Energy estimates for the Einstein-Yang-Mills fields and applications

Abstract

We prove exterior energy estimates for tensorial non-linear wave equations, where the background metric is a perturbation of the Minkowski space-time, and where the derivatives are the Minkowski covariant derivatives. We obtain bounds in the exterior region of the Minkowski space-time, for the weighted $L^2$ norm on each component, separately, of the covariant derivative of the tensorial solutions, and we also control a space-time integral in the exterior of the covariant tangential derivatives of the solutions. As a special application, we use here these energy estimates to prove the exterior stability of the Minkowski space-time, $\mathbb{R}^{1+4}$, as solution to the coupled Einstein-Yang-Mills system associated to any compact Lie group $G$, in the Lorenz gauge and in wave coordinates. The bounds in the exterior for the $L^2$ norm on the covariant derivatives of each component, separately, of the tensor solution, as well as the bound on the space-time integral of the covariant tangential derivatives, are motivated by a problem that we will address in a paper that follows to prove the exterior stability of the $(1+3)$-Minkowski space-time for perturbations governed by the Einstein-Yang-Mills equations.