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Large amplitude stationary solutions of the Morrow model of gas ionization
Focusing solutions of the Vlasov-Poisson system
Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912, USA |
We study smooth, spherically-symmetric solutions to the Vlasov-Poisson system and relativistic Vlasov-Poisson system in the plasma physical case. We construct solutions that initially possess arbitrarily small $ C^k $ norms ($ k \geq 1 $) for the charge densities and the electric fields, but attain arbitrarily large $ L^\infty $ norms of them at some later time.
References:
| [1] |
J. Ben-Artzi, S. Calogero and S. Pankavich,
Arbitrarily large solutions of the Vlasov-Poisson system, SIAM Journal on Mathematical Analysis, 50 (2018), 4311-4326.
doi: 10.1137/17M1142715. |
| [2] |
J. Ben-Artzi, S. Calogero and S. Pankavich, Concentrating solutions of the relativistic Vlasov-Maxwell system, Commun. Math. Sci., 17 (2019), 377–392, arXiv: 1807.02801.
doi: 10.4310/CMS.2019.v17.n2.a4. |
| [3] |
J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.
|
| [4] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
| [5] |
R. Glassey, S. Pankavich and J. Schaeffer,
Decay in time for a one-dimensional two-component plasma, Math. Methods Appl. Sci., 31 (2008), 2115-2132.
doi: 10.1002/mma.1015. |
| [6] |
R. Glassey, S. Pankavich and J. Schaeffer,
On long-time behavior of monocharged and neutral plasma in one and one-half dimensions, Kinetic and Related Models, 2 (2009), 465-488.
doi: 10.3934/krm.2009.2.465. |
| [7] |
R. Glassey, S. Pankavich and J. Schaeffer,
Large time behavior of the relativistic Vlasov-Maxwell system in low space dimension, Differential and Integral Equations, 23 (2010), 61-77.
|
| [8] |
R. Glassey, S. Pankavich and J. Schaeffer,
Time decay for solutions to one-dimensional two component plasma equations, Quarterly of Applied Mathematics, 68 (2010), 135-141.
doi: 10.1090/S0033-569X-09-01143-4. |
| [9] |
R. T. Glassey and J. Schaeffer,
On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.
doi: 10.1007/BF01210740. |
| [10] |
E. Horst,
Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633.
doi: 10.1007/BF02125703. |
| [11] |
R. Illner and G. Rein,
Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case, Math. Methods Appl. Sci., 19 (1996), 1409-1413.
doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2. |
| [12] |
P.-L. Lions and B. Perthame,
Propogation of moments and regularity for the three dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
| [13] |
D. R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983. |
| [14] |
K. Pfaffelmoser,
Global classical solution of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eq., 95 (1992), 281-303.
doi: 10.1016/0022-0396(92)90033-J. |
| [15] |
G. Rein and L. Taegert,
Gravitational collapse and the Vlasov-Poisson system, Annales Henri Poincaré, 17 (2016), 1415-1427.
doi: 10.1007/s00023-015-0424-y. |
show all references
References:
| [1] |
J. Ben-Artzi, S. Calogero and S. Pankavich,
Arbitrarily large solutions of the Vlasov-Poisson system, SIAM Journal on Mathematical Analysis, 50 (2018), 4311-4326.
doi: 10.1137/17M1142715. |
| [2] |
J. Ben-Artzi, S. Calogero and S. Pankavich, Concentrating solutions of the relativistic Vlasov-Maxwell system, Commun. Math. Sci., 17 (2019), 377–392, arXiv: 1807.02801.
doi: 10.4310/CMS.2019.v17.n2.a4. |
| [3] |
J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.
|
| [4] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
| [5] |
R. Glassey, S. Pankavich and J. Schaeffer,
Decay in time for a one-dimensional two-component plasma, Math. Methods Appl. Sci., 31 (2008), 2115-2132.
doi: 10.1002/mma.1015. |
| [6] |
R. Glassey, S. Pankavich and J. Schaeffer,
On long-time behavior of monocharged and neutral plasma in one and one-half dimensions, Kinetic and Related Models, 2 (2009), 465-488.
doi: 10.3934/krm.2009.2.465. |
| [7] |
R. Glassey, S. Pankavich and J. Schaeffer,
Large time behavior of the relativistic Vlasov-Maxwell system in low space dimension, Differential and Integral Equations, 23 (2010), 61-77.
|
| [8] |
R. Glassey, S. Pankavich and J. Schaeffer,
Time decay for solutions to one-dimensional two component plasma equations, Quarterly of Applied Mathematics, 68 (2010), 135-141.
doi: 10.1090/S0033-569X-09-01143-4. |
| [9] |
R. T. Glassey and J. Schaeffer,
On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.
doi: 10.1007/BF01210740. |
| [10] |
E. Horst,
Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633.
doi: 10.1007/BF02125703. |
| [11] |
R. Illner and G. Rein,
Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case, Math. Methods Appl. Sci., 19 (1996), 1409-1413.
doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2. |
| [12] |
P.-L. Lions and B. Perthame,
Propogation of moments and regularity for the three dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
| [13] |
D. R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983. |
| [14] |
K. Pfaffelmoser,
Global classical solution of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eq., 95 (1992), 281-303.
doi: 10.1016/0022-0396(92)90033-J. |
| [15] |
G. Rein and L. Taegert,
Gravitational collapse and the Vlasov-Poisson system, Annales Henri Poincaré, 17 (2016), 1415-1427.
doi: 10.1007/s00023-015-0424-y. |
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