Hysteresis for ferromagnetism: asymptotics of some 2-scale Landau-Lifshitz model

TL;DR

This paper analyzes a two-scale Landau-Lifshitz model for ferromagnetism, demonstrating that under certain conditions, the magnetization dynamics follow equilibrium states, with explicit examples provided for ellipsoidal domains.

Contribution

It introduces a rigorous analysis of the asymptotic behavior of a 2-scale ferromagnetic model incorporating hysteresis and exchange effects in three dimensions.

Findings

Strong solutions follow equilibrium dynamics under stability assumptions

Explicit equilibria and magnetic fields are constructed for ellipsoidal domains

The model captures hysteresis effects in ferromagnetic materials

Abstract

We study a 2-scale version of the Landau-Lifshitz system of ferromagnetism, introduced by Starynkevitch to modelize hysteresis: the response of the magnetization is fast compared to a slowly varying applied magnetic fi eld. Taking the exchange term into account, in space dimension 3, we prove that, under some natural stability assumption on the equilibria of the system, the strong solutions follow the dynamics of these equilibria. We also give explicit examples of relevant equilibria and exterior magnetic fields, when the ferromagnetic medium occupies some ellipsoidal domain.