# American Institute of Mathematical Sciences

February 2018, 11(1): 25-42. doi: 10.3934/krm.2018002

## A non-relativistic model of plasma physics containing a radiation reaction term

 Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany

Received  June 2016 Revised  February 2017 Published  August 2017

While a fully relativistic collisionless plasma is modeled by the Vlasov-Maxwell system a good approximation in the non-relativistic limit is given by the Vlasov-Poisson system. We modify the Vlasov-Poisson system so that damping due to the relativistic effect of radiation reaction is included. We prove existence and uniqueness as well as higher regularity of local classical solutions. Our results also include the higher regularity of classical solutions of the Vlasov-Poisson system depending on the regularity of the initial datum.

Citation: Sebastian Bauer. A non-relativistic model of plasma physics containing a radiation reaction term. Kinetic & Related Models, 2018, 11 (1) : 25-42. doi: 10.3934/krm.2018002
##### References:
 [1] J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differental Equations, 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3. [2] S. Bauer, Post-Newtonian dynamics at order 1. 5 in the Vlasov-Maxwell system, preprint, arXiv: math-ph/0602031. [3] S. Bauer and M. Kunze, The Darwin approximation of the relativistic Vlasov-Maxwell system, Ann. Henri Poincaré, 6 (2005), 283-308. doi: 10.1007/s00023-005-0207-y. [4] S. Bauer and M. Kunze, Radiative friction for charges interacting with the radiation field: Classical many-particle systems, in Analysis, Modeling and Simulation of Multiscale Problems, (2006), 531-551. doi: 10.1007/3-540-35657-6_19. [5] S. Bauer, M. Kunze, G. Rein and A. D. Rendall, Multipole radiation in a collisionless gas coupled to electromagnetism or scalar gravitation, Comm. Math. Phys., 266 (2008), 267-288. doi: 10.1007/s00220-006-0015-3. [6] J. Chen and X. Zhang, Global existence of small amplitude solutions to the Vlasov-Poisson system with radiation damping Internat. J. Math. 26 (2015), 1550098, 19 pp. doi: 10.1142/S0129167X15500986. [7] Z. Chen and X. Zhang, Global existence to the Vlasov-Poisson system and propagation of moments without assumption of finite kinetic energy, Commun. Math. Phys., 343 (2016), 851-879. doi: 10.1007/s00220-016-2616-9. [8] L. E. Fraenkel, Formulae for high derivatives of composite functions, Math. Proc. Cambridge Philos. Soc., 83 (1978), 159-165. doi: 10.1017/S0305004100054402. [9] J. D. Jackson, Classical Electrodynamics Second edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1975. [10] S. Kunimochi, Invariant manifolds in singular perturbation problems for ordinary differential equations, Proc. R. Soc. Lond. A, 116 (1990), 45-78. doi: 10.1017/S0308210500031371. [11] M. Kunze and A. D. Rendall, Simplified models of electromagnetic and gravitational radiation damping, Classical Quantum Gravity, 18 (2001), 3573-3587. doi: 10.1088/0264-9381/18/17/311. [12] M. Kunze and A. D. Rendall, The Vlasov-Poisson system with radiation damping, Ann. Henri Poincaré, 2 (2001), 857-886. doi: 10.1007/s00023-001-8596-z. [13] M. Kunze and H. Spohn, Post-Coulombian dynamics at order $c^-3$, J. Nonlinear Sci., 11 (2001), 321-396. doi: 10.1007/s00332-001-0455-z. [14] H. Lee, The classical limit of the relativistic Vlasov-Maxwell system in two space dimensions, Math. Methods Appl. Sci., 27 (2004), 249-287. doi: 10.1002/mma.424. [15] A. Lindner, Ck-Regularität der Lösungen des Vlasov-Poisson-Systems partieller Differentialgleichungen Diploma thesis, Ludwig Maximilian Universität, München, 1991. [16] 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. [17] G. Rein, Collisionless kinetic equations from astrophysics---the Vlasov-Poisson system, Handbook of differential equations: evolutionary equations, Handb. Differ. Equ, (2007), 383-476. doi: 10.1016/S1874-5717(07)80008-9. [18] J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 104 (1986), 403-421. [19] H. Spohn, Dynamics of Charged Particles and their Radiation Field Cambridge University press, Cambridge, 2004. doi: 10.1017/CBO9780511535178. [20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970. [21] W. Walter, Ordinary Differential Equations Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.

