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

Zdzisław Brzeźniak, Beniamin Goldys and Kim Ngan Le

Abstract

The Landau–Lifshitz–Bloch equation perturbed by a space-dependent noise was proposed in Garanin1991 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.

Keywords: quasilinear stochastic PDE, pathwise uniqueness, Galerkin approximations, method of compactness, invariant measures, weak Feller property.

AMS Subject Classification: Primary 35K59; secondary 35R15, 60H15.