June  2022, 15(3): 341-354. doi: 10.3934/krm.2021053

A toy model for the relativistic Vlasov-Maxwell system

1. 

School of Mathematics, Cardiff University, Cardiff CF24 4AG, Wales, United Kingdom

2. 

Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, USA

3. 

School of Mathematics, Beijing Institute of Technology, Beijing, China

*Corresponding author: Jonathan Ben-Artzi

The authors thank C. Pallard for explaining to them some of the delicate aspects of the Division Lemma, and the anonymous referees whose comments helped improve the presentation of the paper.
The authors thank Claude Bardos and François Golse who proposed this problem over dinner during the workshop "The Cauchy Problem in Kinetic Theory: Recent Progress in Collisionless Models" which was held at Imperial College London in 2015. That workshop was held in honor of Bob Glassey, to whose memory this paper is dedicated.

Received  June 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

Fund Project: The first author acknowledges support from an Engineering and Physical Sciences Research Council Fellowship (EP/N020154/1). The second author acknowledges support from the US National Science Foundation under awards DMS-1911145 and DMS-2107938. The third author acknowledges support from the National Natural Science Foundation of China (11771041, 11831004) and a Marie Sk lodowska-Curie Fellowship (790623)

The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified "toy model" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.

Citation: Jonathan Ben-Artzi, Stephen Pankavich, Junyong Zhang. A toy model for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2022, 15 (3) : 341-354. doi: 10.3934/krm.2021053
References:
[1]

F. BouchutF. Golse and C. Pallard, Classical solutions and the glassey-strauss theorem for the 3D vlasov-maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15.  doi: 10.1007/s00205-003-0265-6.

[2]

P. Gérard and C. Pallard, A mean-field toy model for resonant transport, Kinet. Relat. Models, 3 (2010), 299-309.  doi: 10.3934/krm.2010.3.299.

[3]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.

[4]

R. T. Glassey and J. W. 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.

[5]

R. T. Glassey and J. W. Schaeffer, The "two and one-half dimensional" relativistic vlasov maxwell system, Comm. Math. Phys., 185 (1997), 257-284.  doi: 10.1007/s002200050090.

[6]

R. T. Glassey and W. A. 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.

[7]

R. T. Glassey and W. A. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys., 113 (1987), 191-208.  doi: 10.1007/BF01223511.

[8]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.  doi: 10.3934/cpaa.2002.1.103.

[9]

J. Luk and R. M. Strain, A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014)), 1005-1027.  doi: 10.1007/s00220-014-2108-8.

[10]

C. Nguyen and S. Pankavich, A one-dimensional kinetic model of plasma dynamics with a transport field, Evol. Equ. Control Theory, 3 (2014), 681-698.  doi: 10.3934/eect.2014.3.681.

[11]

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.

[12]

J. W. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 104 (1986), 403-421.  doi: 10.1007/BF01210948.

[13]

J. W. Schaeffer, Global existence of smooth solutions to the vlasov poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.

[14]

S. Wollman, An existence and uniqueness theorem for the Vlasov-Maxwell system, Comm. Pure Appl. Math., 37 (1984), 457-462.  doi: 10.1002/cpa.3160370404.

show all references

References:
[1]

F. BouchutF. Golse and C. Pallard, Classical solutions and the glassey-strauss theorem for the 3D vlasov-maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15.  doi: 10.1007/s00205-003-0265-6.

[2]

P. Gérard and C. Pallard, A mean-field toy model for resonant transport, Kinet. Relat. Models, 3 (2010), 299-309.  doi: 10.3934/krm.2010.3.299.

[3]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.

[4]

R. T. Glassey and J. W. 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.

[5]

R. T. Glassey and J. W. Schaeffer, The "two and one-half dimensional" relativistic vlasov maxwell system, Comm. Math. Phys., 185 (1997), 257-284.  doi: 10.1007/s002200050090.

[6]

R. T. Glassey and W. A. 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.

[7]

R. T. Glassey and W. A. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys., 113 (1987), 191-208.  doi: 10.1007/BF01223511.

[8]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.  doi: 10.3934/cpaa.2002.1.103.

[9]

J. Luk and R. M. Strain, A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014)), 1005-1027.  doi: 10.1007/s00220-014-2108-8.

[10]

C. Nguyen and S. Pankavich, A one-dimensional kinetic model of plasma dynamics with a transport field, Evol. Equ. Control Theory, 3 (2014), 681-698.  doi: 10.3934/eect.2014.3.681.

[11]

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.

[12]

J. W. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 104 (1986), 403-421.  doi: 10.1007/BF01210948.

[13]

J. W. Schaeffer, Global existence of smooth solutions to the vlasov poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.

[14]

S. Wollman, An existence and uniqueness theorem for the Vlasov-Maxwell system, Comm. Pure Appl. Math., 37 (1984), 457-462.  doi: 10.1002/cpa.3160370404.

[1]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153

[2]

Sergiu Klainerman, Gigliola Staffilani. A new approach to study the Vlasov-Maxwell system. Communications on Pure and Applied Analysis, 2002, 1 (1) : 103-125. doi: 10.3934/cpaa.2002.1.103

[3]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (3) : 615-616. doi: 10.3934/krm.2015.8.615

[4]

Mohammad Asadzadeh, Piotr Kowalczyk, Christoffer Standar. On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2019, 12 (1) : 105-131. doi: 10.3934/krm.2019005

[5]

Jin Woo Jang, Robert M. Strain, Tak Kwong Wong. Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus. Kinetic and Related Models, 2022, 15 (4) : 569-604. doi: 10.3934/krm.2021039

[6]

Yunbai Cao, Chanwoo Kim. Glassey-Strauss representation of Vlasov-Maxwell systems in a Half Space. Kinetic and Related Models, 2022, 15 (3) : 385-401. doi: 10.3934/krm.2021034

[7]

Zhiwu Lin. Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach. Kinetic and Related Models, 2022, 15 (4) : 663-679. doi: 10.3934/krm.2021048

[8]

Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic and Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003

[9]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic and Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129

[10]

Robert Glassey, Stephen Pankavich, Jack Schaeffer. Separated characteristics and global solvability for the one and one-half dimensional Vlasov Maxwell system. Kinetic and Related Models, 2016, 9 (3) : 455-467. doi: 10.3934/krm.2016003

[11]

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

[12]

Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic and Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901

[13]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723

[14]

Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic and Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040

[15]

Yemin Chen. Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 889-910. doi: 10.3934/dcds.2008.20.889

[16]

Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic and Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005

[17]

Dayton Preissl, Christophe Cheverry, Slim Ibrahim. Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system. Kinetic and Related Models, 2021, 14 (6) : 1035-1079. doi: 10.3934/krm.2021042

[18]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[19]

Andrea Bondesan, Marc Briant. Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2747-2773. doi: 10.3934/dcds.2021210

[20]

Renjun Duan, Shuangqian Liu, Tong Yang, Huijiang Zhao. Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials. Kinetic and Related Models, 2013, 6 (1) : 159-204. doi: 10.3934/krm.2013.6.159

2020 Impact Factor: 1.432

Article outline

[Back to Top]