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December  2019, 12(6): 1313-1327. doi: 10.3934/krm.2019051

Focusing solutions of the Vlasov-Poisson system

Katherine Zhiyuan Zhang

Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912, USA

Received  February 2019 Revised  May 2019 Published  September 2019

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.

Keywords: Vlasov-Poisson equation, focusing solution, blow-up type behavior, spherically symmetric, kinetic theory.
Mathematics Subject Classification: 35Q83, 35B44, 35B40, 65M25.
Citation: Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
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.  Google Scholar

[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.  Google Scholar

[3] J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Horst, Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633.  doi: 10.1007/BF02125703.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

D. R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3] J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Horst, Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633.  doi: 10.1007/BF02125703.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

D. R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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