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

2016/04/15 by Sari Ghanem, Ghanem, Sari, Dietrich Häfner +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)

paper · pdf · doi:10.48550/arxiv.1604.04477

openalex publication_date 2016/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞ norm of the middle components in the entire exterior of the Schwarzschild black hole, including the event horizon.

Citations

• On uniform decay of the Maxwell fields on black hole space-times

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