Quantized slow blow-up dynamics for the energy-critical corotational wave maps problem

Abstract

We study the blow-up dynamics for the energy-critical 1-corotational wave maps problem with 2-sphere target. In arXiv:0911.0692, Raphaël and Rodnianski exhibited a stable finite time blow-up dynamics arising from smooth initial data. In this paper, we exhibit a sequence of new finite-time blow-up rates (quantized rates), which can still arise from well-localized smooth initial data. We closely follow the strategy of the paper arXiv:1301.1859 by Raphaël and Schweyer, who exhibited a similar construction of the quantized blow-up rates for the harmonic map heat flow. The main difficulty in our wave maps setting stems from the lack of dissipation and its critical nature, which we overcome by a systematic identification of correction terms in higher-order energy estimates.

arXiv comment61 pages, to appear in Anal. PDE

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• On stability of blow up solutions for the critical co-rotational Wave Maps problem
• A refined threshold theorem for (1+2)-dimensional wave maps into surfaces
• Sharp universal rate for stable blow-up of corotational wave maps