# American Institute of Mathematical Sciences

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

## On a charge interacting with a plasma of unbounded mass

 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.
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.

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##### 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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