American Institute of Mathematical Sciences

Advanced
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

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 & 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.  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), 271-292.  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), 613-633. 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), 75-85. 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), 1409-1413. 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), 415-430. 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), to appear. Google Scholar

[9]

B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. P.D.E., 21 (1996), 659-686.  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), 281-303. 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), 269-278. 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, Vol. III, 383-476, Elsevier/North-Holland, Amsterdam, 2007.  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), 199-228. 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), 1313-1335. doi: 10.1080/03605309108820801.  Google Scholar

[15]

J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 34 (2010), 262-277. doi: 10.1002/mma.1354.  Google Scholar

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.  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), 271-292.  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), 613-633. 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), 75-85. 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), 1409-1413. 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), 415-430. 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), to appear. Google Scholar

[9]

B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. P.D.E., 21 (1996), 659-686.  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), 281-303. 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), 269-278. 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, Vol. III, 383-476, Elsevier/North-Holland, Amsterdam, 2007.  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), 199-228. 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), 1313-1335. doi: 10.1080/03605309108820801.  Google Scholar

[15]

J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 34 (2010), 262-277. doi: 10.1002/mma.1354.  Google Scholar

[1]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729
[2]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[3]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004
[4]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[5]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic & Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
[6]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[7]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[8]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[9]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
[10]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
[11]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[12]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021021
[13]

Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032
[14]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723
[15]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
[16]

Trinh T. Nguyen. Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria. Kinetic & Related Models, 2020, 13 (6) : 1193-1218. doi: 10.3934/krm.2020043
[17]

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic & Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005
[18]

Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic & Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729
[19]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic & Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018
[20]

Megan Griffin-Pickering, Mikaela Iacobelli. Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems. Kinetic & Related Models, 2021, 14 (4) : 571-597. doi: 10.3934/krm.2021016

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy