The Landau-Lifshitz-Bloch equation on the thin film

TL;DR

This paper studies the Landau-Lifshitz-Bloch equation in thin ferromagnetic films, establishing weak solutions, deriving a 2D limit, and proposing a model for high-temperature magnetic dynamics in negligible thickness films.

Contribution

It provides a rigorous derivation of the 2D limit equation and introduces a new model for magnetic behavior at high temperatures in thin films.

Findings

Existence of weak solutions for the 3D Landau-Lifshitz-Bloch equation.

Derivation of a 2D limit equation as film thickness tends to zero.

A new equation describing magnetic dynamics at high temperature for thin films.

Abstract

We consider the initial boundary value problem of Landau-Lifshitz-Bloch equation on three-dimensional ferromagnetic films, where the effective field contains the stray field controlled by Maxwell equation and the exchange field contains exchange constant. In this paper, we establish the existence of weak solutions of the equation by using the Faedo-Galerkin approximation method. We also derive its two-dimensional limit equation in a mathematically rigorous way when the film thickness tends to zero under appropriate compactness conditions. Moreover, we obtain an equation that can better describe the magnetic dynamic behavior of ferromagnetic films with negligible thickness at high temperature.