Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior

Abstract

We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant [Formula: see text]. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e.  that a spatially localized energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell–Ipser equation for one component. To treat the Fackerell–Ipser equation, we use a Fourier transform in [Formula: see text]. For the Fackerell–Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.