Linear and nonlinear wave equations on black hole spacetimes

Abstract

In this thesis, I study three problems related to the linear and nonlinear wave equations on black hole spacetimes. These problems are motivated by the nonlinear stability of Kerr spacetime.

First, I prove that sufficiently regular solutions to the wave equation $\Box_g\Phi=0$ on the exterior of the Schwarzschild black hole obey the estimates $|\Phi|\leq C_\delta (t^*)^{-\frac{3}{2}+\delta}$ and $|\partial_t\Phi|\leq C_{\delta} (t^*)^{-2+\delta}$ on a compact region of $r$, including inside the black hole region.

Second, I prove that sufficiently regular solutions to the wave equation $\Box_g\Phi=0$ on the exterior of the sufficiently slowly rotating Kerr black hole also obey the estimates $|\Phi|\leq C_\delta (t^*)^{-\frac{3}{2}+\delta}$. The first two results are proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite $r$ and along the event horizon.

Third, I study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr

spacetimes. I prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method and in particular makes use of the improved decay rates obtained in the first and second results.