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

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.
Citation: Xiuting Li. The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. Evolution Equations and 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.

[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, (). 

[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,, , (). 

[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, (). 

[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,, , (). 

[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.

[1]

Jonathan Ben-Artzi, Stephen Pankavich, Junyong Zhang. A toy model for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2022, 15 (3) : 341-354. doi: 10.3934/krm.2021053

[2]

Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic and Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040

[3]

Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic and Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393

[4]

Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic and Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005

[5]

Dayton Preissl, Christophe Cheverry, Slim Ibrahim. Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system. Kinetic and Related Models, 2021, 14 (6) : 1035-1079. doi: 10.3934/krm.2021042

[6]

Mohammad Asadzadeh, Piotr Kowalczyk, Christoffer Standar. On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2019, 12 (1) : 105-131. doi: 10.3934/krm.2019005

[7]

Stephen Pankavich, Nicholas Michalowski. Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system. Kinetic and Related Models, 2015, 8 (1) : 169-199. doi: 10.3934/krm.2015.8.169

[8]

Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic and Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043

[9]

Mihai Bostan, Thierry Goudon. Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case. Kinetic and Related Models, 2008, 1 (1) : 139-170. doi: 10.3934/krm.2008.1.139

[10]

Jin Woo Jang, Robert M. Strain, Tak Kwong Wong. Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus. Kinetic and Related Models, 2022, 15 (4) : 569-604. doi: 10.3934/krm.2021039

[11]

Dequan Yue, Wuyi Yue. Block-partitioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns. Journal of Industrial and Management Optimization, 2009, 5 (3) : 417-430. doi: 10.3934/jimo.2009.5.417

[12]

Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919

[13]

Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240

[14]

Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045

[15]

Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463

[16]

Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099

[17]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

[18]

Gianluca Favre, Marlies Pirner, Christian Schmeiser. Thermalization of a rarefied gas with total energy conservation: Existence, hypocoercivity, macroscopic limit. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022015

[19]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

[20]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic and Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (142)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]