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On the Formation of Singularities in the Critical O(3) Sigma-Model

Igor Rodnianski·Jacob Sterbenz
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Abstract

We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from R^{2+1} Minkowski space into the sphere 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 is done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k>3. 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) and persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.

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Igor Rodnianski, Jacob Sterbenz. 2008-08-22. On the Formation of Singularities in the Critical O(3) Sigma-Model. https://arxiv.org/abs/math/0605023

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