Global regularity of wave maps VII. Control of delocalised or dispersed solutions

TL;DR

This paper completes the proof of global regularity for 2D wave maps into hyperbolic space by developing perturbation theory and solution synthesis methods for dispersed data, leading to spacetime bounds and scattering results.

Contribution

It introduces a divisible perturbation theory and a solution synthesis approach for dispersed data, finalizing the series' goal of proving global regularity.

Findings

Established perturbation theory for dispersed wave map data

Synthesized solutions from smaller energy components

Proved spacetime bounds and scattering for wave maps

Abstract

This is the final paper in the series \cite{tao:heatwave}, \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} that establishes global regularity for two-dimensional wave maps into hyperbolic targets. In this paper we establish the remaining claims required for this statement, namely a divisible perturbation theory, and a means of synthesising solutions for frequency-delocalised, spatially-dispersed, or spatially-delocalised data out of solutions of strictly smaller energy. As a consequence of the perturbation theory here and the results obtained earlier in the series, we also establish spacetime bounds and scattering properties of wave maps into hyperbolic space.