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The global nonlinear stability of Minkowski space for the massless Einstein-Vlasov system. (English) Zbl 06919598
Summary: Minkowski space is shown to be globally stable as a solution to the Einstein-Vlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the “wave zone”, and then proving a small data semi-global existence result for the characteristic initial value problem for the massless Einstein-Vlasov system in this region. This relies on weighted estimates for the solution which, for the Vlasov part, are obtained by introducing the Sasaki metric on the mass shell and estimating Jacobi fields with respect to this metric by geometric quantities on the spacetime. The stability of Minkowski space result for the vacuum Einstein equations is then appealed to for the remaining regions.

MSC:
35Q83 Vlasov equations
35Q76 Einstein equations
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
35B35 Stability in context of PDEs
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[1] Andréasson, H.: The Einstein-Vlasov system/kinetic theory. Living Rev. Relativ. 14, 4 (2011). doi:10.12942/lrr-2011-4
[2] Bieri, L.: Extensions of the Stability Theorem of the Minkowski Space in General Relativity, Solutions of the Vacuum Einstein Equations. American Mathematical Society, Boston (2009) · Zbl 1172.83001
[3] Choquet-Bruhat, Y, Théreme d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math., 88, 141-225, (1952) · Zbl 0049.19201
[4] Choquet-Bruhat, Y, Problème de Cauchy pour le système intégro-différentiel d’einstein-Liouville, Ann. Inst. Fourier, 21, 181-201, (1971) · Zbl 0208.14303
[5] Choquet-Bruhat, Y., Chruściel, P.: Cauchy Problem with Data on a Characteristic Cone for the Einstein-Vlasov Equations arXiv:1206.0390
[6] Choquet-Bruhat, Y; Geroch, R, Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys., 14, 329-335, (1969) · Zbl 0182.59901
[7] Christodoulou, D.: Notes on the Geometry of Null Hypersurfaces (unpublished)
[8] Christodoulou, D.: The Global Initial Value Problem in General Relativity Ninth Marcel Grossman Meeting (Rome 2000), pp 44-54. World Science Publishing, Singapore (2002) · Zbl 0774.53056
[9] Christodoulou, D.: The Formation of Black Holes in General Relativity. European Mathematical Society Publishing House, Zurich (2009) · Zbl 1197.83004
[10] Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993) · Zbl 0827.53055
[11] Chruściel, P; Paetz, T, The many ways of the characteristic Cauchy problem class, Quantum Gravit., 29, 145006, (2012) · Zbl 1261.83003
[12] Dafermos, M, A note on the collapse of small data self-gravitating massless collisionless matter, J. Hyperbol. Differ. Equ., 3, 589-598, (2006) · Zbl 1115.35135
[13] Dafermos, M., Holzegel, G., Rodnianski, I.: A Scattering Theory Construction of Dynamical Vacuum Black Holes arXiv:1306.5364 · Zbl 1200.35303
[14] Dafermos, M; Rendall, AD, An extension principle for the Einstein-Vlasov system in spherical symmetry, Ann. Henri Poincaré, 6, 1137-1155, (2005) · Zbl 1138.83008
[15] Dafermos, M., Rendall, A.D.: Strong Cosmic Censorship for Surface-Symmetric Cosmological Spacetimes with Collisionless Matter arXiv:gr-qc/0701034 · Zbl 1341.83002
[16] Fajman, D., Joudioux, J., Smulevici, J.: A Vector Field Method for Relativistic Transport Equations with Applications arXiv:1510.04939 · Zbl 1373.35046
[17] Friedrich, H, On the existence of \(n\)-geodesically complete or future complete solutions of einstein’s field equations with smooth asymptotic structure, Commun. Math. Phys., 107, 587-609, (1986) · Zbl 0659.53056
[18] Gudmundsson, S; Kappos, E, On the geometry of tangent bundles, Expo. Math., 20, 1-41, (2002) · Zbl 1007.53027
[19] Hadz̆ić, M., Speck, J.: The Global Future Stability of the FLRW Solutions to the Dust-Einstein System with a Positive Cosmological Constant arXiv:1309.3502 · Zbl 1007.53027
[20] Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Santa Fe, N.M. (eds.) Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, 1984 Lectures in Applied Mathematics, vol. 23, pp. 293-326 (1986) · Zbl 0659.53056
[21] Klainerman, S., Nicolò, F.: The Evolution Problem in General Relativity, vol. 23 Lectures in Applied Mathematics. Birkhäuser, Boston (2003)
[22] Kowalski, O, Curvature of the induced riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew. Math., 250, 124-129, (1971) · Zbl 0222.53044
[23] LeFloch, P.G., Ma, Y.: The Global Nonlinear Stability of Minkowski Space for Self-Gravitating Massive Fields arXiv:1511.03324 · Zbl 1434.83005
[24] Loizelet, J, Solutions globales des équations d’einstein-Maxwell, Ann. Fac. Sci. Toulouse Math., 18, 565-610, (2009) · Zbl 1200.35303
[25] Li, J., Zhu, X.: On the Local Extension of the Future Null Infinity arXiv:1406.0048 · Zbl 1115.35135
[26] Lindblad, H; Rodnianski, I, The global stability of Minkowski space-time in harmonic gauge, Ann. Math., 171, 1401-1477, (2010) · Zbl 1192.53066
[27] Luk, J.: On the Local Existence for the Characteristic Initial Value Problem in General Relativity arXiv:1107.0898
[28] Rendall, AD, Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations, Proc. R. Soc. Lond., 427, 221-239, (1990) · Zbl 0701.35149
[29] Rein, G; Rendall, AD, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Commun. Math. Phys., 150, 561-583, (1992) · Zbl 0774.53056
[30] Ringström, H.: On the Topology and Future Stability of the Universe, Oxford Mathematical Monographs. Oxford University Press, Oxford (2013) · Zbl 1270.83005
[31] Rodnianski, I; Speck, J, The stability of the irrotational Einstein-Euler system with a positive cosmological constant, J. Eur. Math. Soc., 15, 2369-2462, (2013) · Zbl 1294.35164
[32] Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Springer, New York (1977) · Zbl 0373.53001
[33] Sasaki, S, On the differential geometry of tangent bundles, Tohoku Math. J., 10, 338-354, (1958) · Zbl 0086.15003
[34] Speck, J, The global stability of the Minkowski spacetime solution to the Einstein-nonlinear electromagnetic system in wave coordinates, Anal. PDE, 7, 771-901, (2014) · Zbl 1298.35225
[35] Wang, Q.: Global Existence for the Einstein Equations with Massive Scalar Fields. Talk Given at Mathematical Problems in Relativity Workshop, Simons Center, Stony Brook (2015). http://scgp.stonybrook.edu/archives/10300
[36] Zipser, N.: Extensions of the Stability Theorem of the Minkowski Space in General Relativity, Solutions of the Einstein-Maxwell Equations. American Mathematical Society, Boston (2009) · Zbl 1172.83001
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