## 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.

: Primary 35K59; secondary 35R15, 60H15.

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 Tuesday, January 15, 2019