# American Institute of Mathematical Sciences

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September  2016, 9(3): 455-467. doi: 10.3934/krm.2016003

## Separated characteristics and global solvability for the one and one-half dimensional Vlasov Maxwell system

 1 Department of Mathematics, Indiana University, Bloomington, IN 47405 2 Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80002 3 Department of Mathematics Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

Received  September 2015 Revised  January 2016 Published  May 2016

The motion of a collisionless plasma - a high-temperature, low-density, ionized gas - is described by the Vlasov-Maxwell (VM) system. These equations are considered in one space dimension and two momentum dimensions without the assumption of relativistic velocity corrections. The main results are bounds on the spatial and velocity supports of the particle distribution function and uniform estimates on derivatives of this function away from the critical velocity $\vert v_1 \vert = 1$. Additionally, for initial particle distributions that are even in the second velocity argument $v_2$, the global-in-time existence of solutions is shown.
Citation: Robert Glassey, Stephen Pankavich, Jack Schaeffer. Separated characteristics and global solvability for the one and one-half dimensional Vlasov Maxwell system. Kinetic and Related Models, 2016, 9 (3) : 455-467. doi: 10.3934/krm.2016003
##### References:
 [1] 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. [2] R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. [3] R. Glassey, S. Pankavich and J. Schaeffer, Large time behavior of the relativistic vlasov-maxwell system in low space dimension, Differential and Integral Equations, 23 (2010), 61-77. [4] R. Glassey, S. Pankavich and J. Schaeffer, Long-time behavior of monocharged and neutral plasmas in "One and one-half" dimensions, Kinetic and Related Models, 2 (2009), 465-488. doi: 10.3934/krm.2009.2.465. [5] R. Glassey, S. Pankavich and J. Schaeffer, Decay in time for a one-dimensional, two component plasma, Math. Meth. Appl. Sci., 31 (2008), 2115-2132. doi: 10.1002/mma.1015. [6] R. 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. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284. doi: 10.1007/s002200050090. [8] R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions I, Arch. Rational Mech. Anal., 141 (1998), 331-354. doi: 10.1007/s002050050079. [9] R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions II, Arch. Rational Mech. Anal., 141 (1998), 355-374. doi: 10.1007/s002050050079. [10] R. Glassey and W. 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. [11] R. Glassey and W. Strauss, Remarks on collisionless plasmas, Fluids and plasmas: geometry and dynamics, (Boulder, CO., 1983), Contemp. Math., Amer. Math. Soc., Providence, RI, 28 (1984), 269-279. doi: 10.1090/conm/028/751989. [12] P. Lions and P. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273. [13] S. Pankavich and C. Nguyen, A one-dimensional kinetic model of plasma dynamics with a transport field, Evolution Equations and Control Theory, 3 (2014), 681-698. doi: 10.3934/eect.2014.3.681. [14] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J. [15] J. Schaeffer, Global existence of smooth solutions to the vlasov-poisson system in three dimensions, Commun. PDE, 16 (1991), 1313-1335. doi: 10.1080/03605309108820801. [16] N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley: New York, NY, 1967.

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##### References:
 [1] 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. [2] R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. [3] R. Glassey, S. Pankavich and J. Schaeffer, Large time behavior of the relativistic vlasov-maxwell system in low space dimension, Differential and Integral Equations, 23 (2010), 61-77. [4] R. Glassey, S. Pankavich and J. Schaeffer, Long-time behavior of monocharged and neutral plasmas in "One and one-half" dimensions, Kinetic and Related Models, 2 (2009), 465-488. doi: 10.3934/krm.2009.2.465. [5] R. Glassey, S. Pankavich and J. Schaeffer, Decay in time for a one-dimensional, two component plasma, Math. Meth. Appl. Sci., 31 (2008), 2115-2132. doi: 10.1002/mma.1015. [6] R. 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. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284. doi: 10.1007/s002200050090. [8] R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions I, Arch. Rational Mech. Anal., 141 (1998), 331-354. doi: 10.1007/s002050050079. [9] R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions II, Arch. Rational Mech. Anal., 141 (1998), 355-374. doi: 10.1007/s002050050079. [10] R. Glassey and W. 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. [11] R. Glassey and W. Strauss, Remarks on collisionless plasmas, Fluids and plasmas: geometry and dynamics, (Boulder, CO., 1983), Contemp. Math., Amer. Math. Soc., Providence, RI, 28 (1984), 269-279. doi: 10.1090/conm/028/751989. [12] P. Lions and P. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273. [13] S. Pankavich and C. Nguyen, A one-dimensional kinetic model of plasma dynamics with a transport field, Evolution Equations and Control Theory, 3 (2014), 681-698. doi: 10.3934/eect.2014.3.681. [14] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J. [15] J. Schaeffer, Global existence of smooth solutions to the vlasov-poisson system in three dimensions, Commun. PDE, 16 (1991), 1313-1335. doi: 10.1080/03605309108820801. [16] N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley: New York, NY, 1967.
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