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Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations(AMS-210)$
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Jérémie Szeftel and Sergiu Klainerman

Print publication date: 2020

Print ISBN-13: 9780691212425

Published to Princeton Scholarship Online: May 2021

DOI: 10.23943/princeton/9780691212425.001.0001

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Regge-Wheeler Type Equations

Regge-Wheeler Type Equations

Chapter:
(p.600) Chapter Ten Regge-Wheeler Type Equations
Source:
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations
Author(s):

Sergiu Klainerman

Jérémie Szeftel

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691212425.003.0010

This chapter explores estimates for Regge-Wheeler type wave equations used in Theorem M1. It first proves basic Morawetz estimates for ψ‎. The chapter then proves rp-weighted estimates in the spirit of Dafermos and Rodnianski for ψ‎. In particular, it obtains as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without commutating the equation of ψ‎ with derivatives). It also uses a variation of the method of [5] to derive slightly stronger weighted estimates and prove Theorem 5.18 in the case s = 0. Finally, commuting the equation of ψ‎ with derivatives, the chapter completes the proof of Theorem 5.17 by controlling higher order derivatives of ψ‎.

Keywords:   Regge-Wheeler type equations, wave equations, Morawetz estimates, derivatives, weighted estimates, Theorem M1

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