The conformal Einstein field equations with massless Vlasov matter
[Équations de champs conformes d’Einstein couplées à de la matière de Vlasov sans masse]
Annales de l'Institut Fourier, Online first, 44 p.

Nous prouvons la stabilité de l’espace-temps de de Sitter, solution du système d’Einstein–Vlasov avec des particules sans masse. Nous considérons également la stabilité semi-globale de l’espace-temps de Minkowski pour le même système. La preuve de la stabilité repose sur l’usage de techniques conformes, et plus précisément les équations de champs conformes d’Einstein introduites par Friedrich. Nous exploitons l’invariance conforme de l’équation de Vlasov sans masse et adaptons le résultat d’existence locale en temps suffisamment long de Kato au système d’Einstein–Vlasov.

We prove the stability of de Sitter space-time as a solution to the Einstein–Vlasov system with massless particles. The semi-global stability of Minkowski space-time is also addressed. The proof relies on conformal techniques, namely Friedrich’s conformal Einstein field equations. We exploit the conformal invariance of the massless Vlasov equation on the cotangent bundle and adapt Kato’s local existence theorem for symmetric hyperbolic systems to prove a long enough time of existence for solutions of the evolution system implied by the Vlasov equation and the conformal Einstein field equations.

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DOI : https://doi.org/10.5802/aif.3414
Classification : 35L04,  35Q75,  35Q76,  53A30
Mots clés : équation de champs conformes d’Einstein, problème de stabilité, systèmes symétrique hyperbolique, équation de Vlasov
@unpublished{AIF_0__0_0_A10_0,
     author = {Joudioux, J\'er\'emie and Thaller, Maximilian and Valiente Kroon, Juan A.},
     title = {The conformal {Einstein} field equations with massless {Vlasov} matter},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3414},
     language = {en},
     note = {Online first},
}
Joudioux, Jérémie; Thaller, Maximilian; Valiente Kroon, Juan A. The conformal Einstein field equations with massless Vlasov matter. Annales de l'Institut Fourier, Online first, 44 p.

[1] Andréasson, Håkan The Einstein–Vlasov System/Kinetic Theory, Living Reviews in Relativity, Volume 14 (2011) no. 1, 2011-4 | Article | Zbl 1316.83021

[2] Anguige, Keith; Tod, Kenneth P. Isotropic cosmological singularities. II. The Einstein–Vlasov system, Ann. Phys., Volume 276 (1999) no. 2, pp. 294-320 | Article | MR 1710659 | Zbl 1003.83028

[3] Bieri, Lydia; Zipser, Nina Extensions of the stability theorem of the Minkowski space in general relativity, AMS/IP Studies in Advanced Mathematics, 45, American Mathematical Society, International Press, 2009 | MR 2531716 | Zbl 1172.83001

[4] Bigorgne, Léo; Fajman, David; Joudioux, Jérémie; Smulevici, Jacques; Thaller, Maximilian Asymptotic Stability of Minkowski Space-Time with non-compactly supported massless Vlasov matter (2020) (https://arxiv.org/abs/2003.03346, to appear in Archive for Rational Mechanics and Analysis)

[5] Choquet-Bruhat, Yvonne Problème de Cauchy pour le système intégro-différentiel d’Einstein–Liouville, Ann. Inst. Fourier, Volume 21 (1971) no. 3, pp. 181-201 | Article | MR 49 | Zbl 0208.14303

[6] Christodoulou, Demetrios; Klainerman, Sergiu The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, 41, Princeton University Press, 1993 | MR 1316662 | Zbl 0827.53055

[7] Chruściel, Piotr T.; Delay, Erwann Existence of non-trivial, vacuum, asymptotically simple spacetimes, Class. Quant. Grav., Volume 19 (2002) no. 9, p. L71-L79 | Article | MR 1902228 | Zbl 1005.83009

[8] Corvino, Justin; Schoen, Richard M. On the asymptotics for the vacuum Einstein constraint equations, J. Differ. Geom., Volume 73 (2006) no. 2, pp. 185-217 | MR 2225517 | Zbl 1122.58016

[9] Dafermos, Mihalis A note on the collapse of small data self-gravitating massless collisionless matter, J. Hyperbolic Differ. Equ., Volume 3 (2006) no. 4, pp. 589-598 | Article | MR 2289606 | Zbl 1115.35135

[10] Fajman, David; Joudioux, Jérémie; Smulevici, Jacques The Stability of the Minkowski space for the Einstein–Vlasov system (2017) (https://arxiv.org/abs/1707.06141, to appear in Analysis & PDE in 2020)

