Advances in Differential Equations

The Stefan problem with convection and Joule's heating

Meir Shillor and Xiangsheng Xu

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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.

Article information

Adv. Differential Equations, Volume 2, Number 4 (1997), 667-691.

First available in Project Euclid: 23 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R35: Free boundary problems
Secondary: 35K60: Nonlinear initial value problems for linear parabolic equations 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35Q80: PDEs in connection with classical thermodynamics and heat transfer 80A22: Stefan problems, phase changes, etc. [See also 74Nxx]


Xu, Xiangsheng; Shillor, Meir. The Stefan problem with convection and Joule's heating. Adv. Differential Equations 2 (1997), no. 4, 667--691.

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