Struwe-like solutions for the Stochastic Harmonic Map flow

Abstract

We give new results 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 first construct strong solutions that belong locally to the spaces $$C([s,t);H^1)\cap L^2([s,t);H^2)$$ , $$0\le s<t\le T$$ . It that sense, these maps are a counterpart of the so-called “Struwe solutions” of the deterministic model. We then provide 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.