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Zbl 1213.35392
Rodnianski, Igor; Sterbenz, Jacob
On the formation of singularities in the critical $O(3)$ $\sigma $-model. (English)
[J] Ann. Math. (2) 172, No. 1, 187-242 (2010). ISSN 0003-486X; ISSN 1939-8980/e

The authors study a catastrophic instability in the (2+1)-dimensional $O(3)$ sigma model (also known as the wave map flow from (2+1)-dimensional Minkowski space into the sphere $\mathbb S^2$). They establish rigorously and constructively the 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 the maps are restricted to homotopy class $k\geq 4$. The authors uncover an energy concentration mechanism that is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. It is shown 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.
[Helmut Rumpf (Wien)]

MSC 2000:: *35Q75 PDE in relativity
35B35 Stability of solutions of PDE

Keywords: wave map; formation of singularities; energy concentration; self-similarity

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