On stable self-similar blowup for corotational wave maps and equivariant Yang-Mills connections

TL;DR

This paper proves the nonlinear stability of explicit self-similar blowup solutions in corotational wave maps and equivariant Yang-Mills equations, using a novel functional analytic framework in similarity coordinates.

Contribution

It introduces a new analytical framework to study the stability of self-similar blowup solutions in energy-supercritical models.

Findings

Stable self-similar blowup solutions are proven to be asymptotically stable.

The framework applies to models in all energy-supercritical dimensions.

The approach allows evolution of solutions near blowup in regions approaching the light cone.

Abstract

We consider corotational wave maps from Minkowski spacetime into the sphere and the equivariant Yang-Mills equation for all energy-supercritical dimensions. Both models have explicit self-similar finite time blowup solutions, which continue to exist even past the singularity. We prove the nonlinear asymptotic stability of these solutions in spacetime regions that approach the future light cone of the singularity. For this, we develop a general functional analytic framework in adapted similarity coordinates that allows to evolve the stable wave flow near a self-similar blowup solution in such spacetime regions.