[Stabilité non linéaire globale de l'espace-temps de Minkowski pour les champs massifs]
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
Accepté le :
Publié le :
Philippe G. LeFloch 1 ; Yue Ma 2
@article{CRMATH_2016__354_9_948_0,
author = {Philippe G. LeFloch and Yue Ma},
title = {The global nonlinear stability of {Minkowski} space for the {Einstein} equations in the presence of a massive field},
journal = {Comptes Rendus. Math\'ematique},
pages = {948--953},
publisher = {Elsevier},
volume = {354},
number = {9},
year = {2016},
doi = {10.1016/j.crma.2016.07.008},
language = {en},
}
TY - JOUR AU - Philippe G. LeFloch AU - Yue Ma TI - The global nonlinear stability of Minkowski space for the Einstein equations in the presence of a massive field JO - Comptes Rendus. Mathématique PY - 2016 SP - 948 EP - 953 VL - 354 IS - 9 PB - Elsevier DO - 10.1016/j.crma.2016.07.008 LA - en ID - CRMATH_2016__354_9_948_0 ER -
%0 Journal Article %A Philippe G. LeFloch %A Yue Ma %T The global nonlinear stability of Minkowski space for the Einstein equations in the presence of a massive field %J Comptes Rendus. Mathématique %D 2016 %P 948-953 %V 354 %N 9 %I Elsevier %R 10.1016/j.crma.2016.07.008 %G en %F CRMATH_2016__354_9_948_0
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/
[1] Extensions of the Stability Theorem of the Minkowski Space, General Relativity, AMS/IP Stud. Adv. Math., vol. 45, Amer. Math. Soc., International Press, Cambridge, UK, 2009
[2] General Relativity and the Einstein Equations, Oxford Math. Monograph, Oxford University Press, 2009
[3] The Global Nonlinear Stability of the Minkowski Space, Princeton Math. Ser., vol. 41, 1993
[4] Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, Berlin, 1997
[5] Global existence for coupled systems of nonlinear wave and Klein–Gordon equations in three space dimensions, Math. Z., Volume 270 (2012), pp. 487-513
[6] Global existence for nonlinear wave equations, Commun. Pure Appl. Math., Volume 33 (1980), pp. 43-101
[7] Global existence of small amplitude solutions to nonlinear Klein–Gordon equations in four spacetime dimensions, Commun. Pure Appl. Math., Volume 38 (1985), pp. 631-641
[8] The Hyperboloidal Foliation Method, World Scientific Press, Singapore, 2014
[9] The mathematical validity of the
[10] The global nonlinear stability of Minkowski space for self-gravitating massive fields, 2015 (preprint) | arXiv
[11] The global nonlinear stability of Minkowski space for self-gravitating massive fields. The wave-Klein–Gordon model, Commun. Math. Phys. (2016) (in press) | DOI
[12] P.G. LeFloch, Y. Ma, The nonlinear stability of Minkowski spacetime for massive matter in Einstein's theory and
[13] Global existence for the Einstein vacuum equations in wave coordinates, Commun. Math. Phys., Volume 256 (2005), pp. 43-110
[14] The global stability of Minkowski spacetime in harmonic gauge, Ann. Math., Volume 171 (2010), pp. 1401-1477
[15] Normal forms and quadratic nonlinear Klein–Gordon equations, Commun. Pure Appl. Math., Volume 38 (1985), pp. 685-696
- 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
- The Euclidean-hyperboloidal foliation method: application to
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
Commentaires - Politique
Vous devez vous connecter pour continuer.
S'authentifier