American Institute of Mathematical Sciences

Advanced
December  2014, 3(4): 681-698. doi: 10.3934/eect.2014.3.681

A one-dimensional kinetic model of plasma dynamics with a transport field

Charles Nguyen 1, and  Stephen Pankavich 2,

1. 

Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, United States
2. 

Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80002, United States

Received  January 2014 Revised  April 2014 Published  October 2014

Motivated by the fundamental model of a collisionless plasma, the Vlasov-Maxwell (VM) system, we consider a related, nonlinear system of partial differential equations in one space and one momentum dimension. As little is known regarding the regularity properties of solutions to the non-relativistic version of the (VM) equations, we study a simplified system which also lacks relativistic velocity corrections and prove local-in-time existence and uniqueness of classical solutions to the Cauchy problem. For special choices of initial data, global-in-time existence of these solutions is also shown. Finally, we provide an estimate which, independent of the choice of initial data, yields additional global-in-time regularity of the associated field.
Keywords: local-in-time existence, regularity., Vlasov equation, Kinetic theory.
Mathematics Subject Classification: Primary: 35L60, 35Q83; Secondary: 82C22, 82D1.
Citation: Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations & Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681
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.  doi: 10.1007/s00205-003-0265-6.  Google Scholar

[2]

D. Brewer and S. Pankavich, Computational Methods for a One-dimensional Plasma Model with a Transport Field,, SIAM Undergraduate Research Online, 4 (2011), 81.   Google Scholar

[3]

R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Comm. Pure Appl. Math., 42 (1989), 729.  doi: 10.1002/cpa.3160420603.  Google Scholar

[4]

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

[5]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[6]

R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system,, Math. Methods Appl. Sci., 13 (1990), 169.  doi: 10.1002/mma.1670130207.  Google Scholar

[7]

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.  doi: 10.1007/BF00250732.  Google Scholar

[8]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system,, Commun. Pure Appl. Anal., 1 (2002), 103.   Google Scholar

[9]

M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system,, Electron. J. Differential Equations (electronic - 17 pp.), 1 (2005).   Google Scholar

[10]

S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics,, Comm. Partial Differential Equations, 31 (2006), 349.  doi: 10.1080/03605300500358004.  Google Scholar

[11]

S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass,, Math. Methods Appl. Sci., 30 (2007), 529.  doi: 10.1002/mma.796.  Google Scholar

[12]

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

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.  doi: 10.1007/s00205-003-0265-6.  Google Scholar

[2]

D. Brewer and S. Pankavich, Computational Methods for a One-dimensional Plasma Model with a Transport Field,, SIAM Undergraduate Research Online, 4 (2011), 81.   Google Scholar

[3]

R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Comm. Pure Appl. Math., 42 (1989), 729.  doi: 10.1002/cpa.3160420603.  Google Scholar

[4]

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

[5]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[6]

R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system,, Math. Methods Appl. Sci., 13 (1990), 169.  doi: 10.1002/mma.1670130207.  Google Scholar

[7]

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.  doi: 10.1007/BF00250732.  Google Scholar

[8]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system,, Commun. Pure Appl. Anal., 1 (2002), 103.   Google Scholar

[9]

M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system,, Electron. J. Differential Equations (electronic - 17 pp.), 1 (2005).   Google Scholar

[10]

S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics,, Comm. Partial Differential Equations, 31 (2006), 349.  doi: 10.1080/03605300500358004.  Google Scholar

[11]

S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass,, Math. Methods Appl. Sci., 30 (2007), 529.  doi: 10.1002/mma.796.  Google Scholar

[12]

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

[1]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Local existence with mild regularity for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 1011-1041. doi: 10.3934/krm.2013.6.1011
[2]

Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
[3]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[4]

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
[5]

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 & Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003
[6]

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
[7]

Seung-Yeal Ha, Shi Jin, Jinwook Jung. A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks & Heterogeneous Media, 2019, 14 (2) : 317-340. doi: 10.3934/nhm.2019013
[8]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic & Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787
[9]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[10]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[11]

Miroslav Grmela, Michal Pavelka. Landau damping in the multiscale Vlasov theory. Kinetic & Related Models, 2018, 11 (3) : 521-545. doi: 10.3934/krm.2018023
[12]

Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic & Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019
[13]

Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79
[14]

Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
[15]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621
[16]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[17]

Xie Li, Zhaoyin Xiang. Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1465-1480. doi: 10.3934/cpaa.2014.13.1465
[18]

Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311
[19]

Md. Rabiul Haque, Takayoshi Ogawa, Ryuichi Sato. Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space. Communications on Pure & Applied Analysis, 2020, 19 (2) : 677-697. doi: 10.3934/cpaa.2020031
[20]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences