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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin, 1961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity, Submitted, 2014. Google Scholar

[17]

S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system, Submitted, 2014. Google Scholar

[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.  Google Scholar

[19]

J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Communications in Mathematical Physics, 104 (1986), 403-421. doi: 10.1007/BF01210948.  Google Scholar

[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.  Google Scholar

[21]

L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay 78, Vol. 13, Université de Paris-Sud Département de Mathématique, Orsay, 1978.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, Vol. 202, (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin, 1961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity, Submitted, 2014. Google Scholar

[17]

S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system, Submitted, 2014. Google Scholar

[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.  Google Scholar

[19]

J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Communications in Mathematical Physics, 104 (1986), 403-421. doi: 10.1007/BF01210948.  Google Scholar

[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.  Google Scholar

[21]

L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay 78, Vol. 13, Université de Paris-Sud Département de Mathématique, Orsay, 1978.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, Vol. 202, (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  Google Scholar

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