PreprintCC BY 4.0arXiv
Large-Data Global Criteria for Two-Dimensional Critical-Energy Schrödinger Maps
Abstract
We study large-data continuation below the critical energy for the two-dimensional sphere-valued Schrödinger map ∂t u=u×Δu, u:[0,T+)×ℝ2→𝕊2, E(u0)<4π. Working directly with the original equation through the complex tangential fields Wm=∂m u+i u×∂m u and the ambient connections Γα, we prove frequency-envelope bounds governed by the critical norm |∇ u|ℓ2L4t,x. As consequences, T+=∞ if |∇ u|ℓ2L4([0,T+)×ℝ2)<∞, and also if a terminal one-window high-frequency tail 𝒜win(t) satisfies liminft↑ T+𝒜win(t)<ε*. Thus any finite-time failure must force persistent high-frequency concentration on all sufficiently late critical parabolic windows.
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BibTeX
@misc{chung2026largedata,
title = {{Large-Data Global Criteria for Two-Dimensional Critical-Energy Schrödinger Maps}},
author = {Pang-Hung Chung and Dan Han},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02721},
eprint = {2610.02721},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02721},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large-Data Global Criteria for Two-Dimensional Critical-Energy Schrödinger Maps
AU - Pang-Hung Chung
AU - Dan Han
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02721
UR - https://arxiv.org/abs/2610.02721
AB - We study large-data continuation below the critical energy for the two-dimensional sphere-valued Schrödinger map ∂_t u=u×Δu, u:[0,T_+)×ℝ^2→𝕊^2, E(u_0)<4π. Working directly with the original equation through the complex tangential fields W_m=∂_m u+i u×∂_m u and the ambient connections Γ_α, we prove frequency-envelope bounds governed by the critical norm |∇ u|_ℓ^2L^4_t,x. As consequences, T_+=∞ if |∇ u|_ℓ^2L^4([0,T_+)×ℝ^2)<∞, and also if a terminal one-window high-frequency tail 𝒜_win(t) satisfies liminf_t↑ T_+𝒜_win(t)<ε_*. Thus any finite-time failure must force persistent high-frequency concentration on all sufficiently late critical parabolic windows.
ER -