Localized energy estimates of the wave equation on higher dimensional hyperspherical Schwarzschild spacetimes

The purpose of this dissertation is to discuss a robust way to measure dispersion of the linear wave equation on the (n+1)-dimensional Schwarzschild spacetime. One of the greater motivations for studying the higher dimensional Schwarzschild and Kerr spacetimes is to address the question of asymptotic stability of solutions to Einstein's equations. That is, if initial conditions are slightly perturbed, does the solution tend to the unperturbed solution. Even in the simplest case (Minkowski spacetime), establishing nonlinear stability proved to be highly nontrivial. This was originally shown by Christodoulou and Klainerman, and later simplified and generalized by Lindblad and Rodnianski, and Bieri and Zipser, respectively. In considering the Kerr solution, we ask whether solutions to small perturbations of Kerr initial data asymptotically approach perhaps a different member of the Kerr family. Decay estimates are fundamental tools in addressing this question and by studying the linear wave equation on Schwarzschild, we hope to gain some intuition in pursuing this problem. In this thesis we will determine localized energy estimates of the inhomogeneous wave equation box subscript g phi = F on the (n+1)-dimensional Schwarzschild manifold, for n is greater than or equal to 4. An inevitable loss in the estimate arises due to trapped rays on a surface known as the photon sphere. We then modify our technique and improve the estimate at this region

LOCALIZED ENERGY ESTIMATES FOR WAVE EQUATIONS ON (1 + 4)-DIMENSIONAL MYERS-PERRY SPACE-TIMES

Abstract. Localized energy estimates for the wave equation have been increasingly used to prove various other dispersive estimates. This article focuses on proving such localized energy estimates on (1+4)-dimensional Myers-Perry black hole backgrounds with small angular momenta. The Myers-Perry space-times are generalizations of higher dimensional Kerr backgrounds where additional planes of rotation are availabile while still maintaining axial symmetry. Once it is determined that all trapped geodesics have constant r, the method developed by Tataru and the fourth author, which perturbs off of the Schwarzschild case by using a pseudodifferential multiplier, can be adapted. 1