American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.   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), 42 (1989), 729.  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.  doi: 10.1002/mma.1670130207.  Google Scholar

[7]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, $1^{st}$ edition, (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.  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.  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.   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.  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.  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, (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.  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.  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.  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.  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.  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, (1978).   Google Scholar

[22]

N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics,, Wiley, (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.  doi: 10.1512/iumj.1990.39.39009.  Google Scholar

[24]

C. Villani, Hypocoercivity,, Memoirs of the American Mathematical Society, (2009).  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.  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.  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.  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.  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.   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), 42 (1989), 729.  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.  doi: 10.1002/mma.1670130207.  Google Scholar

[7]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, $1^{st}$ edition, (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.  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.  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.   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.  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.  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, (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.  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.  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.  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.  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.  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, (1978).   Google Scholar

[22]

N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics,, Wiley, (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.  doi: 10.1512/iumj.1990.39.39009.  Google Scholar

[24]

C. Villani, Hypocoercivity,, Memoirs of the American Mathematical Society, (2009).  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.  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 - A, 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]

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
[8]

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
[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]

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
[17]

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
[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 - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[20]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences