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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 and 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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

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