Maxwell and Yang-Mills Equations on Curved Black Hole Space-Times

Abstract : In this thesis, we take a systematic study of global regularity of the Maxwell equations and of the Yang-Mills equations on curved black hole space-times. In the first chapter, we write the proof of the non-blow up of the Yang-Mills curvature on arbitrary curved space-times using the Klainerman-Rodnianski parametrix combined with suitable Grönwall type inequalities. While the Chruściel-Shatah argument requires a control on two derivatives of the Yang-Mills curvature, we can get away by controlling only one derivative instead, and write a new gauge independent proof on arbitrary, fixed, sufficiently smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds. In it's sequel, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We show that if we assume that the middle components of the non-stationary solutions of the Maxwell equations verify a Morawetz type estimate supported on a compact region in space around the trapped surface, then we can prove uniform decay properties for the components of the Maxwell fields in the entire exterior of the Schwarzschild black hole, including the event horizon, by making only use of Sobolev inequalities combined with energy estimates using the Maxwell equations directly. This proof does not pass through the scalar wave equation on the Schwarzschild black hole, does not need to separate the middle components for the Maxwell fields, and would then be in particular useful for the non-abelian case of the Yang-Mills equations where the separation of the middle components cannot occur. The last chapter is an opening to different problems in partial differential equations.
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Sari Ghanem. Maxwell and Yang-Mills Equations on Curved Black Hole Space-Times. Mathematics [math]. Université Paris Diderot - Paris 7, Institut de Mathématiques de Jussieu, 2014. English. ⟨tel-01101133⟩

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