Ohm-Hall conduction in hysteresis-free ferromagnetic processes

Augusto Visintin

doi:10.3934/dcdsb.2013.18.551

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2013, Volume 18, Issue 2: 551-563. Doi: 10.3934/dcdsb.2013.18.551

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Ohm-Hall conduction in hysteresis-free ferromagnetic processes

Augusto Visintin^1,

1.
Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento) - Italia

Received: September 30, 2011

Revised: December 31, 2011

Published: February 2013

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Abstract

Electromagnetic processes in a ferromagnetic conductor (e.g., an electric transformer) are here described by coupling 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.

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Mathematics Subject Classification: Primary: 35K60; Secondary: 35R35, 78A25.

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