### 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 language | English (US) |
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Pages (from-to) | 187-242 |

Number of pages | 56 |

Journal | Annals of Mathematics |

Volume | 172 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Mathematics*,

*172*(1), 187-242. https://doi.org/10.4007/annals.2010.172.187

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*Annals of Mathematics*, vol. 172, no. 1, pp. 187-242. https://doi.org/10.4007/annals.2010.172.187

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the formation of singularities in the critical 0(3) σ-model

AU - Rodnianski, Igor

AU - Sterbenz, Jacob

PY - 2010/7/1

Y1 - 2010/7/1

N2 - 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.

AB - 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.

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U2 - https://doi.org/10.4007/annals.2010.172.187

DO - https://doi.org/10.4007/annals.2010.172.187

M3 - Article

VL - 172

SP - 187

EP - 242

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -