Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case

TL;DR

This paper proves the nonlinear stability of certain self-similar Yang-Mills blowup solutions in higher dimensions, extending previous results and providing a general framework for analyzing stability of self-similar solutions in semilinear heat equations.

Contribution

It establishes the nonlinear asymptotic stability of Weinkove's homothetically shrinking solitons in dimensions 5 to 9 and introduces a versatile method for stability analysis of self-similar solutions.

Findings

Proved stability of Weinkove's solutions in higher dimensions

Developed a general spectral stability framework

Extended stability results beyond dimension 5

Abstract

We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of the Weinkove solution under small equivariant perturbations and thus extend a result by the second author and Donninger for $d=5$ to higher dimensions. At the same time, we provide a general framework for proving stability of self-similar blowup solutions to a large class of semilinear heat equations in arbitrary space dimension $d \geq 3$, including a robust and simple method for solving the underlying spectral problems.