A globally stable self-similar blowup profile in energy supercritical Yang-Mills theory

TL;DR

This paper proves the stability of a self-similar blowup solution in energy-supercritical Yang-Mills equations, using hyperboloidal similarity coordinates to analyze the nonlinear stability beyond the singularity.

Contribution

It introduces a stability proof for the self-similar blowup profile in energy-supercritical Yang-Mills theory using hyperboloidal similarity coordinates.

Findings

The self-similar blowup profile is stable under small perturbations.

Growth estimates for free wave evolution are systematically constructed for odd dimensions.

Nonlinear stability beyond the singularity is established.

Abstract

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.