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, Rapha\"el 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 Rapha\"el 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.