Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation

TL;DR

This paper establishes the existence and examines the regularity of weak solutions to a stochastic version of the Landau-Lifshitz-Gilbert equation, which models ferromagnetic materials under random perturbations.

Contribution

It proves the existence of weak martingale solutions for the stochastic Landau-Lifshitz-Gilbert equation with space-dependent noise, including new regularity results even in the deterministic case.

Findings

Existence of weak martingale solutions in a sphere-valued setting

New regularity results for solutions, applicable to deterministic equations

Insights into stochastic perturbations of ferromagnetic models

Abstract

The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space-dependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere $\mathbb S^2$. The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert equation.