Polynomial propagation of moments and global existence for a Vlasov–Poisson system with a point charge
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 373-400.

In this paper, we extend to the case of initial data constituted of a Dirac mass plus a bounded density (with finite moments) the theory of Lions and Perthame [8] for the Vlasov–Poisson equation. Our techniques also provide polynomially growing in time estimates for moments of the bounded density.

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     author = {Desvillettes, Laurent and Miot, Evelyne and Saffirio, Chiara},
     title = {Polynomial propagation of moments and global existence for a {Vlasov{\textendash}Poisson} system with a point charge},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {373--400},
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Desvillettes, Laurent; Miot, Evelyne; Saffirio, Chiara. Polynomial propagation of moments and global existence for a Vlasov–Poisson system with a point charge. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 373-400. doi : 10.1016/j.anihpc.2014.01.001. http://www.numdam.org/articles/10.1016/j.anihpc.2014.01.001/

[1] A.A. Arsenev, Global existence of a weak solution of Vlasov's system of equations, USSR Comput. Math. Math. Phys. 15 (1975), 131 -143

[2] S. Caprino, C. Marchioro, On the plasma-charge model, Kinet. Relat. Models 3 no. 2 (2010), 241 -254 | Zbl

[3] S. Caprino, C. Marchioro, E. Miot, M. Pulvirenti, On the 2D attractive plasma-charge model, Commun. Partial Differ. Equ. 37 no. 7 (2012), 1237 -1272 | Zbl

[4] F. Castella, Propagation of space moments in the Vlasov–Poisson equation and further results, Ann. Inst. Henri Poincaré 16 no. 4 (1999), 503 -533 | EuDML | Numdam | Zbl

[5] Z. Chen, X. Zhang, Sub-linear estimate of large velocity in a collisionless plasma, Commun. Math. Sci. 12 no. 2 (2014), 279 -291 | Zbl

[6] R.J. Di Perna, P.L. Lions, Ordinary differential equations, transport equations and Sobolev spaces, Invent. Math. 98 (1989), 511 -547 | EuDML | Zbl

[7] I. Gasser, P.E. Jabin, B. Perthame, Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. R. Soc. Edinb. A 130 (2000), 1259 -1273 | Zbl

[8] P.L. Lions, B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system, Invent. Math. 105 (1991), 415 -430 | EuDML | Zbl

[9] G. Loeper, Uniqueness of the solution to the Vlasov–Poisson system with bounded density, J. Math. Pures Appl. (9) 86 no. 1 (2006), 68 -79 | Zbl

[10] S. Okabe, T. Ukai, On classical solutions in the large in time of the two-dimensional Vlasov equation, Osaka J. Math. 15 (1978), 245 -261 | Zbl

[11] C. Pallard, Moment propagation for weak solutions to the Vlasov–Poisson system, Commun. Partial Differ. Equ. 37 no. 7 (2012), 1273 -1285 | Zbl

[12] K. Pfaffelmoser, Global existence of the Vlasov–Poisson system in three dimensions for general initial data, J. Differ. Equ. 95 (1992), 281 -303 | Zbl

[13] C. Marchioro, E. Miot, M. Pulvirenti, The Cauchy problem for the 3-D Vlasov–Poisson system with point charges, Arch. Ration. Mech. Anal. 201 (2011), 1 -26 | Zbl

[14] D. Salort, Transport equations with unbounded force fields and application to the Vlasov–Poisson equation, Math. Models Methods Appl. Sci. 19 no. 2 (2009), 199 -228 | Zbl

[15] J. Schaeffer, Global existence of smooth solutions to the Vlasov–Poisson system in three dimensions, Commun. Partial Differ. Equ. 16 no. 8–9 (1991), 1313 -1335 | Zbl

[16] S. Wollman, Global in time solution to the three-dimensional Vlasov–Poisson system, J. Math. Anal. Appl. 176 no. 1 (1996), 76 -91 | Zbl

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