Decay for quasilinear wave equations on cosmological black hole backgrounds

Abstract

This thesis studies the decay and stability of linear and quasilinear wave equations of Schwarzschild--de~Sitter $(M,\Lambda)$ and Kerr de Sitter $(a,M,\Lambda)$ black hole backgrounds, where $a,M,\Lambda$ are respectively the rotation parameter of the black hole, the mass of the black hole and the cosmological constant. Specifically, we study the Klein Gordon equation with mass $\mu^2_{\textit{KG}}\geq 0$ and the quasilinear wave equation which respectively read \begin{equation*} \Box_{g}\psi-\mu^2_{\textit{KG}}\psi=0,\qquad \Box_{g(\nabla\psi)}\psi=\partial\psi\cdot\partial\psi. \end{equation*} These equations have previously been studied extensively by Dyatlov, Hintz Vasy respectively. In the first Chapter of the thesis we use a novel physical space method to prove `relatively non-degenerate' integrated energy estimates for the wave equation on subextremal Schwarzschild de Sitter spacetimes. These are integrated decay statements whose bulk energy density, though degenerate at highest order, is everywhere comparable to the energy density of the boundary fluxes. As a corollary, we prove that solutions of the Klein Gordon equation with $\mu^2_{\textit{KG}}=0$ decay exponentially on the exterior region. The main ingredients are a previous Morawetz estimate of Dafermos Rodnianski and an additional argument based on commutation with a vector field which can be expressed in the form \begin{equation*} r\sqrt{1-\frac{2M}{r}-\frac{\Lambda}{3}r^2}\frac{\partial}{\partial r}, \end{equation*} where $\partial_r$ here denotes the coordinate vector field corresponding to a well chosen system of hyperboloidal coordinates, which we define here. In the limit $\Lambda=0$, our commutation corresponds to the one introduced by Holzegel Kauffman. In the second Chapter of the thesis we give an elementary new argument for global existence and exponential decay of solutions of quasilinear wave equations on Schwarzschild de Sitter black hole backgrounds, for appropriately small initial data. The core of the argument is entirely local, based on time translation invariant energy estimates in spacetime slabs of fixed time length, which are proved by using the `relatively non-degenerate estimates' from the first part of the thesis. Global existence then follows simply by iterating this local result in consecutive spacetime slabs. We infer that an appropriate future energy flux decays exponentially with respect to the energy flux of the initial data. In the third Chapter of the thesis we prove a Morawetz estimate for the Klein Gordon equation on subextremal Kerr de Sitter black hole backgrounds with parameters $(a,M,\Lambda)$, if any of the following conditions hold \begin{equation*} \begin{aligned} & (C_1)~\text{the black hole rotation parameter is sufficiently small},\ & (C_2)~\text{the solution of the Klein Gordon equation is axisymmetric},\ & (C_3)~\text{mode stability on the real axis holds for the Klein--Gordon equation for all rotation parameters up to a}. \end{aligned} \end{equation*} We moreover prove a `relatively non-degenerate estimate' under any of the conditions $(C_1)$, $(C_2)$, $(C_3)$. Specifically, we introduce a novel operator $\mathcal{G}$ that generalizes our previous physical space commutation of the first part of the thesis. Exponential decay is an immediate consequence. We also apply our commutation with $\mathcal{G}$ to the wave equation on any subextremal asymptotically flat Kerr black hole. In the fourth Chapter part of the thesis we prove the stability and exponential decay for the solutions of the quasilinear wave equation on a slowly rotating Kerr de Sitter black hole background. Note that for brevity of the exposition we present the Theorem for case $(C_1)$ of our main Theorem of the third part of the thesis. However, the main Theorem of this part of the thesis can be formulated for any of the cases $(C_1), (C_2),(C_3)$. Similarly to the second part of the thesis we use the result of the `relatively non-degenerate estimates' of the third part and a Cauchy stability result.