On blowup for Yang-Mills fields

TL;DR

This paper investigates the formation of singularities in spherically symmetric Yang-Mills equations in 4 and 5 dimensions, revealing dimension-dependent blowup mechanisms through combined numerical and analytical methods.

Contribution

It provides the first detailed analysis of blowup behavior in critical and supercritical dimensions for Yang-Mills fields, highlighting different self-similar structures.

Findings

Solutions blow up in finite time for large initial data

In 5D, blowup is exactly self-similar

In 4D, blowup is approximately self-similar and involves adiabatic shrinking

Abstract

We study development of singularities for the spherically symmetric Yang-Mills equations in $d+1$ dimensional Minkowski spacetime for $d=4$ (the critical dimension) and $d=5$ (the lowest supercritical dimension). Using combined numerical and analytical methods we show in both cases that generic solutions starting with sufficiently large initial data blow up in finite time. The mechanism of singularity formation depends on the dimension: in $d=5$ the blowup is exactly self-similar while in $d=4$ the blowup is only approximately self-similar and can be viewed as the adiabatic shrinking of the marginally stable static solution. The threshold for blowup and the connection with critical phenomena in the gravitational collapse (which motivated this research) are also briefly discussed.