American Institute of Mathematical Sciences

Advanced
December  2012, 5(4): 729-742. doi: 10.3934/krm.2012.5.729

Time evolution of a Vlasov-Poisson plasma with magnetic confinement

Silvia Caprino 1, , Guido Cavallaro 2, and  Carlo Marchioro 2,

1. 

Dipartimento di Matematica, Università di Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma
2. 

Dipartimento di Matematica "Guido Castelnuovo", Università La Sapienza P.le A. Moro 5, 00185 Roma, Italy, Italy

Received  June 2012 Revised  June 2012 Published  November 2012

We study the time evolution of a Vlasov-Poisson plasma moving in an infinite cylinder, in which it is confined by an unbounded external magnetic field. This field depends only on the distance from the border of the cylinder, is tangent to the border and singular on it. We prove the existence and uniqueness of the solution, and also its confinement inside the cylinder for all times, i.e. the external field behaves like a magnetic mirror. Possible generalizations are discussed.
Keywords: Vlasov-Poisson equation, magnetic confinement..
Mathematics Subject Classification: Primary: 82D10, 35Q99, 76X05; Secondary: 35L6.
Citation: Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729
References:
[1]

J. Batt and G. Rein, Global classical solutions of a periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris, 313 S. I. (1991), 411-416.  Google Scholar

[2]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinetic and Related Models, 3 (2010), 241-254.  Google Scholar

[3]

S. Caprino and C. Marchioro, On a charge interacting with a plasma of unbounded mass, Kinetic and Related Models, 4 (2011), 215-226.  Google Scholar

[4]

S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass, Comm. Part. Diff. Eq., 27 (2002), 791-808.  Google Scholar

[5]

S. Caprino, C. Marchioro, E. Miot and M. Pulvirenti, On the attractive plasma-charge system in 2-d, Comm. Part. Diff. Eq., 37 (2012), 1237-1272. doi: 10.1080/3605302.2011.653032.  Google Scholar

[6]

R. Glassey, "The Cauchy Problem in Kinetic Theory," SIAM: Philadelphia, PA, 1996.  Google Scholar

[7]

P. E. Jabin, The Vlasov-Poisson system with infinite mass and energy, J. Stat. Phys., 103 (2001), 1107-1123.  Google Scholar

[8]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1996), 415-430. Google Scholar

[9]

C. Marchioro, E. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges, Arch. Rat. Mech. Anal., 201 (2011), 1-26. doi: 10.1007/s00205-010-0388-5.  Google Scholar

[10]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 31 (2006), 349-370.  Google Scholar

[11]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Jour. Diff. Eq., 95 (1992), 281-303.  Google Scholar

[12]

G. Rein, Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case, Math. Nachr., 191 (1998), 269-278. doi: 10.1002/mana.19981910114.  Google Scholar

[13]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Mod. Meth. Appl. Sci., 19 (2009), 199-228.  Google Scholar

[14]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Part. Diff. Eq., 16 (1991), 1313-1335.  Google Scholar

[15]

J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 28 (2003), 1057-1084.  Google Scholar

[16]

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

[17]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, Jour. Math. Anal. Appl., 176 (1993), 76-91.  Google Scholar

show all references

References:
[1]

J. Batt and G. Rein, Global classical solutions of a periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris, 313 S. I. (1991), 411-416.  Google Scholar

[2]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinetic and Related Models, 3 (2010), 241-254.  Google Scholar

[3]

S. Caprino and C. Marchioro, On a charge interacting with a plasma of unbounded mass, Kinetic and Related Models, 4 (2011), 215-226.  Google Scholar

[4]

S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass, Comm. Part. Diff. Eq., 27 (2002), 791-808.  Google Scholar

[5]

S. Caprino, C. Marchioro, E. Miot and M. Pulvirenti, On the attractive plasma-charge system in 2-d, Comm. Part. Diff. Eq., 37 (2012), 1237-1272. doi: 10.1080/3605302.2011.653032.  Google Scholar

[6]

R. Glassey, "The Cauchy Problem in Kinetic Theory," SIAM: Philadelphia, PA, 1996.  Google Scholar

[7]

P. E. Jabin, The Vlasov-Poisson system with infinite mass and energy, J. Stat. Phys., 103 (2001), 1107-1123.  Google Scholar

[8]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1996), 415-430. Google Scholar

[9]

C. Marchioro, E. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges, Arch. Rat. Mech. Anal., 201 (2011), 1-26. doi: 10.1007/s00205-010-0388-5.  Google Scholar

[10]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 31 (2006), 349-370.  Google Scholar

[11]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Jour. Diff. Eq., 95 (1992), 281-303.  Google Scholar

[12]

G. Rein, Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case, Math. Nachr., 191 (1998), 269-278. doi: 10.1002/mana.19981910114.  Google Scholar

[13]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Mod. Meth. Appl. Sci., 19 (2009), 199-228.  Google Scholar

[14]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Part. Diff. Eq., 16 (1991), 1313-1335.  Google Scholar

[15]

J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 28 (2003), 1057-1084.  Google Scholar

[16]

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

[17]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, Jour. Math. Anal. Appl., 176 (1993), 76-91.  Google Scholar

[1]

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

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic & Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
[3]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic & Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018
[4]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[5]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic & Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153
[6]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[7]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[8]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[9]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[10]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
[11]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
[12]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004
[13]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021021
[14]

Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032
[15]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic & Related Models, 2015, 8 (3) : 615-616. doi: 10.3934/krm.2015.8.615
[16]

Frédérique Charles, Bruno Després, Benoît Perthame, Rémis Sentis. Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinetic & Related Models, 2013, 6 (2) : 269-290. doi: 10.3934/krm.2013.6.269
[17]

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic & Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005
[18]

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 & Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723
[19]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
[20]

Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic & Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (102)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences