The decay of the $SU(2)$ Yang-Mills fields on the Schwarzschild black hole for spherically symmetric small energy initial data

TL;DR

This paper proves that small, spherically symmetric Yang-Mills fields in the exterior of a Schwarzschild black hole decay uniformly over time, ruling out non-decaying stationary solutions.

Contribution

It establishes decay estimates for Yang-Mills fields on Schwarzschild spacetime under spherical symmetry and small energy assumptions, extending previous methods to this setting.

Findings

Proved Morawetz type decay estimates for Yang-Mills fields.

Established local energy decay and uniform decay of field components.

Showed absence of non-decaying Coulomb solutions under the given Ansatz.

Abstract

We prove uniform decay estimates in the entire exterior of the Schwarzschild black hole for gauge invariant norms on the Yang-Mills fields valued in the Lie algebra associated to the Lie group $SU(2)$. We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. In particular, there don't exist any Coulomb type solutions satisfying this Ansatz. We first prove a Morawetz type estimate for the Yang-Mills fields within this setting, using the Yang-Mills equations directly. We then adapt the proof constructed in previous work by the first author to show local energy decay and uniform decay of the $L^{\infty}$ norm of the middle components in the entire exterior of the Schwarzschild black hole, including the event horizon.