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

Uihyeon Jeong

TL;DR

This work advances the theory of energy-critical wave maps by constructing a family of finite-time blow-up solutions with quantized rates in the 1-corotational setting. Building on modulational techniques, the authors design an approximate blow-up profile $\boldsymbol{Q}_b$ with a high-dimensional modulation vector $b=(b_1,\dots,b_L)$, and derive a coupled system of ODEs for the modulations that selects slow, quantized decay modes. To overcome the lack of dissipation, they introduce radiation corrections and a higher-order Lyapunov functional $\mathcal{E}_{L+1}$ tied to a repulsive conjugated Hamiltonian, enabling monotonic control of the error through a bootstrap argument and a Brouwer fixed-point step. The main result shows the existence of smooth initial data yielding blow-up at time $T$ with rate $\lambda(t) \sim c (T-t)^{\ell} / |\log(T-t)|^{\ell/(\ell-1)}$, and with the radiative part gaining higher regularity, highlighting a robust mechanism for quantized rate blow-up in energy-critical dispersive equations. These methods connect dispersive wave maps to the quantized blow-up phenomena known in parabolic contexts and provide a framework for analyzing unstable directions via codimension-$\ell-1$ modulations and radiation corrections.

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.

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

TL;DR

This work advances the theory of energy-critical wave maps by constructing a family of finite-time blow-up solutions with quantized rates in the 1-corotational setting. Building on modulational techniques, the authors design an approximate blow-up profile with a high-dimensional modulation vector , and derive a coupled system of ODEs for the modulations that selects slow, quantized decay modes. To overcome the lack of dissipation, they introduce radiation corrections and a higher-order Lyapunov functional tied to a repulsive conjugated Hamiltonian, enabling monotonic control of the error through a bootstrap argument and a Brouwer fixed-point step. The main result shows the existence of smooth initial data yielding blow-up at time with rate , and with the radiative part gaining higher regularity, highlighting a robust mechanism for quantized rate blow-up in energy-critical dispersive equations. These methods connect dispersive wave maps to the quantized blow-up phenomena known in parabolic contexts and provide a framework for analyzing unstable directions via codimension- modulations and radiation corrections.

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.
Paper Structure (31 sections, 27 theorems, 424 equations)

This paper contains 31 sections, 27 theorems, 424 equations.

Table of Contents

  1. Introduction
  2. Wave map problem
  3. Corotational symmetry
  4. Harmonic map
  5. Blow-up near $Q$
  6. Main theorem
  7. Notation
  8. Strategy of the proof
  9. Construction of the approximate solution
  10. The linearized dynamics
  11. Admissible functions
  12. $b_1$-admissible functions
  13. Control of the extra growth
  14. Adapted norms of $b_1$ admissible functions
  15. Approximate blow-up profiles
  16. ...and 16 more sections

Key Result

Theorem 1.1

There exists a constant $\varepsilon_0>0$ such that for all 1-corotational initial data $(u_0,\dot{u}_0)$ with the corresponding solutions to eq:wm1 blow up in finite time $0<T=T(u_0,\dot{u}_0)<\infty$ as follows: for some $(u^*,\dot{u}^*) \in \mathcal{H}$, with the universal blow up speed: Here, $\mathcal{H}$, $\mathcal{H}^2$ are given by def:H sobolev, def:H2 sobolev.

Theorems & Definitions (73)

  • Theorem 1.1: Stable blow-up for 1-corotational wave maps RaphaelRodnianski2012IHESKim2023CMP
  • Remark 1.1: $1$-corotational symmetry
  • Theorem 1.2: Quantized blow-up for 1-corotational wave maps
  • Remark 1.2: Further regularity of asymptotic profile
  • Remark 1.3: Quantized blow-up
  • Remark 1.4: Instability of blow-up
  • Definition 2.1: Admissible functions
  • Remark 2.1
  • Lemma 2.2: Action of $\boldsymbol{H}$ and $\boldsymbol{H}^{-1}$ on admissible functions
  • proof
  • ...and 63 more