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March  2016, 5(1): 135-145. doi: 10.3934/eect.2016.5.135

The energy conservation for weak solutions to the relativistic Nordström-Vlasov system

Xiuting Li 1,

1. 

School of Automation, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received  October 2015 Revised  January 2016 Published  March 2016

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.
Keywords: Relativistic Nordström-Vlasov system, energy conservation., weak solution.
Mathematics Subject Classification: 83D05, 85A05, 35Q75, 35Q8.
Citation: Xiuting Li. The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. Evolution Equations & Control Theory, 2016, 5 (1) : 135-145. doi: 10.3934/eect.2016.5.135
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity, Inc. NY, Oxford University Press, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[11]

S. Friedrich, Global small solutions of the Vlasov-Nordström system,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Miot, A uniqueness criterion for unbounded solutions to the Vlaosv-Poisson system,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[11]

S. Friedrich, Global small solutions of the Vlasov-Nordström system,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Miot, A uniqueness criterion for unbounded solutions to the Vlaosv-Poisson system,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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