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

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.