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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

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

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.
Keywords: unbounded velocities., Vlasov-Poisson equation, unbounded mass, magnetic confinement.
Mathematics Subject Classification: Primary: 82D10, 35Q99, 76X05; Secondary: 35L6.
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 & 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. doi: 10.1007/s002050100150. Google Scholar

[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. doi: 10.1081/PDE-120002874. Google Scholar

[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. doi: 10.3934/krm.2012.5.729. Google Scholar

[4]

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

[5]

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

[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. doi: 10.1016/j.jmaa.2015.02.012. Google Scholar

[7]

R. Glassey, The Cauchy Problem in Kinetic Theory,, SIAM, (1996). doi: 10.1137/1.9781611971477. Google Scholar

[8]

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

[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. doi: 10.3934/krm.2015.8.153. 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. doi: 10.1080/03605300500358004. 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. doi: 10.1016/0022-0396(92)90033-J. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

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. doi: 10.1007/s002050100150. Google Scholar

[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. doi: 10.1081/PDE-120002874. Google Scholar

[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. doi: 10.3934/krm.2012.5.729. Google Scholar

[4]

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

[5]

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

[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. doi: 10.1016/j.jmaa.2015.02.012. Google Scholar

[7]

R. Glassey, The Cauchy Problem in Kinetic Theory,, SIAM, (1996). doi: 10.1137/1.9781611971477. Google Scholar

[8]

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

[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. doi: 10.3934/krm.2015.8.153. 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. doi: 10.1080/03605300500358004. 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. doi: 10.1016/0022-0396(92)90033-J. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

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