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Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry

Published online by Cambridge University Press:  24 October 2008

Gerhard Rein
Gerhard Rein
Affiliation:
Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany
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Abstract

The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 4 , May 1996 , pp. 739 - 762
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Christodoulou, D.. Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys. 93 (1984), 171–195.CrossRefGoogle Scholar
[2]Eardley, D., Liang, E. and Sachs, R.. Velocity-dominated singularities in irrotational dust cosmologies. J. Math. Phys. 13 (1972), 99–106.CrossRefGoogle Scholar
[3]Pfaffelmoser, K.. Global classical solutions of the Vlasov-Poisson system in three dimensions for generai initial data. J. Diff. Eqns. 95 (1992), 281–303.CrossRefGoogle Scholar
[4]Rein, G.. Static solutions of the spherically symmetric Vlasov-Einstein system. Math. Proc. Cambridge Phil. Soc. 115 (1994), 559–570.CrossRefGoogle Scholar
[5]Rein, G. and Rendall, A. D.. Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Commun. Math. Phys. 150 (1992), 561–583.CrossRefGoogle Scholar
[6]Rein, G. and Rendall, A. D.. The Newtonian limit of the spherically symmetric Vlasov-Einstein system. Commun. Math. Phys. 150 (1992), 585–591.CrossRefGoogle Scholar
[7]Rein, G. and Rendall, A. D.. Smooth static solutions of the spherically symmetric Vlasov-Einstein system. Ann. de l'Inst. H. Poincaré, Physique Théorique 59 (1993), 383–397.Google Scholar
[8]Rein, G. and Rendall, A. D.. Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting. Arch. Rational Mech. Anal. 126 (1994), 183–201.CrossRefGoogle Scholar
[9]Rein, G., Rendall, A. D. and Schaeffer, J.. A regularity theorem for the spherically symmetric Vlasov-Einstein system. Commun. Math. Phys. 168 (1995), 467–478.CrossRefGoogle Scholar
[10]Rendall, A. D.. Global properties of locally spatially homogeneous cosmological models with matter. (Preprint 1994), gr-qc/9409009.Google Scholar
[11]Kendall, A. D.. Crushing singularities with spherical and plane symmetry. In preparation.Google Scholar
[12]Schaeffer, J.. Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions. Commun. Part. Diff. Eqns. 16 (1991), 1313–1335.CrossRefGoogle Scholar