On the formation of singularities in the critical 0(3) σ-model

Igor Rodnianski, Jacob Sterbenz

Research output: Contribution to journalArticle

58 Citations (Scopus)

Abstract

We study the phenomena of energy concentration for the critical 0(3) sigma model, also known as the wave map flow from [R{double-struck}2+1 Minkowski space into the sphere S{double-struck}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 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.

Original languageEnglish (US)
Pages (from-to)187-242
Number of pages56
JournalAnnals of Mathematics
Volume172
Issue number1
DOIs
StatePublished - Jul 1 2010

Fingerprint

Singularity
Self-focusing
Symmetry Reduction
Sigma Models
Harmonic Maps
Minkowski Space
Shrinking
Small Perturbations
Equivariant
Sobolev Spaces
Blow-up
Homotopy
Energy
Model
Context
Class
Symmetry
Perturbation

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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On the formation of singularities in the critical 0(3) σ-model. / Rodnianski, Igor; Sterbenz, Jacob.

In: Annals of Mathematics, Vol. 172, No. 1, 01.07.2010, p. 187-242.

Research output: Contribution to journalArticle

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