Pathwise Solvability and Bubbling in 2D Stochastic Landau-Lifshitz-Gilbert Equations

Ben Goldys, Chunxi Jiao and Christof Melcher

Abstract

We investigate the stochastic Landau-Lifshitz-Gilbert (LLG) equation on a periodic 2D domain, driven by infinite-dimensional Gaussian noise in a Sobolev class. We establish strong local well-posedness in the energy space and characterize blow-up at random times in terms of energy concentration at small scales (bubbling). By iteration, we construct pathwise global weak solutions, with energy evolving as a càdlàg process, and prove uniqueness within this class. These results offer a stochastic counterpart to the deterministic concept of Struwe solutions. The approach relies on a transformation that leads to a magnetic Landau-Lifshitz-Gilbert equation with random gauge coefficients.

Keywords: Stochastic Landau-Lifshitz equation, Multiplicative noise, Struwe solutions, Bubbling, Doss-Sussman type transformation.

AMS Subject Classification: Primary 60H15; secondary 35A02, 35A21, 35K59, 35Q60.