×

Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge. (English) Zbl 1323.35178

The aim of this article is to propose new estimate for moments of the three-dimensional Vlasov-Poisson system. The new estimates improve the existing theory in the following directions. First, the authors can now consider initial data which are the sum of a Dirac mass and a continuous density which does not vanish in the vicinity of the Dirac mass. Second, the authors can now bound the higher-order momentums of the distribution by polynomials with respect to time in an explicit computable way in the case when the initial datum is not compactly supported. The new work improves the existing theory by P. L. Lions and B. Perthame [Invent. Math. 105, No. 2, 415–430 (1991; Zbl 0741.35061)].

MSC:

35Q83 Vlasov equations
35B45 A priori estimates in context of PDEs
82D10 Statistical mechanics of plasmas

Citations:

Zbl 0741.35061
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arsenev, A. A., Global existence of a weak solution of Vlasov’s system of equations, USSR Comput. Math. Math. Phys., 15, 131-143 (1975)
[2] Caprino, S.; Marchioro, C., On the plasma-charge model, Kinet. Relat. Models, 3, 2, 241-254 (2010) · Zbl 1193.82045
[3] Caprino, S.; Marchioro, C.; Miot, E.; Pulvirenti, M., On the 2D attractive plasma-charge model, Commun. Partial Differ. Equ., 37, 7, 1237-1272 (2012) · Zbl 1263.82051
[4] Castella, F., Propagation of space moments in the Vlasov-Poisson equation and further results, Ann. Inst. Henri Poincaré, 16, 4, 503-533 (1999) · Zbl 1011.35034
[5] Chen, Z.; Zhang, X., Sub-linear estimate of large velocity in a collisionless plasma, Commun. Math. Sci., 12, 2, 279-291 (2014) · Zbl 1295.35082
[6] Di Perna, R. J.; Lions, P. L., Ordinary differential equations, transport equations and Sobolev spaces, Invent. Math., 98, 511-547 (1989) · Zbl 0696.34049
[7] Gasser, I.; Jabin, P. E.; Perthame, B., Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. R. Soc. Edinb. A, 130, 1259-1273 (2000) · Zbl 0984.35102
[8] Lions, P. L.; Perthame, B., Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105, 415-430 (1991) · Zbl 0741.35061
[9] Loeper, G., Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. (9), 86, 1, 68-79 (2006) · Zbl 1111.35045
[10] Okabe, S.; Ukai, T., On classical solutions in the large in time of the two-dimensional Vlasov equation, Osaka J. Math., 15, 245-261 (1978) · Zbl 0405.35002
[11] Pallard, C., Moment propagation for weak solutions to the Vlasov-Poisson system, Commun. Partial Differ. Equ., 37, 7, 1273-1285 (2012) · Zbl 1387.76115
[12] Pfaffelmoser, K., Global existence of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95, 281-303 (1992) · Zbl 0810.35089
[13] Marchioro, C.; Miot, E.; Pulvirenti, M., The Cauchy problem for the \(3-D\) Vlasov-Poisson system with point charges, Arch. Ration. Mech. Anal., 201, 1-26 (2011) · Zbl 1321.76081
[14] Salort, D., Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19, 2, 199-228 (2009) · Zbl 1165.82011
[15] Schaeffer, J., Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Commun. Partial Differ. Equ., 16, 8-9, 1313-1335 (1991) · Zbl 0746.35050
[16] Wollman, S., Global in time solution to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl., 176, 1, 76-91 (1996) · Zbl 0814.35105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.