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The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. (English) Zbl 1350.83004

Summary: We study the Cauchy problem of the relativistic Nordström-Vlasov system. Under some additional conditions, total energy for weak solutions with BV scalar field are shown to be conserved.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
85A05 Galactic and stellar dynamics
35Q75 PDEs in connection with relativity and gravitational theory
35Q83 Vlasov equations
83C40 Gravitational energy and conservation laws; groups of motions
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References:

[1] L. Ambrosio, <em>Functions of Bounded Variation and Free Discontinuity</em>,, Inc. NY (2000) · Zbl 0957.49001
[2] F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation,, Arch. Rational Mech. Anal., 157, 75 (2001) · Zbl 0979.35032 · doi:10.1007/PL00004237
[3] F. Bouchut, Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system,, Rev. Mat.Iberoamericana., 20, 865 (2004) · Zbl 1064.35097 · doi:10.4171/RMI/409
[4] S. Calogero, Global weak solutions to the Nordström-Vlasov system,, J. Differential Equations., 204, 323 (2004) · Zbl 1060.35027 · doi:10.1016/j.jde.2004.02.011
[5] S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system,, Commun. Math. Phys., 266, 343 (2006) · Zbl 1123.35080 · doi:10.1007/s00220-006-0029-x
[6] S. Calogero, Spherically symmetric steady states of galactic dynamics in scalar gravity,, Class. Quantum Grav., 20, 1729 (2003) · Zbl 1030.83018 · doi:10.1088/0264-9381/20/9/310
[7] S. Calogero, On classical solutions of the Nordström-Vlasov system,, Comm. Partial Diff. Eqs., 28, 1863 (2003) · Zbl 1060.35141 · doi:10.1081/PDE-120025488
[8] R. J. Diperna, Global weak solutions of Vlasov-Mxwell systems,, Comm. Pure Appl. Math., 42, 729 (1989) · Zbl 0698.35128 · doi:10.1002/cpa.3160420603
[9] R. J. Diperna, Ordinary differential equations, transport theory and Sobolev spaces,, Amer. Math. Soc., 98, 511 (1989) · Zbl 0696.34049 · doi:10.1007/BF01393835
[10] L. C. Evans, <em>Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics</em>,, American Mathematical Society (1998) · Zbl 0902.35002
[11] S. Friedrich, Global small solutions of the Vlasov-Nordström system,, preprint
[12] P.-L. Lions, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105, 415 (1991) · Zbl 0741.35061 · doi:10.1007/BF01232273
[13] G. Loeper, Uniqueness of the solution to Vlasov-Poisson system with bounded density,, J. Math. Pures Appl., 86, 68 (2006) · Zbl 1111.35045 · doi:10.1016/j.matpur.2006.01.005
[14] E. Miot, A uniqueness criterion for unbounded solutions to the Vlaosv-Poisson system,, <a href= · Zbl 1357.82041
[15] C. Pallard, On global smooth solutions to the 3D Vlasov-Nordström system,, Ann. I. H. Poincaré, 23, 85 (2006) · Zbl 1092.85001 · doi:10.1016/j.anihpc.2005.02.001
[16] G. Rein, Global weak solutions to the relativistic Vlasov-Maxwell system revisted,, Comm. Math. Sci., 2, 145 (2004) · Zbl 1090.35168 · doi:10.4310/CMS.2004.v2.n2.a1
[17] G. Rein, Collisionless kinetic equation from astrophysics-the Vlasov-Poisson system,, in: Handbook of Differential Equations: Evolutionary Equations, 3, 383 (2007) · Zbl 1193.35230 · doi:10.1016/S1874-5717(07)80008-9
[18] R. Sospedra-Alfonso, On the energy conservation by weak solutions of the relativistic Vlasov-Maxwell system,, Comm. Math. Sci., 8, 901 (2010) · Zbl 1206.35232 · doi:10.4310/CMS.2010.v8.n4.a6
[19] S. L. Shapiro, Scalar gravitation: A laboratory for numerical relativity,, Phys. Rev. D., 47, 1529 (1993) · doi:10.1103/PhysRevD.47.1529
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