Extension principles for the Einstein Yang--Mills system

We prove the local existence theorem and establish an extension principle for the spherically symmetric Einstein Yang–Mills system (SSEYM) with $H^1$ data. This in addition implies Cauchy stability for the system. In contrast to a massless scalar field, the purely magnetic Yang–Mills field in spherical symmetry satisfies a wave-type equation with a singular potential. As a consequence, the proof of Christodoulou [10], based on an $L^{\mathrm{\infty }}-L^{\mathrm{\infty }}$ estimate, fails in the Yang–Mills context. Instead, we employ an $L^2$-based method, which is valid for both massless and massive scalar matter fields as well.