Global, Non-Scattering Solutions to the Energy Critical Yang-Mills Problem

TL;DR

This paper constructs global solutions to a reduced scalar wave equation from Yang-Mills theory, featuring solitons with asymptotically constant scale coupled to large radiation, including cases with minimal decay assumptions.

Contribution

It introduces a novel method to construct solutions with solitons and radiation without assuming fixed soliton scale, extending previous wave map techniques.

Findings

Existence of solutions with solitons and large radiation

Construction of solutions with minimal decay assumptions

One-parameter families of solutions for various soliton scales

Abstract

We consider the Yang-Mills problem on $\mathbb{R}^{1+4}$ with gauge group $SO(4)$. In an appropriate equivariant reduction, this Yang-Mills problem reduces to a single scalar semilinear wave equation. This semilinear equation admits a one-parameter family of solitons, each of which is a re-scaling of a fixed solution. In this work, we construct a class of solutions, each of which consists of a soliton whose length scale is asymptotically constant, coupled to large radiation, plus corrections which slowly decay to zero in the energy norm. Our class of solutions includes ones for which the radiation component is only "logarithmically" better than energy class. As such, the solutions are not constructed by apriori assuming the length scale to be constant. Instead, we use an approach similar to a previous work of the author regarding wave maps. In the setup of this work, the soliton length scale asymptoting to a constant is a necessary condition for the radiation profile to have finite energy. An interesting point of our construction is that, for each radiation profile, there exist one-parameter families of solutions consisting of the radiation profile coupled to a soliton, which has any asymptotic value of the length scale.