Well-posedness and invariant measures for the stochastically perturbed Landau-Lifshitz-Baryakhtar equation

TL;DR

This paper investigates the well-posedness and invariant measures of the stochastic Landau-Lifshitz-Baryakhtar equation in bounded domains, establishing existence, uniqueness, and invariant measures for solutions in various dimensions.

Contribution

It provides the first comprehensive analysis of well-posedness and invariant measures for the stochastic Landau-Lifshitz-Baryakhtar equation across different dimensions and initial data conditions.

Findings

Unique local solutions in 1D, 2D, and 3D for initial data in H^1.

Global solutions and invariant measures for small initial data in 1D.

Existence of martingale solutions in 3D due to non-uniqueness.

Abstract

In this paper, we study the initial-boundary value problem for the stochastic Landau-Lifshitz-Baryakhtar (SLLBar) equation with Stratonovich-type noise in bounded domains $\mathcal{O}\subset\mathbb{R}^d$, $d=1,2,3$. Our main results can be briefly described as follows: (1) for $d=1,2,3$ and any $\mathbf{u}_0\in\mathbb{H}^1$, the SLLBar equation admits a unique local-in-time pathwise weak solution; (2) for $d=1$ and small-data $\mathbf{u}_0\in\mathbb{H}^1$, the SLLBar equation has a unique global-in-time pathwise weak solution and at least one invariant measure; (3) for $d=1,2$ and small-data $\mathbf{u}_0\in\mathbb{L}^2$, the SLLBar equation possesses a unique global-in-time pathwise very weak solution and at least one invariant measure, while for $d=3$ only the existence of martingale solution is obtained due to the loss of pathwise uniqueness.