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August  2022, 15(4): 721-727. doi: 10.3934/krm.2021021

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 Published  August 2022 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 and Related Models, 2022, 15 (4) : 721-727. 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.

[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.

[3]

J. BattM. 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. 

[4]

J. R. BurganM. R. FeixE. 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.

[5]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM: Philadelphia, 1996. doi: 10.1137/1.9781611971477.

[6]

R. GlasseyS. 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.

[7]

R. GlasseyS. Pankavich and J. Schaeffer, Large time behavior of the relativistic Vlasov Maxwell system in low space dimension, Diff. Integral Equations, 23 (2010), 61-77. 

[8]

R. GlasseyS. 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.

[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.

[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. Meth. 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, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.

[13]

S. Pankavich, Exact large time behavior of Spherically-Symmetric plasmas, https://arXiv.org/abs/2006.11447

[14]

B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations, 21 (1996), 659-686. 

[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.

[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.

[17]

J. Schaeffer, Large-time behavior of a one-dimensional monocharged plasma, Differential Integral Equations, 20 (2007), 277-292. 

show all references

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.

[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.

[3]

J. BattM. 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. 

[4]

J. R. BurganM. R. FeixE. 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.

[5]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM: Philadelphia, 1996. doi: 10.1137/1.9781611971477.

[6]

R. GlasseyS. 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.

[7]

R. GlasseyS. Pankavich and J. Schaeffer, Large time behavior of the relativistic Vlasov Maxwell system in low space dimension, Diff. Integral Equations, 23 (2010), 61-77. 

[8]

R. GlasseyS. 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.

[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.

[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. Meth. 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, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.

[13]

S. Pankavich, Exact large time behavior of Spherically-Symmetric plasmas, https://arXiv.org/abs/2006.11447

[14]

B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations, 21 (1996), 659-686. 

[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.

[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.

[17]

J. Schaeffer, Large-time behavior of a one-dimensional monocharged plasma, Differential Integral Equations, 20 (2007), 277-292. 

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