Comptes Rendus
Mathematical physics
The global nonlinear stability of Minkowski space for the Einstein equations in the presence of a massive field
[Stabilité non linéaire globale de l'espace-temps de Minkowski pour les champs massifs]
Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 948-953.

Nous généralisons la méthode du feuiletage hyperboloïdal introduite par les auteurs en 2014 pour traiter des systèmes quasilinéaires couplant des équations d'ondes et des équations de Klein–Gordon. Dans cette nouvelle formulation, nous réussissons à traiter des termes métriques d' « interaction forte ». Nous appliquons cette méthode pour démontrer la stabilité non linéaire de l'espace de Minkowski pour les équations d'Einstein des champs scalaires massifs auto-gravitants. En suivant un travail de Lindblad et de Rodnianski, nous analysons la structure des équations d'Einstein en coordonnées d'ondes, qui constituent précisément un système d'équations d'ondes quasi linéaires avec « interaction forte ».

We provide a significant extension of the Hyperboloidal Foliation Method introduced by the authors in 2014 in order to establish global existence results for systems of quasilinear wave equations posed on a curved space, when wave equations and Klein–Gordon equations are coupled. This method is based on a (3+1) foliation (of the interior of a future light cone in Minkowski spacetime) by spacelike and asymptotically hyperboloidal hypersurfaces. In the new formulation of the method, we succeed to cover wave-Klein–Gordon systems containing “strong interaction” terms at the level of the metric, and then generalize our method in order to establish a new existence theory for the Einstein equations of general relativity. Following pioneering work by Lindblad and Rodnianski on the Einstein equations in wave coordinates, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields.

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DOI : 10.1016/j.crma.2016.07.008

Philippe G. LeFloch 1 ; Yue Ma 2

1 Laboratoire Jacques-Louis-Lions, Centre national de la recherche scientifique, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris, France
2 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049 Shaanxi, People's Republic of China
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Philippe G. LeFloch; Yue Ma. The global nonlinear stability of Minkowski space for the Einstein equations in the presence of a massive field. Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 948-953. doi : 10.1016/j.crma.2016.07.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.07.008/

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  • Philippe G. LeFloch; Yue Ma Nonlinear stability of self-gravitating massive fields, Annals of PDE, Volume 10 (2024) no. 2, p. 217 (Id/No 16) | DOI:10.1007/s40818-024-00172-1 | Zbl:7949341
  • Philippe G. LeFloch; Yue Ma The Euclidean-hyperboloidal foliation method: application to f(R) modified gravity, General Relativity and Gravitation, Volume 56 (2024) no. 5, p. 63 (Id/No 66) | DOI:10.1007/s10714-024-03250-8 | Zbl:1551.83099
  • Philippe G. LeFloch; Yue Ma Nonlinear stability of self-gravitating massive fields. A wave-Klein-Gordon model, Classical and Quantum Gravity, Volume 40 (2023) no. 15, p. 36 (Id/No 154001) | DOI:10.1088/1361-6382/acde31 | Zbl:1517.83063
  • Philippe G. LeFloch; Yue Ma Einstein-Klein-Gordon spacetimes in the harmonic near-Minkowski regime, Portugaliae Mathematica, Volume 79 (2022) no. 3-4, pp. 343-393 | DOI:10.4171/pm/2084 | Zbl:1515.83035
  • Shijie Dong; Philippe Gerard LeFloch; Zoe Wyatt Global evolution of the U(1) Higgs boson: nonlinear stability and uniform energy bounds, Annales Henri Poincaré, Volume 22 (2021) no. 3, pp. 677-713 | DOI:10.1007/s00023-020-00955-9 | Zbl:1469.35182
  • Patenou Jean Baptiste Characteristic Cauchy problem on the light cone for the Einstein-Vlasov system in temporal gauge, Classical and Quantum Gravity, Volume 38 (2021) no. 18, p. 42 (Id/No 185009) | DOI:10.1088/1361-6382/ac186f | Zbl:1482.83006
  • Alexandru D. Ionescu; Benoit Pausader Global regularity of solutions of the Einstein-Klein-Gordon system: a review, Quarterly of Applied Mathematics, Volume 78 (2020) no. 2, pp. 277-303 | DOI:10.1090/qam/1555 | Zbl:1439.35469
  • Zoe Wyatt The weak null condition and Kaluza-Klein spacetimes, Journal of Hyperbolic Differential Equations, Volume 15 (2018) no. 2, pp. 219-258 | DOI:10.1142/s0219891618500091 | Zbl:1394.35500
  • Jean Baptiste Patenou Cauchy problem on a characteristic cone for the Einstein-Vlasov system. I: The initial data constraints, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 355 (2017) no. 2, pp. 187-192 | DOI:10.1016/j.crma.2016.11.018 | Zbl:1360.53026
  • Philippe G. LeFloch; Yue Ma The global nonlinear stability of Minkowski space for self-gravitating massive fields. The wave-Klein-Gordon model, Communications in Mathematical Physics, Volume 346 (2016) no. 2, pp. 603-665 | DOI:10.1007/s00220-015-2549-8 | Zbl:1359.83003

Cité par 10 documents. Sources : zbMATH

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