Finite-time singularity of the stochastic harmonic map flow

Abstract

We investigate the influence of an infinite dimensional Gaussian noise on the bubbling phenomenon for the stochastic harmonic map flow u(t, ·) : D 2 → S 2 , from the two-dimensional unit disc onto the sphere. The diffusion term is assumed to have range one pointwisely in the tangent space T u(t,x) S 2 , so that the noise preserves the 1-corotational symmetry of solutions. Under the assumption that its space-correlation is of trace class (in some appropriate Hilbert space), we prove that the noise generates blow-up with positive probability. This scenario happens no matter how we choose the initial data, provided it fulfills the latter symmetry assumption.

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

“…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. This scenario happens no matter how the initial datum is chosen, in contrast with the deterministic scenario where the initial datum determines which solution will explode.…”

“…(For spatially-discretized versions of LLG, this was rigorously established in [50], the generator in question being a multi-dimensional variant of the Laplace-Beltrami operator). See also the introduction in [35], for a related discussion.…”