Existence of a unique solution and invariant measures for the stochastic Landau--Lifshitz--Bloch equation

Abstract

The Landau--Lifshitz--Bloch equation perturbed by a space-dependent noise was proposed in Garanin 1991 as a model for evolution of spins in ferromagnatic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain $D\subset \mathbb R^d$, $d=1,2,3$, we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases $d=1,2$ we prove uniqueness of pathwise solutions and the existence of invariant measures.