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 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.
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 ».
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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 theory of modified gravity, 2014 | arXiv
[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 -gravity, in preparation.
[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
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