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Quasilinear wave
equations on asymptotically flat spacetimes with applications
to Kerr black holes
Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski and Martin Taylor |
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Vol. 19 (2026), No. 5, 909–1028
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Abstract |
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We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating Kerr family of black holes in general relativity. There are two distinguishing aspects of our approach. The first aspect is its dyadically localised nature: The nontrivial part of the analysis is reduced entirely to time-translation-invariant -weighted estimates, in the spirit of Dafermos and Rodnianski (2010b), to be applied on dyadic time-slabs which for large are outgoing. Global existence and decay then both immediately follow by elementary iteration on consecutive such time-slabs, without further global bootstrap. The second, and more fundamental, aspect is our direct use of a “black box” linear inhomogeneous energy estimate on exactly stationary metrics, together with a novel but elementary physical-space top-order identity that need not capture the structure of trapping and is robust to perturbation. In the specific example of Kerr black holes, the required linear inhomogeneous estimate can then be quoted directly from the literature (Dafermos et al. (2016)), while the additional top-order physical-space identity can be shown easily in many cases (we include in the Appendix a proof for the Kerr case , which can in fact be understood in this context simply as a perturbation of Schwarzschild). In particular, the approach circumvents the need either for producing a purely physical-space identity capturing trapping or for a careful analysis of the commutation properties of frequency projections with the wave operator of time-dependent metrics. |
Keywords
nonlinear wave equations, black holes, general relativity
nonlinear stability
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Mathematical Subject Classification
Primary: 35L05, 83C05
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Milestones
Received: 8 August 2023
Revised: 8 October 2024
Accepted: 8 January 2025
Published: 21 May 2026
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Authors |
| Mihalis Dafermos | |
| Department of Mathematics Princeton University Princeton, NJ United States |
DPMMS University of Cambridge Cambridge United Kingdom |
| Gustav Holzegel | |
| Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Münster Germany |
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| Igor Rodnianski | |
| Department of Mathematics Princeton University Princeton, NJ United States |
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| Martin Taylor | |
| Department of Mathematics Imperial College London London United Kingdom |
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| © 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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