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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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Decay Estimates for α‎ and α‎ (Theorems M2, M3)

Decay Estimates for α‎ and α‎ (Theorems M2, M3)

Chapter:
(p.264) Chapter Six Decay Estimates for α‎ and α‎ (Theorems M2, M3)
Source:
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations
Author(s):

Sergiu Klainerman

Jérémie Szeftel

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691212425.003.0006

This chapter investigates the proof for Theorems M2 and M3. It relies on the decay of q to prove the decay estimates for α‎ and α‎. More precisely, the chapter relies on the results of Theorem M1 to prove Theorem M2 and M3. In Theorem M1, decay estimates are derived for q defined with respect to the global frame constructed in Proposition 3.26. To recover α‎ from q, the chapter derives a transport equation for α‎ where q is on the RHS. It then derives as Teukolsky–Starobinsky identity a parabolic equation for α‎. The chapter also improves bootstrap assumptions for α‎.

Keywords:   Theorem M2, Theorem M3, decay estimates, global frame, transport equation, Teukolsky–Starobinsky identity, parabolic equation, bootstrap assumptions

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