Annals of PDE

, 4:15 | Cite as

A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

  • Y. Angelopoulos
  • S. AretakisEmail author
  • D. Gajic


We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner–Nordström black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds.


Wave equation Asymptotics Black holes Stability problems 



We would like to thank Mihalis Dafermos and Georgios Moschidis for several insightful discussions. S. Aretakis acknowledges support through NSF Grant DMS-1600643 and a Sloan Research Fellowship. D. Gajic acknowledges support by the European Research Council Grant No. ERC-2011-StG 279363-HiDGR.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA
  3. 3.Department of MathematicsUniversity of Toronto ScarboroughTorontoCanada
  4. 4.Department of MathematicsUniversity of TorontoTorontoCanada
  5. 5.Department of MathematicsImperial College LondonLondonUK
  6. 6.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

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