Morawetz Estimate for Linearized Gravity in Schwarzschild

The equations governing the perturbations of the Schwarzschild metric satisfy the Regge–Wheeler–Zerilli–Moncrief system. Applying the technique introduced in Andersson and Blue (Ann Math 182(2):787–853, 2015), we prove an integrated local energy decay estimate for both the Regge–Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the rp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^p$$\end{document} hierarchy estimates (Dafermos and Rodnianski, in: Exner (ed) XVIth international congress on mathematical physics, World Scientic, London, pp 421–433, 2009; Schlue in Anal PDE 6:515–600, 2013) to prove that both the Regge–Wheeler and Zerilli variables decay as t-32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-\frac{3}{2}}$$\end{document} in fixed regions of r.