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

MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons20682

Andersson,  Lars
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1310.2664.pdf
(Preprint), 505KB

s0219891615500204.pdf
(beliebiger Volltext), 572KB

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Zitation

Andersson, L., & Blue, P. (2015). Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior. Journal of Hyperbolic Differential Equations, 12(4), 689-743. doi:10.1142/S0219891615500204.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-0029-007B-9
Zusammenfassung
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.