American Institute of Mathematical Sciences

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March  2011, 4(1): 215-226. doi: 10.3934/krm.2011.4.215

On a charge interacting with a plasma of unbounded mass

Silvia Caprino 1, and  Carlo Marchioro 2,

1. 

Dipartimento di Matematica, Università di Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma
2. 

Dipartimento di Matematica, Università La Sapienza, Piazzale Aldo Moro 2, 00185 Roma

Received  September 2010 Revised  November 2010 Published  January 2011

We consider a positive Vlasov-Helmholtz plasma in interaction with a positive point charge in $\R^2$ and we prove an existence and uniqueness theorem for this system without any assumption on the decay at infinity of the spatial density.
Keywords: infinitely extended systems., Vlasov-Helmholtz plasma.
Mathematics Subject Classification: Primary: 82D10, 35Q99; Secondary: 35L6.
Citation: Silvia Caprino, Carlo Marchioro. On a charge interacting with a plasma of unbounded mass. Kinetic and Related Models, 2011, 4 (1) : 215-226. doi: 10.3934/krm.2011.4.215
References:
[1]

P. Butta', G. Ferrari and C. Marchioro, Speedy motions of a body immersed in an infinitely extended medium, Jour. Stat. Phys., 140 (2010), 1182-1194.

[2]

E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rational Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.

[3]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinetic and Related Problems, 3 (2010), 241-254.

[4]

S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass, Commun. PDE, 27 (2002), 791-808. doi: 10.1081/PDE-120002874.

[5]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I, Math. Meth. Appl. Sci., 3 (1981), 229-248. doi: 10.1002/mma.1670030117.

[6]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II, Math. Meth. Appl. Sci., 4 (1982), 19-32. doi: 10.1002/mma.1670040104.

[7]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1996), 415-430. doi: 10.1007/BF01232273.

[8]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, Jour. de Math. Pure Appl., 86 (2006), 68-79.

[9]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case, Phys. D, 74 (1994), 268-300. doi: 10.1016/0167-2789(94)90198-8.

[10]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case, Phys. D, 79 (1994), 41-76. doi: 10.1016/0167-2789(94)90037-X.

[11]

C. Marchioro, E. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges, Arch. Rational Mech. Anal. (2010) in press.

[12]

S. Okabe and T. Ukai, On classical solutions in the large in time for the two-dimensional Vlasov's equation, Osaka Jour. Math., 15 (1978), 245-261.

[13]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics, Comm. PDE, 31 (2006), 349-370. doi: 10.1080/03605300500358004.

[14]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Jour. Diff. Eq., 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.

[15]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Mod. Meth. Appl. Sci., 19 (2009), 199-228. doi: 10.1142/S0218202509003401.

[16]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Commun. PDE., 16 (1991), 1313-1335.

[17]

S. Wollman, Global in time solutions to the two-dimensional Vlasov-Poisson system, Commun. Pure Appl. Math., 33 (1980), 173-197. doi: 10.1002/cpa.3160330205.

[18]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, Jour. Math. Anal. Appl., 176 (1993), 76-91. doi: 10.1006/jmaa.1993.1200.

[19]

Y. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data, Comm. Pure Appl. Math., 47 (1994), 1365-1401. doi: 10.1002/cpa.3160471004.

show all references

References:
[1]

P. Butta', G. Ferrari and C. Marchioro, Speedy motions of a body immersed in an infinitely extended medium, Jour. Stat. Phys., 140 (2010), 1182-1194.

[2]

E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rational Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.

[3]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinetic and Related Problems, 3 (2010), 241-254.

[4]

S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass, Commun. PDE, 27 (2002), 791-808. doi: 10.1081/PDE-120002874.

[5]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I, Math. Meth. Appl. Sci., 3 (1981), 229-248. doi: 10.1002/mma.1670030117.

[6]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II, Math. Meth. Appl. Sci., 4 (1982), 19-32. doi: 10.1002/mma.1670040104.

[7]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1996), 415-430. doi: 10.1007/BF01232273.

[8]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, Jour. de Math. Pure Appl., 86 (2006), 68-79.

[9]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case, Phys. D, 74 (1994), 268-300. doi: 10.1016/0167-2789(94)90198-8.

[10]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case, Phys. D, 79 (1994), 41-76. doi: 10.1016/0167-2789(94)90037-X.

[11]

C. Marchioro, E. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges, Arch. Rational Mech. Anal. (2010) in press.

[12]

S. Okabe and T. Ukai, On classical solutions in the large in time for the two-dimensional Vlasov's equation, Osaka Jour. Math., 15 (1978), 245-261.

[13]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics, Comm. PDE, 31 (2006), 349-370. doi: 10.1080/03605300500358004.

[14]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Jour. Diff. Eq., 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.

[15]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Mod. Meth. Appl. Sci., 19 (2009), 199-228. doi: 10.1142/S0218202509003401.

[16]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Commun. PDE., 16 (1991), 1313-1335.

[17]

S. Wollman, Global in time solutions to the two-dimensional Vlasov-Poisson system, Commun. Pure Appl. Math., 33 (1980), 173-197. doi: 10.1002/cpa.3160330205.

[18]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, Jour. Math. Anal. Appl., 176 (1993), 76-91. doi: 10.1006/jmaa.1993.1200.

[19]

Y. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data, Comm. Pure Appl. Math., 47 (1994), 1365-1401. doi: 10.1002/cpa.3160471004.

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