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

Sebastian Bauer

doi:10.3934/krm.2018002

Article Contents

2018, Volume 11, Issue 2: 1-18. Doi: 10.3934/krm.2018002

This issue Previous Article Non-contraction of intermediate admissible discontinuities for 3-D planar isentropic magnetohydrodynamics Next Article A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures

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

Sebastian Bauer

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

Received: June 2016

Revised: February 2017

Published: April 2018

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary:35B30, 35L60, 78A40;Secondary:82C21, 82C22, 82D10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	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.
	S. Bauer, Post-Newtonian dynamics at order 1. 5 in the Vlasov-Maxwell system, arXiv: math-ph/0602031.
	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.
	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, Springer, (2006), 531–551.
	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.
	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.
	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.
	L. E. Fraenkel , Formulae for high derivatives of composite functions, Math. Proc. Cambridge Philos. Soc., 83 (1978) , 159-165. doi: 10.1017/S0305004100054402.
	J. D. Jackson, Classical Electrodynamics Second edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1975.
	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.
	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.
	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.
	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.
	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.
	A. Lindner, $C^k$-Regularität der Lösungen des Vlasov-Poisson-Systems partieller Differentialgleichungen Diploma thesis, Ludwig Maximilian Universität, München, 1991.
	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.
	G. Rein, Collisionless kinetic equations from astrophysics—the Vlasov-Poisson system, in Handbook of differential equations: evolutionary equations, Handb. Differ. Equ. , Elsevier, 3 (2007), 383–476.
	J. Schaeffer , The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 104 (1986) , 403-421.
	H. Spohn, Dynamics of Charged Particles and their Radiation Field Cambridge University press, Cambridge, 2004.
	E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970.
	W. Walter, Ordinary Differential Equations Springer-Verlag, New York, 1998.