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Asymptotic properties of linear field equations in anti-de sitter space

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Title: Asymptotic properties of linear field equations in anti-de sitter space
Authors: Holzegel, G
Luk, J
Smulevici, J
Warnick, C
Item Type: Journal Article
Abstract: We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.
Issue Date: 4-Nov-2019
Date of Acceptance: 18-Aug-2019
URI: http://hdl.handle.net/10044/1/75110
DOI: 10.1007/s00220-019-03601-6
ISSN: 0010-3616
Publisher: Springer (part of Springer Nature)
Start Page: 1125
End Page: 1178
Journal / Book Title: Communications in Mathematical Physics
Volume: 374
Copyright Statement: © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor/Funder: Commission of the European Communities
Funder's Grant Number: FP7-ERC-2013-StG-337488
Keywords: Science & Technology
Physical Sciences
Physics, Mathematical
Physics
WAVE-EQUATION
CONTROLLABILITY
STABILITY
DYNAMICS
DECAY
Science & Technology
Physical Sciences
Physics, Mathematical
Physics
WAVE-EQUATION
CONTROLLABILITY
STABILITY
DYNAMICS
DECAY
0101 Pure Mathematics
0105 Mathematical Physics
0206 Quantum Physics
Mathematical Physics
Publication Status: Published online
Open Access location: https://link.springer.com/article/10.1007/s00220-019-03601-6
Online Publication Date: 2019-11-04
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences
Mathematics