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

A toy model for the relativistic Vlasov-Maxwell system

Jonathan Ben-Artzi 1,, , Stephen Pankavich 2, and  Junyong Zhang 3,

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.

Keywords: Kinetic theory, Vlasov-Maxwell system, global existence.
Mathematics Subject Classification: Primary: 35Q83, 35B40; Secondary: 82D10.
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. Bouchut, F. 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. Bouchut, F. 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.

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