Renormalization and blow up for the critical Yang-Mills problem

TL;DR

This paper constructs finite-time blow-up solutions for the energy-critical Yang-Mills equations in 4+1 dimensions, using a novel rescaling approach that involves a continuum of rates modified by logarithmic factors.

Contribution

It introduces a new family of blow-up solutions for the critical Yang-Mills problem with a continuum of rates, including logarithmic corrections, based on a modified self-similar ansatz.

Findings

Existence of blow-up solutions with rates exceeding 3/2

Blow-up solutions involve a logarithmic modification of the self-similar rate

The linearized operator has a zero energy eigenvalue, affecting blow-up dynamics

Abstract

We consider the Yangs-Mills equations in 4+1 dimensions. This is the energy critical case and we show that it admits a family of solutions which blow up in finite time. They are obtained by the spherically symmetric ansatz in the SO(4) gauge group and result by rescaling of the instanton solution. The rescaling is done via a prescribed rate which in this case is a modification of the self-similar rate by a power of |log t|. The powers themselves take any value exceeding 3/2 and thus form a continuum of distinct rates leading to blow-up. The methods are related to the authors' previous work on wave maps and the energy critical semi-linear equation. However, in contrast to these equations, the linearized Yang-Mills operator (around an instanton) exhibits a zero energy eigenvalue rather than a resonance. This turns out to have far-reaching consequences, amongst which are a completely different family of rates leading to blow-up (logarithmic rather than polynomial corrections to the self-similar rate).