The stochastic Landau--Lifshitz--Baryakhtar equation: Global solution and invariant measure

TL;DR

This paper establishes the existence, uniqueness, and long-term behavior of solutions to a complex stochastic PDE modeling magnetic spins in ferromagnetic materials, including invariant measures and stability analysis.

Contribution

It proves the existence of a unique strong solution to the stochastic LLBar equation, its convergence to the LLB equation, and the existence of invariant measures, advancing understanding of stochastic magnetic models.

Findings

Existence of a unique strong solution in 1-3 dimensions.

Convergence of solutions to the LLB equation as exchange relaxation vanishes.

Existence of invariant measures and stability results at high temperatures.

Abstract

The Landau--Lifshitz--Baryakhtar (LLBar) equation perturbed by both additive and multiplicative noises is a system of fourth order stochastic PDEs which models the evolution of magnetic spin fields in ferromagnetic materials at elevated temperatures, taking into account longitudinal damping, long-range interactions, spin current, and noise-induced phenomena at high temperatures. In this paper, we show the existence of a unique pathwise solution (which is analytically strong) to the stochastic LLBar equation posed in a bounded domain $\mathscr{D}\subset \mathbb{R}^d$, where $d=1,2,3$. We also prove the convergence of this pathwise solution to that of the stochastic Landau--Lifshitz--Bloch (LLB) equation in the limit of vanishing exchange relaxation parameter. Finally, we show the Feller property of the Markov semigroup associated with the strong solution, and prove the existence of nontrivial invariant measures. For temperatures above the Curie temperature, exponential stability of the solution and uniqueness of invariant measure are obtained under certain dissipativity conditions.