Ohm-Hall conduction in hysteresis-free ferromagnetic processes

Abstract

Electromagnetic processes in a ferromagnetic conductor(e.g., an electric transformer) are here described bycoupling the Maxwell equations with nonlinear constitutive laws of the form$$\vec B \in \mu_0\vec H + {\mathcal M}(x) \vec H/|\vec H|,\qquad\vec J = \sigma(x) \big( \vec E + \vec E_a(x,t) + h(x)\vec J \!\times\! \vec B \big).$$Here $\vec E_a$ stands for an applied electromotive force; the saturation${\mathcal M}(x)$, the conductivity $\sigma(x)$ and the Hall coefficient $h(x)$ are also prescribed.The first relation accounts for hysteresis-free ferromagnetism,the second one for the Ohm law and the Hall effect. This model leads to the formulation of an initial-value problem for a doubly-nonlinear parabolic-hyperbolic system in the whole $R^3$. Existence of a weak solution is proved, via approximation by time-discretization, derivation of a priori estimates, and passage to the limit.This final step rests upon a time-dependent extension of the Murat and Tartar div-curl lemma, and on compactness by strict convexity.