The decay of Maxwell and Yang-Mills fields on a black hole
First, I will introduce the Yang-Mills equations and present their relation to the Einstein vacuum equations. I will then explain briefly the vector field method and introduce the energy estimates. Thereafter, I will expose the gauge independent proof of the global non-blow-up of the Yang-Mills curvature, valued in any Lie algebra, on arbitrary, fixed, sufficiently smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds. Then, I will start motivating the far more challenging problem which is that of proving uniform bounds for the Yang-Mills fields on space-times admitting a black hole. For this, I will introduce the Schwarzschild black hole and then present a new proof of decay for the Maxwell fields which does not pass through the decoupling of the middle-components, assuming a certain Morawetz type estimate for the middle components. Finally, I will present a joint work with Dietrich Häfner, that consists in proving decay for spherically symmetric SU(2) Yang-Mills fields in the domain of outer-communication of the Schwarzschild black hole. This is done by proving in this setting, a Morawetz type estimate that is stronger than the one assumed in my previous work. We then prove uniform decay estimates in the entire exterior region of the black hole, including the event horizon, for gauge invariant norms on the Yang-Mills curvature generated from such initial data, including the pointwise norm of the so-called middle components.