On Landau-Lifshitz’ equations for ferromagnetism

Abstract

According to the classical theory of Weiβ, on a microscopic scale a ferromagnetic body is magnetically saturated (|M| =M ⁰: constant) and is composed by uniformly magnetized regions separated by thin transition layers.

At equilibrium this corresponds to the minimization of the magnetic energy functional

under the above constraint; this problem has at least one solutionM of classH ¹, which in general is not unique. The evolution is governed by Landau-Lifshitz’ equations

$$\begin{gathered} \frac{{\partial M}}{{\partial t}} = \lambda _1 M \times H^e = \lambda _2 M \times \left( {M \times H^e } \right) \left( {\lambda _1 ,\lambda _2 : constant; \lambda _2 > 0} \right) \hfill \\ H^e = - \frac{{\partial e_{mag} \left( M \right)}}{{\partial M}} \left( {e_{mag} : density of magnetic energy} \right) \hfill \\ \end{gathered} $$

; these are coupled with Maxwell’s equations. An existence result based on the energy estimate and some complementary properties are proved for the corresponding variational formulation. Finally the magnetostrictive effect is included.