On the formation of singularities in the critical $O(3)$ $\sigma$-model

Abstract

We study the phenomena of energy concentration for the critical $O(3)$ sigma model, also known as the wave map flow from $\mathbb{R}^{2+1}$ Minkowski space into the sphere $\mathbb{S}^2$. We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis are done in the context of the $k$-equivariant symmetry reduction, and we restrict to maps with homotopy class $k\geqslant 4$. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) in that it persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.