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 $t$. 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 localised 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 $t$. For the Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.

50 pages. v3 minor typographical changes

References in corpus (4)

• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• Errata for ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'', ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'', and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation''
• Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem
• Local energy decay for Maxwell fields part I: Spherically symmetric black-hole backgrounds

Cited by in corpus (9)

• Superradiance on the Reissner-Nordstrom metric
• Pointwise decay for the Maxwell field on black hole space-times
• Linear Stability of the Non-Extreme Kerr Black Hole
• Asymptotics for the wave equation on differential forms on Kerr-de Sitter space
• A geometric description of Maxwell field in a Kerr spacetime
• Asymptotic completeness for superradiant Klein-Gordon equations and applications to the De Sitter Kerr metric
• Refined Error Estimates for the Riccati Equation with Applications to the Angular Teukolsky Equation
• Local energy decay for Maxwell fields part I: Spherically symmetric black-hole backgrounds
• Lectures on Linear Stability of Rotating Black Holes