1997 The Stefan problem with convection and Joule's heating
Meir Shillor, Xiangsheng Xu
Adv. Differential Equations 2(4): 667-691 (1997). DOI: 10.57262/ade/1366741153


We establish the existence and partial regularity of a capacity solution to a coupled, degenerate, strongly nonlinear system of PDE's which models the melting of a solid due to volume electric heating. The system generalizes the usual Stefan problem, the evolutionary thermistor problem, and the spot welding problem. We allow temperature dependence for the electrical conductivity---which may lead to degeneracy---and take fully into account the flow of the fluid, which we model with the Navier-Stokes system. Existence is proved by considering a sequence of approximate problems, for which a priori estimates are obtained. Then the limit provides a capacity solution for the original problem. The approximate problems are obtained by smoothing, time-retardation and penalization. Of special interest is the fact that the set where the material is above its melting temperature is open, since only there the Navier-Stokes equations hold. The question of the behavior of solutions in mushy regions, regions where the temperature is identically the melting temperature, is left open.


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Meir Shillor. Xiangsheng Xu. "The Stefan problem with convection and Joule's heating." Adv. Differential Equations 2 (4) 667 - 691, 1997. https://doi.org/10.57262/ade/1366741153


Published: 1997
First available in Project Euclid: 23 April 2013

zbMATH: 1023.35528
MathSciNet: MR1441861
Digital Object Identifier: 10.57262/ade/1366741153

Primary: 35R35
Secondary: 35K60 , 35Q30 , 35Q80 , 80A22

Rights: Copyright © 1997 Khayyam Publishing, Inc.


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Vol.2 • No. 4 • 1997
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