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.