The stochastic LandauLifshitzBaryakhtar equation: Global solution and invariant measure
Abstract
The LandauLifshitzBaryakhtar (LLBar) equation perturbed by a spacedependent noise 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, longrange interactions, and noiseinduced phenomena at high temperatures. In this paper, we show the existence of a martingale 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 pathwise uniqueness of the solution, which implies the existence of a unique probabilistically strong solution. Finally, we show the Feller property of the Markov semigroup associated with the strong solution, which implies the existence of invariant measures.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.14112
 arXiv:
 arXiv:2405.14112
 Bibcode:
 2024arXiv240514112G
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs;
 60H15