×

Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature. (English) Zbl 0734.35159

Summary: 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 \(L_ 1\) bound for fractional time derivatives.

MSC:

35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1093/imamat/30.1.57 · Zbl 0544.35088
[2] Brezis, J. Math. Pure Appl 58 pp 153– (1979)
[3] DOI: 10.1016/0362-546X(84)90111-1 · Zbl 0557.35129
[4] Almgren, Mem. AMS 4 pp 165– (1976)
[5] Visintin, Mathematical Models for Phase Change Problems (1988) · Zbl 0656.73043
[6] Visintin, Material Instabilities in Continuums Mechanics (1988)
[7] Friedman, Variational Principles and Free Boundary Problems (1982)
[8] Meirmanov, On a classical solution of the multidimensional Stefan problem for quasilinear parabolic equations. Mat. Sb. 112 pp 170– (1980)
[9] Rogers, Free Boundary Problems, Theory and Applications I (1983)
[10] DOI: 10.2748/tmj/1178229399 · Zbl 0571.35109
[11] Gurtin, Arch. Rat. Mec. An. 96 pp 199– (1986)
[12] Glusti, Minimal Surfaces and Functions of Bounded Variation (1984)
[13] Visintin, Report of IAN of CNR (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.