Measure-valued solutions to a generalized Landau–Lifshitz–Bloch equation

Abstract

This paper considers a generalized version of the Landau–Lifshitz–Bloch equation. We prove existence of measure-valued solutions for the model in a bounded domain of ${\mathbb{R}}^d$ $(d\ge 1)$. The main difficulties in this study are due to the loss of compactness in the equation and the presence of a nonlinear term of type $\mathbf{u}\wedge \mathrm{div}(a(\nabla \mathbf{u}))$ which does not satisfy the monotonicity assumption in the sense of Leray–Lions. We use a compactness result proved in Landes et al. [Ark. Mat. 25 (1987), 29–40] and the concept of measure-valued solutions which are appropriate to solve this problem.