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Existence of global weak solutions for a class of quasilinear equations describing Joule's heating
Author(s) -
Bień Marian
Publication year - 1999
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199910)22:15<1275::aid-mma39>3.0.co;2-7
Subject(s) - joule heating , mathematics , galerkin method , degenerate energy levels , maxwell's equations , heat equation , mathematical analysis , a priori and a posteriori , a priori estimate , extension (predicate logic) , work (physics) , joule (programming language) , energy (signal processing) , thermodynamics , physics , finite element method , philosophy , statistics , epistemology , quantum mechanics , computer science , programming language
The existence of global weak solutions to the degenerate problem describing Joule's heating in a current and heat conductive medium is proved via the Galerkin method. The existence proof proceeds by a sequence of a priori estimates, which may be achieved by the standard method. Under suitable hypotheses on the electrical conductivity the boundedness in L ∞ ( Q T ) of the absolute temperature of the medium is established by the method of Stampacchia. The paper is an extension of the work by Cimatti [3,4], Shi et al . [12] and Xu [16]. This extension consists of employing the equations for the electric field E , derived from Maxwell's equations, instead of the equation for the electric potential, with the appropriate modification in the energy balance equation. Copyright © 1999 John Wiley & Sons, Ltd.