On two problems of electrical heating of conductors

Abstract

Abstract. The thermal effects of the currents induced in a massive conductor by an external slowly varying magnetic field are studied with regard to existence and uniqueness of solutions. In the first part a theorem of existence of solution is also given for the thermistor problem with a current limiting device.1. Introduction. In this paper we study various problems related to the Joule heating of a conductor. In Sec. 2 a theorem of existence is given for a version of the "thermistor problem" first proposed in [3] and then studied for special boundary conditions in [1], The boundary conditions here considered are quite general and the same methods apply even to more general situations. In Sec. 3 a model is proposed for describing the Joule heating due to the eddy currents in a very long cylinder when the external variable magnetic field is parallel to the axis of the cylinder. A theorem of existence is given for this problem.

A non-local thermistor problem

“…This non-local problem has received much less attention. Noteworthy are the results found in Cimatti [6,20] and Howison, Rodrigues and Shillor [8].…”

“…The mathematical aspects of the system given by (1.1), (1.2), (1.5)'and (1.6) have recently received considerable attention. The question of the existence of solutions was studied [5][6][7][8] under the assumption that the conductivities are uniformly bounded away from zero and from infinity, and [9][10][11][12] in the case that the conductivities are allowed to go to zero if the temperature goes to infinity. The uniqueness problem has also been investigated in [10].…”

“…However, in Cimatti [6,20] it is explicitly assumed that f a k/cr = 00, and that the boundary conditions are of a somewhat more special nature. In Howison et al [8], a is assumed bounded above and below by positive constants and, furthermore, it is also assumed that k = 1.…”

“…In Howison et al [8], a is assumed bounded above and below by positive constants and, furthermore, it is also assumed that k = 1. We do draw, however, on some of the ideas of these other authors [6,8,20]. We recall that the Kirchoff transformation w = T(u) with T(u) = j%k(£)dE, may always be employed to reduce system (1.1)-(1.2) to one for (w,<p) with k= 1.…”

Global solutions of Maxwell's equations in an electromagnetic field with a temperature-dependent electrical conductivity

“…He is also grateful to Professor John R. Ockendon for providing me with reference [8]. He wants to thank the referees for the several valuable suggestions.…”