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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

Robert Glassey 1, , Stephen Pankavich 2, and  Jack Schaeffer 3,

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.
Keywords: Kinetic theory, global existence., Vlasov-Maxwell.
Mathematics Subject Classification: Primary: 35L60, 35Q83, 35Q99; Secondary: 82C21, 82C22, 82D1.
Citation: Robert Glassey, Stephen Pankavich, Jack Schaeffer. Separated characteristics and global solvability for the one and one-half dimensional Vlasov Maxwell system. Kinetic & 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.  Google Scholar

[2]

R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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]

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

[2]

R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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