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 \[ \partial_t u=u\timesΔu, \qquad u:[0,T_+)\times\mathbb{R}^2\to\mathbb{S}^2, \qquad E(u_0)<4π. \] Working directly with the original equation through the complex tangential fields $W_m={\partial }_{m}u+\mathrm{i}u\times {\partial }_{m}u$ and the ambient connections ${\Gamma }_{\alpha }$, we prove frequency-envelope bounds governed by the critical norm $\Vert \nabla u{\Vert }_{{\ell }^2{L}_{t,x}^4}$. As consequences, $T_{+}=\infty$ if $\Vert \nabla u{\Vert }_{{\ell }^2{L}^4([0,T_{+})\times {\mathbb{R}}^2)}<\infty$, and also if a terminal one-window high-frequency tail ${\mathcal{A}}_{\mathrm{win}}(t)$ satisfies ${\mathrm{liminf}}_{t\uparrow T_{+}}{\mathcal{A}}_{\mathrm{win}}(t)<{\epsilon }_{\ast }$. Thus any finite-time failure must force persistent high-frequency concentration on all sufficiently late critical parabolic windows.