Finite-time singularity of the stochastic harmonic map flow

TL;DR

This paper demonstrates that infinite-dimensional Gaussian noise can induce finite-time blow-up in the stochastic harmonic map flow from a disk to a sphere, regardless of initial data, under certain symmetry and correlation conditions.

Contribution

It proves that trace-class Gaussian noise can cause finite-time singularities in the stochastic harmonic map flow while preserving symmetry.

Findings

Noise induces blow-up with positive probability

Blow-up occurs regardless of initial data under symmetry

Noise preserves 1-corotational symmetry

Abstract

We investigate the influence of an infinite dimensional Gaussian noise on the bubbling phenomenon for the stochastic harmonic map flow $u(t,\cdot ):\mathbb{D}^2\to\mathbb{S}^2$, from the two-dimensional unit disc onto the sphere. The diffusion term is assumed to have range one pointwisely in the tangent space $T_{u(t,x)}\mathbb{S}^2$, 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.