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

We study the blow-up dynamics for the energy-critical 1-corotational wave map problem with target the 2-sphere.Raphal and Rodnianski (Publ.Math.Inst.Hautes tudes Sci.115 (2012), 1-122) exhibited 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 Raphal and Schweyer (Anal.PDE 7:8 (2014), 1713-1805), who exhibited a similar construction of the quantized blow-up rates for the harmonic map heat flow.The main difficulty in our wave map 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.1. Introduction 2415 2. Construction of the approximate solution 2425 3. The trapped solutions 2447 4. Proof of the main theorem 2465 Appendix A. Coercive properties 2469 Appendix B. Interpolation estimates 2472 Appendix C. Leibniz rule for A k 2473 Appendix D. Monotonicity for the intermediate energy 2474