Global oscillatory solutions for the Yang-Mills heat flow

TL;DR

This paper studies the long-term behavior of solutions to the Yang-Mills heat flow in four dimensions, revealing for the first time the existence of solutions with oscillatory asymptotic behavior as time approaches infinity.

Contribution

It demonstrates the existence of oscillatory solutions in the Yang-Mills heat flow, expanding understanding of possible long-time dynamics beyond blow-up and blow-down.

Findings

Global solutions can exhibit oscillatory behavior at infinity.

Long-time dynamics are determined by initial data with specific decay.

First example of oscillatory behavior in Yang-Mills heat flow.

Abstract

We investigate the long-time dynamics for the global solution of the $SO(4)$-equivariant Yang-Mills heat flow (YMHF) with structure group $SU(2)$ in space dimension $4$. For a class of initial data with specific decay at spatial infinity, we prove that the long-time dynamics of YMHF can be described by the initial data in a unified manner. As a consequence, the global solutions can exhibit blow-up, blow-down, and more exotically, {\it oscillatory} asymptotic behavior at time infinity. This seems to be the first example of Yang-Mills heat flows with oscillatory behavior as $t\to \infty$.