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

Jeong, Uihyeon

doi:10.2140/apde.2025.18.2415

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

Uihyeon Jeong

Vol. 18 (2025), No. 10, 2415–2480

DOI: 10.2140/apde.2025.18.2415

Abstract

We study the blow-up dynamics for the energy-critical 1-corotational wave map problem with target the $2$ -sphere. Raphaël and Rodnianski (Publ. Math. Inst. Hautes Études 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 Raphaël 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.

Keywords

wave maps, blow-up, corotational symmetry, harmonic map

Mathematical Subject Classification

Primary: 35B44

Secondary: 35L71, 37K40, 58E20

Milestones

Received: 26 December 2023

Revised: 21 August 2024

Accepted: 4 November 2024

Published: 4 November 2025

Authors

Uihyeon Jeong
Department of Mathemetical Sciences Korea Advanced Institute of Science and Technology Daejeon South Korea

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Vol. 18, No. 10, 2025