A one-dimensional kinetic model of plasma dynamics with a transport field

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December 2014, 3(4): 681-698. doi: 10.3934/eect.2014.3.681

A one-dimensional kinetic model of plasma dynamics with a transport field

Charles Nguyen ^1, and Stephen Pankavich ^2,

1.	Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, United States
2.	Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80002, United States

Received January 2014 Revised April 2014 Published October 2014

Abstract
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Motivated by the fundamental model of a collisionless plasma, the Vlasov-Maxwell (VM) system, we consider a related, nonlinear system of partial differential equations in one space and one momentum dimension. As little is known regarding the regularity properties of solutions to the non-relativistic version of the (VM) equations, we study a simplified system which also lacks relativistic velocity corrections and prove local-in-time existence and uniqueness of classical solutions to the Cauchy problem. For special choices of initial data, global-in-time existence of these solutions is also shown. Finally, we provide an estimate which, independent of the choice of initial data, yields additional global-in-time regularity of the associated field.

Keywords: local-in-time existence, regularity., Vlasov equation, Kinetic theory.

Mathematics Subject Classification: Primary: 35L60, 35Q83; Secondary: 82C22, 82D1.

Citation: Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations and Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681

References:

[1]	F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6.
[2]	D. Brewer and S. Pankavich, Computational Methods for a One-dimensional Plasma Model with a Transport Field, SIAM Undergraduate Research Online, 4 (2011), 81-104.
[3]	R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.
[4]	P. Gerard and C. Pallard, A mean-field toy model for resonant transport, Kinet. Relat. Models, 3 (2010), 299-309. doi: 10.3934/krm.2010.3.299.
[5]	R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.
[6]	R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179. doi: 10.1002/mma.1670130207.
[7]	R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90. doi: 10.1007/BF00250732.
[8]	S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.
[9]	M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system, Electron. J. Differential Equations (electronic - 17 pp.), 1 (2005), 17pp.
[10]	S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 31 (2006), 349-370. doi: 10.1080/03605300500358004.
[11]	S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass, Math. Methods Appl. Sci., 30 (2007), 529-548. doi: 10.1002/mma.796.
[12]	N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley: New York, NY, 1967.

show all references

References:

[1]	F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6.
[2]	D. Brewer and S. Pankavich, Computational Methods for a One-dimensional Plasma Model with a Transport Field, SIAM Undergraduate Research Online, 4 (2011), 81-104.
[3]	R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.
[4]	P. Gerard and C. Pallard, A mean-field toy model for resonant transport, Kinet. Relat. Models, 3 (2010), 299-309. doi: 10.3934/krm.2010.3.299.
[5]	R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.
[6]	R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179. doi: 10.1002/mma.1670130207.
[7]	R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90. doi: 10.1007/BF00250732.
[8]	S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.
[9]	M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system, Electron. J. Differential Equations (electronic - 17 pp.), 1 (2005), 17pp.
[10]	S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 31 (2006), 349-370. doi: 10.1080/03605300500358004.
[11]	S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass, Math. Methods Appl. Sci., 30 (2007), 529-548. doi: 10.1002/mma.796.
[12]	N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley: New York, NY, 1967.

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