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Global weak solutions to the Nordström-Vlasov system. (English) Zbl 1060.35027
Summary: The Nordström–Vlasov system is a Lorentz invariant model for a self-gravitating collisionless gas. We establish suitable a priori bounds on the solutions of this system, which together with energy estimates and the smoothing effect of ”momentum averaging” yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion for classical solutions which was derived recently. The weak solutions are shown to preserve mass.

35D05 Existence of generalized solutions of PDE (MSC2000)
35B60 Continuation and prolongation of solutions to PDEs
35Q75 PDEs in connection with relativity and gravitational theory
83C40 Gravitational energy and conservation laws; groups of motions
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
Full Text: DOI arXiv
[1] H. Andréasson, The Einstein-Vlasov system/kinetic theory, Living Reviews in Relativity, lrr-2002-7, http://relativity.livingreviews.org/Articles/lrr-2002-7/index.html.
[2] Calogero, S., Spherically symmetric steady states of galactic dynamics in scalar gravity, Class. quantum grav., 20, 1729-1741, (2003) · Zbl 1030.83018
[3] S. Calogero, H. Lee, The non-relativistic limit of the Nordström-Vlasov system, math-ph/0309030.
[4] Calogero, S.; Rein, G., On classical solutions of the nordström – vlasov system, Commun. partial differential equations, 28, 1863-1885, (2003) · Zbl 1060.35141
[5] DiPerna, R.J.; Lions, P.-L., Global weak solutions of vlasov – maxwell systems, Comm. pure appl. math., 42, 6, 729-757, (1989) · Zbl 0698.35128
[6] Glassey, R.T., The Cauchy problem in kinetic theory, (1996), SIAM Philadelphia · Zbl 0372.35009
[7] Golse, F.; Lions, P.-L.; Perthame, B.; Sentis, R., Regularity of the moments of the solution of a transport equation, J. funct. anal., 76, 1, 110-125, (1988) · Zbl 0652.47031
[8] Kruse, K.; Rein, G., A stability result for the relativistic vlasov – maxwell system, Arch. rational mech. anal., 121, 2, 187-203, (1992) · Zbl 0841.35120
[9] Kunzinger, M.; Rein, G.; Steinbauer, R.; Teschl, G., Global weak solutions of the relativistic vlasov – klein – gordon system, Commun. math. phys., 238, 1-2, 367-378, (2003) · Zbl 1033.81028
[10] Nordström, G., Zur theorie der gravitation vom standpunkt des relativitätsprinzips, Ann. phys. lpz., 42, 533, (1913) · JFM 44.0890.02
[11] G. Rein, Selfgravitating Systems in Newtonian theory—the Vlasov-Poisson System, Vol. 41, Banach Center Publications, Part I, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1997, pp. 179-194. · Zbl 0893.35130
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