solids. In the case of one space dimension there are numerous results available concerning the existence, uniqueness, and stability of the solution [c.f. 6]. However the case of several space variables is considerably more difficult. This is due in large part to the fact that the geometry of the problem can become quite complicated, and smooth initial and boundary data do not necessarily lead to smooth solutions. In particular, under heating, a connected solid can melt into two (or more) disconnected solids, thus leading to a problem in which the free boundary varies in a discontinuous manner. These difficulties have motivated several researchers to look for “weak” solutions to the Stefan problem [c.f. 1, 3, 5]. Although this approach is quite general and leads to numerical schemes for solving the problem under consideration, there are several drawbacks to this method, among them being the fact that no information is obtained concerning the structure of the interphase boundary, nor is there much information on the regularity of these weak solutions.

MSC classification

Secondary: 35K05: Heat equation 80A20: Heat and mass transfer, heat flow
Type
Research Article
Information
Mathematika , Volume 21 , Issue 2 , December 1974 , pp. 282 - 286
Copyright
Copyright © University College London 1974

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References

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