Stable blow-up solutions for the $SO(d)$-equivariant supercritical Yang-Mills heat flow

TL;DR

This paper constructs stable, finite-time blow-up solutions for the supercritical $SO(d)$-equivariant Yang-Mills heat flow in high dimensions, revealing quantized blow-up rates and stability properties.

Contribution

It introduces a family of smooth solutions exhibiting finite-time blow-up with quantized rates and analyzes their stability in high-dimensional supercritical settings.

Findings

Constructed smooth blow-up solutions with universal profiles.

Identified quantized blow-up rates depending on dimension.

Proved stability of solutions under initial data perturbations.

Abstract

We consider the $SO(d)$-equivariant Yang-Mills heat flow \begin{equation*} \partial_t u-\partial_r^2 u-\frac{(d-3)}{r}\partial_r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{equation*} in dimensions $d>10.$ We construct a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of a universal profile \begin{equation*} u(t,r)\sim Q\left(\frac{r}{\lambda(t)}\right), \end{equation*}where $Q$ is a stationary state of the equation and the blow-up rates are quantized by \begin{equation*} \lambda(t)\sim c_{u}(T-t)^{\frac{l}{\gamma}},\,\,\,l\,\,\,\text{is any positive integer},\,\,\,\gamma=\gamma(d)=\frac{d-4-\sqrt{(d-6)^2-12}}{2}. \end{equation*} Moreover, such solutions are in fact $(l-1)$-codimension stable under pertubation of the initial data.