Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system

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March 2015, 8(1): 169-199. doi: 10.3934/krm.2015.8.169

Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system

Stephen Pankavich ^1, and Nicholas Michalowski ^2,

1.	Department of Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois St, Golden, CO 80401, United States
2.	Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, United States

Received May 2014 Revised August 2014 Published December 2014

Abstract
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In a recent paper Calogero and Alcántara [Kinet. Relat. Models, 4 (2011), pp. 401-426] derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the ``one and one-half'' dimensional version of this problem with nonlinear electromagnetic interactions - the relativistic Vlasov-Maxwell-Fokker-Planck system - and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.

Keywords: Vlasov, Kinetic theory, regularity., Fokker-Planck equation, global existence.

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

Citation: Stephen Pankavich, Nicholas Michalowski. Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system. Kinetic and Related Models, 2015, 8 (1) : 169-199. doi: 10.3934/krm.2015.8.169

References:

[1]	J. Alcántara and S. Calogero, On a relativistic Fokker-Planck equation in kinetic theory, Kinetic and Related Models, 4 (2011), 401-426. doi: 10.3934/krm.2011.4.401.
[2]	F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Archive for Rational Mechanics and Analysis, 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6.
[3]	M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1007-1025. doi: 10.1142/S0218202511005222.
[4]	P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 19 (1986), 519-542.
[5]	R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Communications on Pure and Applied Mathematics 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.
[6]	R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Mathematical Methods in the Applied Sciences, 13 (1990), 169-179. doi: 10.1002/mma.1670130207.
[7]	R. T. Glassey, The Cauchy Problem in Kinetic Theory, $1^{st}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.
[8]	R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Archive for Rational Mechanics and Analysis, 92 (1986), 59-90. doi: 10.1007/BF00250732.
[9]	F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, Journal of Functional Analysis, 244 (2007), 95-118. doi: 10.1016/j.jfa.2006.11.013.
[10]	S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Communications on Pure and Applied Analysis, 1 (2002), 103-125.
[11]	R. Lai, On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system, Mathematical Methods in the Applied Sciences, 18 (1995), 1013-1040. doi: 10.1002/mma.1670181302.
[12]	R. Lai, On the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Mathematical Methods in the Applied Sciences, 21 (1998), 1287-1296. doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G.
[13]	J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin, 1961.
[14]	P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Inventiones Mathematicae, 105 (1991), 415-430. doi: 10.1007/BF01232273.
[15]	S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Communications in Partial Differential Equations, 31 (2006), 349-370. doi: 10.1080/03605300500358004.
[16]	S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity, Submitted, 2014.
[17]	S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system, Submitted, 2014.
[18]	K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Journal of Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.
[19]	J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Communications in Mathematical Physics, 104 (1986), 403-421. doi: 10.1007/BF01210948.
[20]	J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Communications in Partial Differential Equations, 16 (1991), 1312-1335. doi: 10.1080/03605309108820801.
[21]	L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay 78, Vol. 13, Université de Paris-Sud Département de Mathématique, Orsay, 1978.
[22]	N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley, New York, NY, 1967.
[23]	H. D. Victory, Jr. and B. P. O'Dwyer, On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana University Mathematics Journal, 39 (1990), 105-156. doi: 10.1512/iumj.1990.39.39009.
[24]	C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, Vol. 202, (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.
[25]	T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM Journal on Mathematical Analysis, 42 (2010), 459-488. doi: 10.1137/090755796.

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References:

[1]	J. Alcántara and S. Calogero, On a relativistic Fokker-Planck equation in kinetic theory, Kinetic and Related Models, 4 (2011), 401-426. doi: 10.3934/krm.2011.4.401.
[2]	F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Archive for Rational Mechanics and Analysis, 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6.
[3]	M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1007-1025. doi: 10.1142/S0218202511005222.
[4]	P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 19 (1986), 519-542.
[5]	R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Communications on Pure and Applied Mathematics 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.
[6]	R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Mathematical Methods in the Applied Sciences, 13 (1990), 169-179. doi: 10.1002/mma.1670130207.
[7]	R. T. Glassey, The Cauchy Problem in Kinetic Theory, $1^{st}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.
[8]	R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Archive for Rational Mechanics and Analysis, 92 (1986), 59-90. doi: 10.1007/BF00250732.
[9]	F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, Journal of Functional Analysis, 244 (2007), 95-118. doi: 10.1016/j.jfa.2006.11.013.
[10]	S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Communications on Pure and Applied Analysis, 1 (2002), 103-125.
[11]	R. Lai, On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system, Mathematical Methods in the Applied Sciences, 18 (1995), 1013-1040. doi: 10.1002/mma.1670181302.
[12]	R. Lai, On the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Mathematical Methods in the Applied Sciences, 21 (1998), 1287-1296. doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G.
[13]	J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin, 1961.
[14]	P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Inventiones Mathematicae, 105 (1991), 415-430. doi: 10.1007/BF01232273.
[15]	S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Communications in Partial Differential Equations, 31 (2006), 349-370. doi: 10.1080/03605300500358004.
[16]	S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity, Submitted, 2014.
[17]	S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system, Submitted, 2014.
[18]	K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Journal of Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.
[19]	J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Communications in Mathematical Physics, 104 (1986), 403-421. doi: 10.1007/BF01210948.
[20]	J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Communications in Partial Differential Equations, 16 (1991), 1312-1335. doi: 10.1080/03605309108820801.
[21]	L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay 78, Vol. 13, Université de Paris-Sud Département de Mathématique, Orsay, 1978.
[22]	N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley, New York, NY, 1967.
[23]	H. D. Victory, Jr. and B. P. O'Dwyer, On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana University Mathematics Journal, 39 (1990), 105-156. doi: 10.1512/iumj.1990.39.39009.
[24]	C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, Vol. 202, (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.
[25]	T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM Journal on Mathematical Analysis, 42 (2010), 459-488. doi: 10.1137/090755796.

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