[11] Fajman, David; Joudioux, Jérémie; Smulevici, Jacques A vector field method for relativistic transport equations with applications, Anal. PDE, Volume 10 (2017) no. 7, pp. 1539-1612 | Article | MR 3683922 | Zbl 1373.35046

[12] Friedrich, Helmut On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Commun. Math. Phys., Volume 107 (1986) no. 4, pp. 587-609 | Article | MR 868737 | Zbl 0659.53056

[13] Friedrich, Helmut On the global existence and the asymptotic behavior of solutions to the Einstein–Maxwell–Yang–Mills equations, J. Differ. Geom., Volume 34 (1991) no. 2, pp. 275-345 | MR 1131434 | Zbl 0737.53070

[14] Hübner, Peter General relativistic scalar-field models and asymptotic flatness, Class. Quant. Grav., Volume 12 (1995) no. 3, pp. 791-808 | Article | MR 1326039 | Zbl 0822.53054

[15] Kato, Tosio The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., Volume 58 (1975) no. 3, pp. 181-205 | Article | MR 0390516 | Zbl 0343.35056

[16] Klainerman, Sergiu; Nicolò, Francesco The evolution problem in general relativity, Progress in Mathematical Physics, 25, Birkhäuser, 2003 | Article | MR 1946854 | Zbl 1010.83004

[17] LeFloch, Philippe G.; Ma, Yue The global nonlinear stability of Minkowski space for self-gravitating massive fields, Commun. Math. Phys., Volume 346 (2016) no. 2, pp. 603-665 | Article | MR 3535896 | Zbl 1359.83003

[18] Lindblad, Hans; Taylor, Martin Global Stability of Minkowski Space for the Einstein–Vlasov System in the Harmonic Gauge, Arch. Ration. Mech. Anal., Volume 235 (2020) no. 1, Lindblad2019, pp. 517-633 | Article | MR 4062482 | Zbl 1434.35234

[19] Lindquist, Richard W. Relativistic transport theory, Ann. Phys., Volume 37 (1966) no. 3, pp. 487-518 | Article | Zbl 0142.23902

[20] Lübbe, Christian; Valiente Kroon, Juan Antonio The extended conformal Einstein field equations with matter: the Einstein–Maxwell field, J. Geom. Phys., Volume 62 (2012) no. 6, pp. 1548-1570 | Article | MR 2911223 | Zbl 1239.53018

[21] Lübbe, Christian; Valiente Kroon, Juan Antonio On the conformal structure of the extremal Reissner–Nordström spacetime, Class. Quant. Grav., Volume 31 (2014) no. 17, 175015 | Zbl 1304.83035

[22] Penrose, Roger; Rindler, Wolfgang Spinors and space-time. Volume 1: Two-spinor calculus and relativistic fields, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1984 | MR 917488 | Zbl 0538.53024

[23] Penrose, Roger; Rindler, Wolfgang Spinors and space-time. Vol. 2: Spinor and twistor methods in space-time geometry, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1986 | MR 944085 | Zbl 0591.53002

[24] Rendall, Alan D. An introduction to the Einstein–Vlasov system, Mathematics of gravitation. Part I: Lorentzian geometry and Einstein equations. Proceedings of the workshop on mathematical aspects of theories of gravitation (Banach Center Publications), Volume 41, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1997, pp. 35-68 | MR 1466508 | Zbl 0892.35148

[25] Ringström, Hans On the topology and future stability of the universe, Oxford Mathematical Monographs, Oxford University Press, 2013 | Article | MR 3186493 | Zbl 1270.83005

[26] Sarbach, Olivier; Zannias, Thomas The geometry of the tangent bundle and the relativistic kinetic theory of gases, Class. Quant. Grav., Volume 31 (2014) no. 8, 085013 | Zbl 1295.83035

[27] Speck, Jared The nonlinear stability of the trivial solution to the Maxwell–Born–Infeld system, J. Math. Phys., Volume 53 (2012) no. 8, 083703 | Article | MR 3012654 | Zbl 1278.83026

[28] Taylor, Martin The global nonlinear stability of Minkowski space for the massless Einstein–Vlasov system, Ann. PDE, Volume 3 (2017) no. 1, 9 | Article | MR 3629140 | Zbl 06919598

[29] Valiente Kroon, Juan Antonio Conformal methods in general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2016 | Article | MR 3585918 | Zbl 1368.83004

[30] Wollman, Stephen Local existence and uniqueness theory of the Vlasov–Maxwell system, J. Math. Anal. Appl., Volume 127 (1987) no. 1, pp. 103-121 | Article | MR 904213 | Zbl 0645.35013