Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T08:27:32.905Z Has data issue: false hasContentIssue false

Solutions for the two-phase Stefan problem with the Gibbs–Thomson Law for the melting temperature

Published online by Cambridge University Press:  16 July 2009

Stephan Luckhaus
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, 5300 Bonn, West Germany

Abstract

The coupling of the Stefan equation for the heat flow with the Gibbs–Thomson law relating the melting temperature to the mean curvature of the phase interface is considered. Solutions, global in time, are constructed which satisfy the natural a priori estimates. Mathematically the main difficulty is to prove a certain regularity in time for the temperature and the indicator function of the phase separately. A capacity type estimate is used to give an L1 bound for fractional time derivatives.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Almgren, F. J. 1976 Existence and regularity almost everywhere of elliptic variational problems with constraints. Mem. AMS 4, 165.Google Scholar
Berger, A. E. & Rogers, J. C. W. 1984 Some properties of the nonlinear semigroup for the problem μt–Δf(μ) = 0. Nonlinear Anal. 8, 909939.Google Scholar
Brezis, H. & Crandall, M. G. 1979 Uniqueness of solutions of the initial value problem for μt–Δ(φ(μ)) = 0. J. Math. Pure Appl. 58, 153163.Google Scholar
Chadam, J. & Ortoleva, P. 1983 The stabilizing effect of surface tension on the development of planar free boundaries in single phase Cauchy Stefan problems. JIMA 30, 5766.Google Scholar
Friedman, A. 1982 Variational Principles and Free Boundary Problems. Wiley.Google Scholar
Glusti, E. 1984 Minimal Surfaces and Functions of Bounded Variation. Birkhäuser.Google Scholar
Gurtin, M. 1986 On the two-phase Stefan problem with interfacial energy and entropy. Arch. Rat. Mec. An. 96, 199241.CrossRefGoogle Scholar
Hanzawa, , Ei, Ichu 1981 Classical solution of the Stefan problem. Tohoku Math.J. 33, 297335.CrossRefGoogle Scholar
Meirmanov, A. M. 1980 On a classical solution of the multidimensional Stefan problem for quasilinear parabolic equations. Mat. Sb. 112, 170192.Google Scholar
Rogers, J. C. W. 1983 The Stefan problem with surface tension. In Free Boundary Problems, Theory and Applications I (ed. Fasanao, A. & Primicerio, M.). Pitman.Google Scholar
Visintin, A. 1984 Stefan problem with surface tension. Report of IAN of CNR. Pavia.Google Scholar
Visintin, A. 1988a Surface tension effects in phase transitions. In Material Instabilities in Continuums Mechanics (ed. Ball, J.). Clarendon.Google Scholar
Visintin, A. 1988b Stefan problem with surface tension. In Mathematical Models for Phase Change Problems (ed. Rodrigues, J. F.). Birkhäuser.Google Scholar