[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 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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@article{CRMATH_2016__354_9_948_0,
author = {LeFloch, Philippe G. and Ma, Yue},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.07.008/}
}
TY - JOUR AU - LeFloch, Philippe G. AU - Ma, Yue 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2016.07.008/ DO - 10.1016/j.crma.2016.07.008 LA - en ID - CRMATH_2016__354_9_948_0 ER -
%0 Journal Article %A LeFloch, Philippe G. %A Ma, Yue %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 %U http://www.numdam.org/articles/10.1016/j.crma.2016.07.008/ %R 10.1016/j.crma.2016.07.008 %G en %F CRMATH_2016__354_9_948_0
LeFloch, Philippe G.; Ma, Yue. The global nonlinear stability of Minkowski space for the Einstein equations in the presence of a massive field. Comptes Rendus. Mathématique, Tome 354 (2016) no. 9, pp. 948-953. doi : 10.1016/j.crma.2016.07.008. http://www.numdam.org/articles/10.1016/j.crma.2016.07.008/
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