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On the formation of singularities in the critical $O(3)$ $\sigma $-model. (English) Zbl 1213.35392
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.

35Q75PDEs in connection with relativity and gravitational theory
35B35Stability of solutions of PDE
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