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##### References:
 [1] J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differental Equations, 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3. [2] S. Bauer, Post-Newtonian dynamics at order 1. 5 in the Vlasov-Maxwell system, preprint, arXiv: math-ph/0602031. [3] S. Bauer and M. Kunze, The Darwin approximation of the relativistic Vlasov-Maxwell system, Ann. Henri Poincaré, 6 (2005), 283-308. doi: 10.1007/s00023-005-0207-y. [4] S. Bauer and M. Kunze, Radiative friction for charges interacting with the radiation field: Classical many-particle systems, in Analysis, Modeling and Simulation of Multiscale Problems, (2006), 531-551. doi: 10.1007/3-540-35657-6_19. [5] S. Bauer, M. Kunze, G. Rein and A. D. Rendall, Multipole radiation in a collisionless gas coupled to electromagnetism or scalar gravitation, Comm. Math. Phys., 266 (2008), 267-288. doi: 10.1007/s00220-006-0015-3. [6] J. Chen and X. Zhang, Global existence of small amplitude solutions to the Vlasov-Poisson system with radiation damping Internat. J. Math. 26 (2015), 1550098, 19 pp. doi: 10.1142/S0129167X15500986. [7] Z. Chen and X. Zhang, Global existence to the Vlasov-Poisson system and propagation of moments without assumption of finite kinetic energy, Commun. Math. Phys., 343 (2016), 851-879. doi: 10.1007/s00220-016-2616-9. [8] L. E. Fraenkel, Formulae for high derivatives of composite functions, Math. Proc. Cambridge Philos. Soc., 83 (1978), 159-165. doi: 10.1017/S0305004100054402. [9] J. D. Jackson, Classical Electrodynamics Second edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1975. [10] S. Kunimochi, Invariant manifolds in singular perturbation problems for ordinary differential equations, Proc. R. Soc. Lond. A, 116 (1990), 45-78. doi: 10.1017/S0308210500031371. [11] M. Kunze and A. D. Rendall, Simplified models of electromagnetic and gravitational radiation damping, Classical Quantum Gravity, 18 (2001), 3573-3587. doi: 10.1088/0264-9381/18/17/311. [12] M. Kunze and A. D. Rendall, The Vlasov-Poisson system with radiation damping, Ann. Henri Poincaré, 2 (2001), 857-886. doi: 10.1007/s00023-001-8596-z. [13] M. Kunze and H. Spohn, Post-Coulombian dynamics at order $c^-3$, J. Nonlinear Sci., 11 (2001), 321-396. doi: 10.1007/s00332-001-0455-z. [14] H. Lee, The classical limit of the relativistic Vlasov-Maxwell system in two space dimensions, Math. Methods Appl. Sci., 27 (2004), 249-287. doi: 10.1002/mma.424. [15] A. Lindner, Ck-Regularität der Lösungen des Vlasov-Poisson-Systems partieller Differentialgleichungen Diploma thesis, Ludwig Maximilian Universität, München, 1991. [16] 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. [17] G. Rein, Collisionless kinetic equations from astrophysics---the Vlasov-Poisson system, Handbook of differential equations: evolutionary equations, Handb. Differ. Equ, (2007), 383-476. doi: 10.1016/S1874-5717(07)80008-9. [18] J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 104 (1986), 403-421. [19] H. Spohn, Dynamics of Charged Particles and their Radiation Field Cambridge University press, Cambridge, 2004. doi: 10.1017/CBO9780511535178. [20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970. [21] W. Walter, Ordinary Differential Equations Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.
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