# American Institute of Mathematical Sciences

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

## Growth estimates and uniform decay for a collisionless plasma

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

Received  January 2011 Revised  February 2011 Published  April 2011

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.
Citation: Christophe Pallard. Growth estimates and uniform decay for a collisionless plasma. Kinetic & 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. Poincar\'e Anal. Non Lin\'eaire, 2 (1985), 101. Google Scholar [2] J. Batt, M. Kunze and G. Rein, On the asymptotic behaviour of a one-dimensional monocharged plasma,, Adv. Differential Equations, 3 (1998), 271. Google Scholar [3] R. Glassey, "The Cauchy Problem in Kinetic Theory,", SIAM, (1996). Google Scholar [4] E. Horst, Symmetric plasmas and their decay,, Comm. Math. Phys., 126 (1990), 613. doi: 10.1007/BF02125703. Google Scholar [5] E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 16 (1993), 75. doi: 10.1002/mma.1670160202. Google Scholar [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. doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2. Google Scholar [7] P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1991), 415. doi: 10.1007/BF01232273. Google Scholar [8] C. Pallard, A note on the growth of velocities in a collisionless plasma,, Math. Meth. Appl. Sci. (2011), (2011). Google Scholar [9] B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations,, Comm. P.D.E., 21 (1996), 659. Google Scholar [10] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Diff. Eqns., 95 (1992), 281. doi: 10.1016/0022-0396(92)90033-J. Google Scholar [11] G. Rein, Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case,, Math. Nachr., 191 (1998), 269. doi: 10.1002/mana.19981910114. Google Scholar [12] G. Rein, Collisionless kinetic equations from astrophysics -- the Vlasov-Poisson system,, Handbook of differential equations: evolutionary equations, III (2007), 383. Google Scholar [13] D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation,, Math. Models Methods Appl. Sci., 19 (2009), 199. doi: 10.1142/S0218202509003401. Google Scholar [14] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Comm. P.D.E., 16 (1991), 1313. doi: 10.1080/03605309108820801. Google Scholar [15] J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 34 (2010), 262. doi: 10.1002/mma.1354. 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. Poincar\'e Anal. Non Lin\'eaire, 2 (1985), 101. Google Scholar [2] J. Batt, M. Kunze and G. Rein, On the asymptotic behaviour of a one-dimensional monocharged plasma,, Adv. Differential Equations, 3 (1998), 271. Google Scholar [3] R. Glassey, "The Cauchy Problem in Kinetic Theory,", SIAM, (1996). Google Scholar [4] E. Horst, Symmetric plasmas and their decay,, Comm. Math. Phys., 126 (1990), 613. doi: 10.1007/BF02125703. Google Scholar [5] E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 16 (1993), 75. doi: 10.1002/mma.1670160202. Google Scholar [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. doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2. Google Scholar [7] P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1991), 415. doi: 10.1007/BF01232273. Google Scholar [8] C. Pallard, A note on the growth of velocities in a collisionless plasma,, Math. Meth. Appl. Sci. (2011), (2011). Google Scholar [9] B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations,, Comm. P.D.E., 21 (1996), 659. Google Scholar [10] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Diff. Eqns., 95 (1992), 281. doi: 10.1016/0022-0396(92)90033-J. Google Scholar [11] G. Rein, Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case,, Math. Nachr., 191 (1998), 269. doi: 10.1002/mana.19981910114. Google Scholar [12] G. Rein, Collisionless kinetic equations from astrophysics -- the Vlasov-Poisson system,, Handbook of differential equations: evolutionary equations, III (2007), 383. Google Scholar [13] D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation,, Math. Models Methods Appl. Sci., 19 (2009), 199. doi: 10.1142/S0218202509003401. Google Scholar [14] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Comm. P.D.E., 16 (1991), 1313. doi: 10.1080/03605309108820801. Google Scholar [15] J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 34 (2010), 262. doi: 10.1002/mma.1354. Google Scholar
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