Struwe-like solutions for the Stochastic Harmonic Map Flow

TL;DR

This paper establishes well-posedness and uniqueness criteria for the two-dimensional stochastic harmonic map flow, extending deterministic Struwe solutions to the stochastic setting under trace-class noise conditions.

Contribution

It introduces strong solutions for the stochastic harmonic map flow, extending the concept of Struwe solutions to stochastic models with a new uniqueness criterion.

Findings

Constructed strong solutions in specified function spaces.

Extended deterministic Struwe solutions to stochastic case.

Provided a natural criterion for solution uniqueness.

Abstract

We give a new result on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau-Lifshitz-Gilbert model for thermal fluctuations in micromagnetics. We construct strong solutions that belong locally to the spaces $C([s,t);H^1)\cap L^2([s,t);H^2)$, $0\leq s<t\leq T$. It that sense, these maps are a counterpart of the so-called "Struwe solutions" of the deterministic model. We also give a natural criterion of uniqueness that extends A.\ Freire's Theorem to the stochastic case. Both results are obtained under the condition that the noise term has a trace-class covariance in space.