Propagation of velocity moments and uniqueness for the magnetized Vlasov-Poisson system
Résumé
In this paper we present two results regarding the three-dimensional Vlasov-Poisson system in the full space with a general bounded magnetic field. First, we study the propagation of velocity moments for solutions to the system. We rely on Pallard's optimal result regarding the unmagnetized Vlasov-Poisson system and we combine it with an induction procedure depending on the cyclotron frequency T_c = 1/||B||_\infty. This induction procedure, similar to the one used by the author in the case of a constant magnetic field, is necessary because we can only get satisfactory estimates on a small time scale compared to the cyclotron frequency. Second, we manage to extend a result by Miot regarding uniqueness for Vlasov-Poisson to the magnetized case. This result relied heavily on the second-order structure of the Cauchy problem for the characteristics. The main difficulty in the magnetized case is that we lose this second-order structure.
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