2006/04/30 by Igor Rodnianski, Jacob Sterbenz, Rodnianski, Igor +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods for differential equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math/0605023
openalex publication_date 2006/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from R2+1 Minkowski space into the sphere S2. 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.