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Asymptotic properties of linear field equations in anti-de sitter space
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Holzegel2020_Article_AsymptoticPropertiesOfLinearFi.pdf | Published version | 790.71 kB | Adobe PDF | View/Open |
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 |