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 ) ∩ L 2 ([s, t); H 2 ), 0 ≤ s < t ≤ 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.

A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations

“…with µ = 0). This path was followed by one of the present authors in [34,35], leading to a two-dimensional theory which carries some similarities with the deterministic model studied by Struwe in the 80's ( [53]). In [35], it was shown for d = 2 that equivariant solutions can blow-up in finite time, and that the probability of such an event is positive provided the noise has a certain symmetry properties.…”

Finite-time singularity of the stochastic harmonic map flow

“…It is natural to expect that the solution constructed above actually lives in H 2 , up to the singular time. It can be shown, according to a bootstrap argument (see [30] for details in a slightly different setting), that provided sup t∈[0,σ) |∆u(t)| L 2 < ∞ for some stopping time σ ∈ (0, T ], then u| [0,σ] has arbitrary regularity in space (with respect to what is allowed by the data u 0 , φ). Consequently, in (1.19), blow-up happens also for β * = 2.…”

Existence, uniqueness and regularity for the stochastic Ericksen-Leslie equation

“…Strategy of the proof of Theorem 1.2. The existence part generalizes arguments for the deterministic case from [19,15,24] to the SPDE (SEL), and benefits from [14]. As it turns out, solutions are arbitrary regular in the space-like variable (as permitted by the data), as could be easily seen by a higher order generalization of Theorem 5.1.…”

“…In addition, we will see that the solutions of (SEL) are unique under some integrability property which, to the best of our knowledge, is new even in the deterministic context (it suffices to let ν = 0 in (SEL)). Thanks to a classical interpolation inequality (see (2.1)), the partially regular solutions constructed above fulfill (correspondingly) the integrability condition (1.16), which implies in turn that these solutions are unique in their class (as in the case of the stochastic harmonic map flow in [14]).…”

“…We are interested in a special class of local strong solutions, which is uniquely caracterized by the "forward bubbling" property. In [14], these were referred to as"Struwe solutions", in analogy with the deterministic theory in [30].…”