Nonlinear Wave Dynamics in Black Hole Spacetimes

TL;DR

This thesis develops a method for proving global boundedness and decay of nonlinear wave equations on black hole spacetimes, with applications to problems including stability of Kerr black holes.

Contribution

It introduces a new robust method for Morawetz estimates and decay hierarchies applicable to various black hole spacetime problems.

Findings

Proved decay and boundedness for nonlinear waves on Minkowski, Schwarzschild, and Kerr spacetimes.

Extended the method to axisymmetric Kerr spacetimes with subextremal rotation.

Progressed toward stability results for slowly rotating Kerr black holes.

Abstract

This manuscript is a lightly reformatted version of my 2017 PhD thesis. I am posting it on arXiv at the request of my advisor, Sergiu Klainerman, who noted that it has been useful to some students. The content largely reflects the thesis in its original form. This thesis details a method for proving global boundedness and decay results for nonlinear wave equations on black hole spacetimes. The method is applied to five example problems of increasing difficulty. The first problem, which addresses the semilinear wave equation on Minkowski space, is quite simple and should be accessible to a reader who is still new to the field of partial differential equations. The final problem, which was posed by Ionescu and Klainerman in [IK14], constitutes a step toward proving stability for slowly rotating Kerr black holes. The remaining intermediate problems are: a semilinear wave equation on the Schwarzschild spacetime, a semilinear wave equation on any subextremal Kerr spacetime with the additional assumption of axisymmetry, and a restriction of the final problem to the Schwarzschild case. The method used in this thesis is based on a few particular developments that may be useful for other related problems. These include: a new method for constructing Morawetz-type estimates that is fairly robust (insofar as it may be successfully applied to all five problems), a strategy based on a decay hierarchy for energy estimates on uniformly spacelike hypersurfaces using, in particular, a notion of weak decay, and a technique for handling certain terms with factors that are singular on an axis of symmetry.