Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system

TL;DR

This paper analyzes the long-term behavior of solutions to the 2D corotational wave map system and 4D radially symmetric Yang-Mills equations with sub-critical energy, showing they either are static harmonic maps or scatter over time.

Contribution

It establishes the asymptotic scattering behavior for solutions with energy below or equal to the minimal harmonic map energy in these systems.

Findings

Solutions with minimal energy harmonic maps are static.

Solutions with less than minimal energy harmonic maps scatter over time.

The paper characterizes the dichotomy between static solutions and scattering.

Abstract

We describe the asymptotic behavior as time goes to infinity of solutions of the 2 dimensional corotational wave map system and of solutions to the 4 dimensional, radially symmetric Yang-Mills equation, in the critical energy space, with data of energy smaller than or equal to a harmonic map of minimal energy. An alternative holds: either the data is the harmonic map and the soltuion is constant in time, or the solution scatters in infinite time.