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\geq1) . 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 \boldsymbol{u}\wedge\mathrm{div}(a(\nabla\boldsymbol{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.