# American Institute of Mathematical Sciences

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

 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.
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
 [1] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [2] John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371 [3] Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056 [4] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 [5] Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028 [6] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [7] Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016 [8] Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485 [9] Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169 [10] Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016 [11] Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044 [12] Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008 [13] Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 [14] Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028 [15] Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic & Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393 [16] Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124 [17] Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681 [18] Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135 [19] Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215 [20] Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

2020 Impact Factor: 1.432