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

JOURNAL OF GEOMETRY AND PHYSICS

卷:123

Article

Ghanem, Sari¹  Hafner, Dietrich¹ 

[1] Univ Grenoble Alpes, Inst Fourier, 100 Rue Math, F-38610 Gieres, France

关键词: Schwarzschild black hole;    Yang-Mills equation;    Local energy decay;   

DOI  :  10.1016/j.geomphys.2017.09.005

来源: Elsevier

【 摘 要 】

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 do not 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. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free