Growth estimates and uniform decay for a collisionless plasma

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June 2011, 4(2): 549-567. doi: 10.3934/krm.2011.4.549

Growth estimates and uniform decay for a collisionless plasma

Christophe Pallard ^1,

1.	Laboratoire de Mathématiques, Université Paris-Sud 11, 91405 Orsay, France

Received January 2011 Revised February 2011 Published April 2011

Abstract
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We consider the classical Vlasov-Poisson system in three space dimensions in the electrostatic case. For smooth solutions starting from compactly supported initial data, an estimate on velocities is derived, showing an upper bound with a growth rate no larger than $(t\ln t)^{6/25}$. As a consequence, a decay estimate is obtained for the electric field in the $L^\infty$ norm.

Keywords: collisionless plasma., Vlasov-Poisson system.

Mathematics Subject Classification: Primary: 82D10; Secondary: 35B40, 76X0.

Citation: Christophe Pallard. Growth estimates and uniform decay for a collisionless plasma. Kinetic and Related Models, 2011, 4 (2) : 549-567. doi: 10.3934/krm.2011.4.549

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.
[2]	J. Batt, M. Kunze and G. Rein, On the asymptotic behaviour of a one-dimensional monocharged plasma, Adv. Differential Equations, 3 (1998), 271-292.
[3]	R. Glassey, "The Cauchy Problem in Kinetic Theory," SIAM, 1996.
[4]	E. Horst, Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633. doi: 10.1007/BF02125703.
[5]	E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Meth. Appl. Sci., 16 (1993), 75-85. doi: 10.1002/mma.1670160202.
[6]	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.
[7]	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.
[8]	C. Pallard, A note on the growth of velocities in a collisionless plasma, Math. Meth. Appl. Sci. (2011), to appear.
[9]	B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. P.D.E., 21 (1996), 659-686.
[10]	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.
[11]	G. Rein, Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case, Math. Nachr., 191 (1998), 269-278. doi: 10.1002/mana.19981910114.
[12]	G. Rein, Collisionless kinetic equations from astrophysics -- the Vlasov-Poisson system, Handbook of differential equations: evolutionary equations, Vol. III, 383-476, Elsevier/North-Holland, Amsterdam, 2007.
[13]	D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19 (2009), 199-228. doi: 10.1142/S0218202509003401.
[14]	J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. P.D.E., 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.
[15]	J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 34 (2010), 262-277. doi: 10.1002/mma.1354.

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.
[2]	J. Batt, M. Kunze and G. Rein, On the asymptotic behaviour of a one-dimensional monocharged plasma, Adv. Differential Equations, 3 (1998), 271-292.
[3]	R. Glassey, "The Cauchy Problem in Kinetic Theory," SIAM, 1996.
[4]	E. Horst, Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633. doi: 10.1007/BF02125703.
[5]	E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Meth. Appl. Sci., 16 (1993), 75-85. doi: 10.1002/mma.1670160202.
[6]	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.
[7]	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.
[8]	C. Pallard, A note on the growth of velocities in a collisionless plasma, Math. Meth. Appl. Sci. (2011), to appear.
[9]	B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. P.D.E., 21 (1996), 659-686.
[10]	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.
[11]	G. Rein, Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case, Math. Nachr., 191 (1998), 269-278. doi: 10.1002/mana.19981910114.
[12]	G. Rein, Collisionless kinetic equations from astrophysics -- the Vlasov-Poisson system, Handbook of differential equations: evolutionary equations, Vol. III, 383-476, Elsevier/North-Holland, Amsterdam, 2007.
[13]	D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19 (2009), 199-228. doi: 10.1142/S0218202509003401.
[14]	J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. P.D.E., 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.
[15]	J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 34 (2010), 262-277. doi: 10.1002/mma.1354.

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