Quantized slow blowup dynamics for the energy-critical Schrödinger map flow
Abstract
We study the Cauchy problem for the energy-critical Schrödinger map into the 2-sphere: ∂tu = u×Δu, (t,x)∈ℝ1+2, u(t,x)∈𝕊2 for equivariant initial data of homotopy degree k=1. Given any integer L∈ℕ+, we exhibit a set of initial data arbitrarily close to the ground state harmonic map in the energy norm such that the corresponding solution blows up in finite time with a quantized rate λ(t)∼ ((T−t)L)/(|log(T−t)|2L/(2L−1)) as t↑ T. Our proof is inspired by the techniques developed for the energy-critical harmonic map heat flow in a similar context by Raphaël and Schweyer [Anal. PDE 7 (2014), 1713--1805]. However, its adaptation to a Schrödinger-type equation requires the introduction of a new energy-Virial-Morawetz Lyapunov functional tailored to quantized slow blowup dynamics.
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BibTeX
@misc{jendrej2026quantized,
title = {{Quantized slow blowup dynamics for the energy-critical Schrödinger map flow}},
author = {Jacek Jendrej and Jitao Xu and Xu Yuan and Lifeng Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20313},
eprint = {2610.20313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20313},
}
RIS
TY - UNPB
TI - Quantized slow blowup dynamics for the energy-critical Schrödinger map flow
AU - Jacek Jendrej
AU - Jitao Xu
AU - Xu Yuan
AU - Lifeng Zhao
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.20313
UR - https://arxiv.org/abs/2610.20313
AB - We study the Cauchy problem for the energy-critical Schrödinger map into the 2-sphere: ∂_tu = u×Δu, (t,x)∈ℝ^1+2, u(t,x)∈𝕊^2 for equivariant initial data of homotopy degree k=1. Given any integer L∈ℕ_+, we exhibit a set of initial data arbitrarily close to the ground state harmonic map in the energy norm such that the corresponding solution blows up in finite time with a quantized rate λ(t)∼ ((T−t)^L)/(|log(T−t)|^2L/(2L−1)) as t↑ T. Our proof is inspired by the techniques developed for the energy-critical harmonic map heat flow in a similar context by Raphaël and Schweyer [Anal. PDE 7 (2014), 1713--1805]. However, its adaptation to a Schrödinger-type equation requires the introduction of a new energy-Virial-Morawetz Lyapunov functional tailored to quantized slow blowup dynamics.
ER -