Stable blowup for the supercritical Yang-Mills heat flow

TL;DR

This paper proves the nonlinear stability of a self-similar blowup solution for the supercritical Yang-Mills heat flow in five dimensions, demonstrating finite-time blowup and convergence to the blowup profile under small perturbations.

Contribution

It establishes the nonlinear asymptotic stability of an explicit self-similar blowup solution for the Yang-Mills heat flow in a supercritical setting.

Findings

Solutions blow up in finite time under certain initial conditions.

Blowup solutions converge to a self-similar profile in Sobolev and $L^{ty}$ norms.

Stability holds under small perturbations in a suitable topology.

Abstract

In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.