# American Institute of Mathematical Sciences

doi: 10.3934/krm.2021021
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## On time decay for the spherically symmetric Vlasov-Poisson system

 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA, https://www.cmu.edu/math/people/faculty/schaeffer-j.html

In memory of Robert Glassey

Received  April 2021 Early access June 2021

A collisionless plasma is modeled by the Vlasov-Poisson system. Solutions in three space dimensions that have smooth, compactly supported initial data with spherical symmetry are considered. An improved field estimate is presented that is based on decay estimates obtained by Illner and Rein. Then some estimates are presented that ensure only particles with sufficiently small velocity can be found within a certain (time dependent) ball.

Citation: Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic & Related Models, doi: 10.3934/krm.2021021
##### References:
 [1] C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar [2] J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342–364. doi: 10.1016/0022-0396(77)90049-3.  Google Scholar [3] J. Batt, M. Kunze and G. Rein, On the asymptotic behavior of a one-dimensional, monocharged plasma and a rescaling method, Adv. Differential Equations, 3 (1998), 271-292.   Google Scholar [4] J. R. Burgan, M. R. Feix, E. Fijalkow and A. Munier, Self-similar and asymptotic solutions for a one-dimensional Vlasov beam, J. Plasma Physics, 29 (1983), 139-142.  doi: 10.1017/S0022377800000635.  Google Scholar [5] R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM: Philadelphia, 1996. doi: 10.1137/1.9781611971477.  Google Scholar [6] R. Glassey, S. Pankavich and J. Schaeffer, Decay in time for a one-dimensional, two component plasma, Math. Meth. Appl. Sci., 31 (2008), 2115-2132.  doi: 10.1002/mma.1015.  Google Scholar [7] R. Glassey, S. Pankavich and J. Schaeffer, Large time behavior of the relativistic Vlasov Maxwell system in low space dimension, Diff. Integral Equations, 23 (2010), 61-77.   Google Scholar [8] R. Glassey, S. Pankavich and J. Schaeffer, On long-time behavior of monocharged and neutral plasma in one and one-half dimensions, Kinet. Relat. Models, 2 (2009), 465-488.  doi: 10.3934/krm.2009.2.465.  Google Scholar [9] E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov-equation, parts Ⅰ and Ⅱ, Math. Meth. appl. Sci., 3 (1981), 229–248, and 4 (1982), 19–32. doi: 10.1002/mma.1670030117.  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. Meth. 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, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar [13] S. Pankavich, Exact large time behavior of Spherically-Symmetric plasmas, https://arXiv.org/abs/2006.11447 Google Scholar [14] B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations, 21 (1996), 659-686.   Google Scholar [15] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eqns., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar [16] G. Rein, Collisionless Kinetic Equations from Astrophysics – The Vlasov-Poisson System, in Handbook of Differential Equations, Evolutionary Equations, Vol. Ⅲ, 383–476, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007. doi: 10.1016/S1874-5717(07)80008-9.  Google Scholar [17] J. Schaeffer, Large-time behavior of a one-dimensional monocharged plasma, Differential Integral Equations, 20 (2007), 277-292.   Google Scholar

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##### References:
 [1] C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar [2] J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342–364. doi: 10.1016/0022-0396(77)90049-3.  Google Scholar [3] J. Batt, M. Kunze and G. Rein, On the asymptotic behavior of a one-dimensional, monocharged plasma and a rescaling method, Adv. Differential Equations, 3 (1998), 271-292.   Google Scholar [4] J. R. Burgan, M. R. Feix, E. Fijalkow and A. Munier, Self-similar and asymptotic solutions for a one-dimensional Vlasov beam, J. Plasma Physics, 29 (1983), 139-142.  doi: 10.1017/S0022377800000635.  Google Scholar [5] R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM: Philadelphia, 1996. doi: 10.1137/1.9781611971477.  Google Scholar [6] R. Glassey, S. Pankavich and J. Schaeffer, Decay in time for a one-dimensional, two component plasma, Math. Meth. Appl. Sci., 31 (2008), 2115-2132.  doi: 10.1002/mma.1015.  Google Scholar [7] R. Glassey, S. Pankavich and J. Schaeffer, Large time behavior of the relativistic Vlasov Maxwell system in low space dimension, Diff. Integral Equations, 23 (2010), 61-77.   Google Scholar [8] R. Glassey, S. Pankavich and J. Schaeffer, On long-time behavior of monocharged and neutral plasma in one and one-half dimensions, Kinet. Relat. Models, 2 (2009), 465-488.  doi: 10.3934/krm.2009.2.465.  Google Scholar [9] E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov-equation, parts Ⅰ and Ⅱ, Math. Meth. appl. Sci., 3 (1981), 229–248, and 4 (1982), 19–32. doi: 10.1002/mma.1670030117.  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. Meth. 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, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar [13] S. Pankavich, Exact large time behavior of Spherically-Symmetric plasmas, https://arXiv.org/abs/2006.11447 Google Scholar [14] B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations, 21 (1996), 659-686.   Google Scholar [15] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eqns., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar [16] G. Rein, Collisionless Kinetic Equations from Astrophysics – The Vlasov-Poisson System, in Handbook of Differential Equations, Evolutionary Equations, Vol. Ⅲ, 383–476, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007. doi: 10.1016/S1874-5717(07)80008-9.  Google Scholar [17] J. Schaeffer, Large-time behavior of a one-dimensional monocharged plasma, Differential Integral Equations, 20 (2007), 277-292.   Google Scholar
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