December  2016, 9(4): 657-686. doi: 10.3934/krm.2016011

A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror

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

Received  October 2015 Revised  March 2016 Published  September 2016

We study the time evolution of a single species positive plasma, confined in a cylinder and having infinite charge. We extend the result of a previous work by the same authors, for a plasma density having compact support in the velocities, to the case of a density having unbounded support and gaussian decay in the velocities.
Citation: Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic and Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
References:
[1]

E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rat. Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.

[2]

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. doi: 10.1081/PDE-120002874.

[3]

S. Caprino, G. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement, Kinetic and Related Models, 5 (2012), 729-742. doi: 10.3934/krm.2012.5.729.

[4]

S. Caprino, G. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164. doi: 10.1137/130916527.

[5]

S. Caprino, G. Cavallaro and C. Marchioro, Remark on a magnetically confined plasma with infinite charge, Rend. Mat. Appl., 35 (2014), 69-98.

[6]

S. Caprino, G. Cavallaro and C. Marchioro, On a Vlasov-Poisson plasma confined in a torus by a magnetic mirror, J. Math. Anal. Appl., 427 (2015), 31-46. doi: 10.1016/j.jmaa.2015.02.012.

[7]

R. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[8]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.

[9]

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

[10]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 31 (2006), 349-370. doi: 10.1080/03605300500358004.

[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. doi: 10.1016/0022-0396(92)90033-J.

[12]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Part. Diff. Eq., 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.

[13]

J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186.

[14]

J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354.

[15]

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.

[16]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl., 176 (1993), 76-91. doi: 10.1006/jmaa.1993.1200.

show all references

References:
[1]

E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rat. Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.

[2]

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. doi: 10.1081/PDE-120002874.

[3]

S. Caprino, G. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement, Kinetic and Related Models, 5 (2012), 729-742. doi: 10.3934/krm.2012.5.729.

[4]

S. Caprino, G. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164. doi: 10.1137/130916527.

[5]

S. Caprino, G. Cavallaro and C. Marchioro, Remark on a magnetically confined plasma with infinite charge, Rend. Mat. Appl., 35 (2014), 69-98.

[6]

S. Caprino, G. Cavallaro and C. Marchioro, On a Vlasov-Poisson plasma confined in a torus by a magnetic mirror, J. Math. Anal. Appl., 427 (2015), 31-46. doi: 10.1016/j.jmaa.2015.02.012.

[7]

R. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[8]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.

[9]

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

[10]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 31 (2006), 349-370. doi: 10.1080/03605300500358004.

[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. doi: 10.1016/0022-0396(92)90033-J.

[12]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Part. Diff. Eq., 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.

[13]

J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. Part. Diff. Eq., 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186.

[14]

J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354.

[15]

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.

[16]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl., 176 (1993), 76-91. doi: 10.1006/jmaa.1993.1200.

[1]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic and Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Silvia Caprino, Carlo Marchioro. On a charge interacting with a plasma of unbounded mass. Kinetic and Related Models, 2011, 4 (1) : 215-226. doi: 10.3934/krm.2011.4.215

[7]

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

[8]

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

[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]

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

[11]

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

[12]

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

[13]

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

[14]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic and Related Models, 2022, 15 (4) : 721-727. doi: 10.3934/krm.2021021

[15]

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

[16]

Xianglong Duan. Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data. Kinetic and Related Models, 2022, 15 (1) : 119-146. doi: 10.3934/krm.2021049

[17]

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

[18]

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

[19]

C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934/cpaa.2011.10.1663

[20]

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

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (164)
  • HTML views (0)
  • Cited by (8)

[Back to Top]