Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions

TL;DR

This paper advances the understanding of wave map regularity by reducing the minimal-energy blowup problem to showing bounded entropy for certain classes of solutions, contributing to the proof of global regularity.

Contribution

It further reduces the minimal-energy blowup problem to entropy bounds for frequency and spatially delocalized solutions, progressing toward the full regularity proof.

Findings

Reduction of blowup problem to entropy bounds for specific solutions

Demonstration of bounded entropy for frequency-delocalized solutions

Progress towards proving global regularity of wave maps

Abstract

In the previous papers in this series, the global regularity conjecture for wave maps from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic space $\H^m$ was reduced to the problem of constructing a minimal-energy blowup solution which is almost periodic modulo symmetries in the event that the conjecture fails. In this paper, we show that this problem can be reduced further, to that of showing that solutions at the critical energy which are either frequency-delocalised, spatially-dispersed, or spatially-delocalised have bounded ``entropy''. These latter facts will be demonstrated in the final paper in this series.