The Landau–Lifshitz–Bloch Equation in the Thin Film

In this paper, we study the initial-boundary value problem of the Landau–Lifshitz–Bloch equation in three-dimensional ferromagnetic films, where the effective field contains the stray field controlled by the Maxwell equation, and the exchange field contains exchange constant. Firstly, we establish the existence of weak solutions of the equation by using the Faedo–Galerkin approximation. We also derive its two-dimensional limit equation in a mathematically rigorous way when the film thickness tends towards 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 temperatures.