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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

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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley: New York, NY, 1967. Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley: New York, NY, 1967. Google Scholar

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