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The energy conservation for weak solutions to the relativistic Nordström-Vlasov system
| 1. | School of Automation, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China |
References:
| [1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity, Inc. NY, Oxford University Press, 2000. |
| [2] |
F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rational Mech. Anal., 157 (2001), 75-90.
doi: 10.1007/PL00004237. |
| [3] |
F. Bouchut, F. Golse and C. Pallard, Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system, Rev. Mat.Iberoamericana., 20 (2004), 865-892.
doi: 10.4171/RMI/409. |
| [4] |
S. Calogero and G. Rein, Global weak solutions to the Nordström-Vlasov system, J. Differential Equations., 204 (2004), 323-338.
doi: 10.1016/j.jde.2004.02.011. |
| [5] |
S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system, Commun. Math. Phys., 266 (2006), 343-353.
doi: 10.1007/s00220-006-0029-x. |
| [6] |
S. Calogero, Spherically symmetric steady states of galactic dynamics in scalar gravity, Class. Quantum Grav., 20 (2003), 1729-1741.
doi: 10.1088/0264-9381/20/9/310. |
| [7] |
S. Calogero and G. Rein, On classical solutions of the Nordström-Vlasov system, Comm. Partial Diff. Eqs., 28 (2003), 1863-1885.
doi: 10.1081/PDE-120025488. |
| [8] |
R. J. Diperna and P.-L. Lions, Global weak solutions of Vlasov-Mxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.
doi: 10.1002/cpa.3160420603. |
| [9] |
R. J. Diperna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Amer. Math. Soc., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
| [10] |
L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. |
| [11] |
S. Friedrich, Global small solutions of the Vlasov-Nordström system, preprint, arXiv:math/0407023v1. |
| [12] |
P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
| [13] |
G. Loeper, Uniqueness of the solution to Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.
doi: 10.1016/j.matpur.2006.01.005. |
| [14] |
E. Miot, A uniqueness criterion for unbounded solutions to the Vlaosv-Poisson system, arXiv:1409.6988v1. |
| [15] |
C. Pallard, On global smooth solutions to the 3D Vlasov-Nordström system, Ann. I. H. Poincaré, 23 (2006), 85-96.
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 (2004), 145-158.
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, {Elsevier}, 3 (2007), 383-476.
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 (2010), 901-908.
doi: 10.4310/CMS.2010.v8.n4.a6. |
| [19] |
S. L. Shapiro and S. A. Teukolsky, Scalar gravitation: A laboratory for numerical relativity, Phys. Rev. D., 47 (1993), 1529-1540.
doi: 10.1103/PhysRevD.47.1529. |
show all references
References:
| [1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity, Inc. NY, Oxford University Press, 2000. |
| [2] |
F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rational Mech. Anal., 157 (2001), 75-90.
doi: 10.1007/PL00004237. |
| [3] |
F. Bouchut, F. Golse and C. Pallard, Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system, Rev. Mat.Iberoamericana., 20 (2004), 865-892.
doi: 10.4171/RMI/409. |
| [4] |
S. Calogero and G. Rein, Global weak solutions to the Nordström-Vlasov system, J. Differential Equations., 204 (2004), 323-338.
doi: 10.1016/j.jde.2004.02.011. |
| [5] |
S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system, Commun. Math. Phys., 266 (2006), 343-353.
doi: 10.1007/s00220-006-0029-x. |
| [6] |
S. Calogero, Spherically symmetric steady states of galactic dynamics in scalar gravity, Class. Quantum Grav., 20 (2003), 1729-1741.
doi: 10.1088/0264-9381/20/9/310. |
| [7] |
S. Calogero and G. Rein, On classical solutions of the Nordström-Vlasov system, Comm. Partial Diff. Eqs., 28 (2003), 1863-1885.
doi: 10.1081/PDE-120025488. |
| [8] |
R. J. Diperna and P.-L. Lions, Global weak solutions of Vlasov-Mxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.
doi: 10.1002/cpa.3160420603. |
| [9] |
R. J. Diperna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Amer. Math. Soc., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
| [10] |
L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. |
| [11] |
S. Friedrich, Global small solutions of the Vlasov-Nordström system, preprint, arXiv:math/0407023v1. |
| [12] |
P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
| [13] |
G. Loeper, Uniqueness of the solution to Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.
doi: 10.1016/j.matpur.2006.01.005. |
| [14] |
E. Miot, A uniqueness criterion for unbounded solutions to the Vlaosv-Poisson system, arXiv:1409.6988v1. |
| [15] |
C. Pallard, On global smooth solutions to the 3D Vlasov-Nordström system, Ann. I. H. Poincaré, 23 (2006), 85-96.
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 (2004), 145-158.
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, {Elsevier}, 3 (2007), 383-476.
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 (2010), 901-908.
doi: 10.4310/CMS.2010.v8.n4.a6. |
| [19] |
S. L. Shapiro and S. A. Teukolsky, Scalar gravitation: A laboratory for numerical relativity, Phys. Rev. D., 47 (1993), 1529-1540.
doi: 10.1103/PhysRevD.47.1529. |
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