Tail-Driven Multi-Bubble Dynamics for the 2-Equivariant Harmonic-Map Heat Flow

Abstract

We study the late-time dynamics of global finite-energy 2-equivariant harmonic-map heat flow from R² to S². Building on the continuous multi-bubble decomposition of Jendrej–Lawrie and a preceding scalar-sign rigidity and bubble-number selection result, we establish a triangular asymptotic mechanism specific to the critical D=2 regime. The outermost bubble scale is driven by critical far-field memory through an m=2 Gustafson–Nakanishi–Tsai-type normal-form mechanism. In the scalar Hasimoto convention q(v)=v_r−2 sin(v)/r, the resulting outer coefficient is −1/π. In contrast, every inner logarithmic gap satisfies a universal recursive clock law with asymptotic coefficient 16/π. The proof combines a scale-uniform weighted Rellich estimate, global square-integrability of the bare multi-bubble tower tension, a full-profile Hasimoto radiation variable, a zero-moment critical bridge, finite-cutoff renormalization of the non-L² 1/r current, and a reduction of the exact multi-bubble radiation equation to the free l=1 flow before the final critical heat-kernel/Hankel extraction. The main theorem is stated for global solutions. The resulting critical D=2 dynamics have a triangular structure: far-field critical tail → outermost scale → recursive inner-scale cascade.