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Global existence of solutions of the spherically symmetric Vlasov- Einstein system with small initial data. (English) Zbl 0774.53056
The coupled Einstein-Vlasov system describes a self-gravitating collisionless gas within general relativity. The present paper establishes existence of global spherically symmetric and asymptotically flat solutions to small initial data. The proof consists in the construction of a local solution of the Cauchy problem and in some continuation argument. As one conclusion, cosmic censorship holds for the Einstein-Vlasov system.

53Z05 Applications of differential geometry to physics
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
58J47 Propagation of singularities; initial value problems on manifolds
Full Text: DOI
[1] Bardos, C., Degond, P.: Global existence for the Vlasov-Poisson equation in three space variables with small initial data. Ann. Inst. H. Poincaré (Analyse non linéaire)2, 101–118 (1985) · Zbl 0593.35076
[2] Batt, J.: Global symmetric solutions of the initial value problem of stellar dynamics. J. Diff. Eq.25, 342–364 (1977) · Zbl 0366.35020 · doi:10.1016/0022-0396(77)90049-3
[3] Choquet-Bruhat, Y.: Problème de Cauchy pour le système integro differentiel d’Einstein-Liouville. Ann. Inst. Fourier, Grenoble21, 181–201 (1971) · Zbl 0208.14303
[4] Christodoulou, D.: Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys.93, 171–195 (1984) · doi:10.1007/BF01223743
[5] Christodoulou, D.: The problem of a self-gravitating scalar field. Commun. Math. Phys.105, 337–361 (1986) · Zbl 0608.35039 · doi:10.1007/BF01205930
[6] Christodoulou, D.: Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large. Commun. Math. Phys.106, 587–621 (1986) · Zbl 0613.53047 · doi:10.1007/BF01463398
[7] Christodoulou, D.: The structure and uniqueness of generalized solutions of the spherically symmetric Einstein-scalar equations. Commun. Math. Phys.109, 591–611 (1987) · Zbl 0613.53048 · doi:10.1007/BF01208959
[8] Christodoulou, D.: A mathematical theory of gravitational collapse. Commun. Math. Phys.109, 613–647 (1987) · Zbl 0613.53049 · doi:10.1007/BF01208960
[9] Christodoulou, D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math.44, 339–373 (1991) · Zbl 0728.53061 · doi:10.1002/cpa.3160440305
[10] Christodoulou, D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Preprint, Courant Institute, 1991 · Zbl 0853.35122
[11] Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minskowski space. To appear in Ann. of Math. · Zbl 0827.53055
[12] Ehlers, J.: Survey of general relativity theory. In: Israel, W. (ed.) Relativity, Astrophysics and Cosmology. Dordrecht: Reidel 1973
[13] Glassey, R., Strauss, W.: Singularity formation in a collisionless plasma could only occur at high velocities. Arch. Rat. Mech. Anal.92, 59–90 (1986) · Zbl 0595.35072 · doi:10.1007/BF00250732
[14] Hartman, P.: Ordinary differential equations. Boston: Birkhäuser 1982 · Zbl 0476.34002
[15] Hawking, S. W., Ellis, G.F.R.: The large-scale structure of space-time. Cambridge: Cambridge University Press 1973 · Zbl 0265.53054
[16] John, F.: Formation of singularities in elastic waves. Lecture notes in physics,195, Berlin, Heidelberg, New York: Springer 194–210 (1984)
[17] Lions, P.-L., Perthame, B.: Propagation of moments and regularity for the three dimensional Vlasov-Poisson system. Invent. Math.105, 415–430 (1991) · Zbl 0741.35061 · doi:10.1007/BF01232273
[18] Oppenheimer, J.R., Snyder, H.: On continued gravitational contraction. Phys. Rev.56, 455–459 (1939) · Zbl 0022.28104 · doi:10.1103/PhysRev.56.455
[19] Pfaffelmoser, K.: Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Diff. Eq.95, 281–303 (1992) · Zbl 0810.35089 · doi:10.1016/0022-0396(92)90033-J
[20] Rein, G.: Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics. Commun. Math. Phys.135, 41–78 (1991) · Zbl 0722.35091 · doi:10.1007/BF02097656
[21] Rein, G., Rendall, A.D.: The Newtonian limit of the spherically symmetric Vlasov-Einsein system. Commun. Math. Phys.150, 585–591 (1992) · Zbl 0772.53063 · doi:10.1007/BF02096963
[22] Rendall, A.D.: On the choice of matter model in general relativity. Preprint, 1992 · Zbl 0754.76098
[23] Schaeffer, J.: Global existence of smooth solutions of the Vlasov-Poisson system in three dimensions. Commun. Partial Diff. Eq.16, 1313–1336 (1991) · Zbl 0746.35050 · doi:10.1080/03605309108820801
[24] Sideris, T.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys.101, 475–485 (1985) · Zbl 0606.76088 · doi:10.1007/BF01210741
[25] Yodzis, P., Seifert, H.-J., Müller zum Hagen, H.: On the occurrence of naked singularities in general relativity. Commun. Math. Phys.34, 135–148 (1973) · doi:10.1007/BF01646443
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