Hyperboloidal similarity coordinates and a globally stable blowup profile for supercritical wave maps

TL;DR

This paper introduces hyperboloidal similarity coordinates to analyze supercritical wave maps, demonstrating the stability of explicit blowup solutions and enabling continuation beyond singularity in a novel coordinate framework.

Contribution

It develops a new coordinate system for tracking wave map evolution through blowup, proving stability and continuation results in the supercritical setting.

Findings

Proves asymptotic stability of explicit blowup solutions.

Introduces a coordinate system for tracking evolution past blowup.

Establishes continuation beyond singularity.

Abstract

We consider co-rotational wave maps from (1+3)-dimensional Minkowski space into the three-sphere. This model exhibits an explicit blowup solution and we prove the asymptotic nonlinear stability of this solution in the whole space under small perturbations of the initial data. The key ingredient is the introduction of a novel coordinate system that allows one to track the evolution past the blowup time and almost up to the Cauchy horizon of the singularity. As a consequence, we also obtain a result on continuation beyond blowup.