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

By Igor Rodnianski, Jacob Sterbenz. 2006

157 citations recorded by the source corpus.

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 are done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k ≥ 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.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6f25669efc3c8f2f788c3a83bd8528df95e7e8d9

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In Mathematical Physics

In Nuclear and High Energy Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Asymptotic decomposition for semilinear Wave and equivariant wave map equations (2015, methodology): “Indeed, singular solutions in the form of a shrinking soliton plus a residue term have been constructed for the energy critical wave equation in [28] for 3 + 1 dimensional case and in [20] for the 4 + 1 dimensional case, for the 2 + 1 equivariant dimensional wave map equation in [26, 30], and for th”
Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton (2014, methodology): “For the energy critical wave map problem, after the work [42], a complete description of a generic finite-time blow-up dynamics (log correction to the self-similar speed) was given by Raphaël and Rodnianski [39], while unstable regimes with different speeds were constructed by Krieger, Schlag and Ta”
Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation (2009, background): “[Str02, Str03], and the recent preprint of Sterbenz and Tataru [ST09] for the general case of solutions without any special invariant properties. We also refer to the works of Rodnianski and Sterbenz [RS], Krieger, Schlag and Tataru [KST08] and Rapha¨el and Rodnianski [RR] for the construction of bl”