Propagation of velocity moments and uniqueness for the magnetized Vlasov-Poisson system

TL;DR

This paper studies how velocity moments propagate in the 3D Vlasov-Poisson system with magnetic fields and extends uniqueness results to magnetized cases, using classical and induction methods.

Contribution

It introduces a new approach combining classical moment techniques with induction for magnetic fields and extends uniqueness results to the magnetized Vlasov-Poisson system.

Findings

Propagation of velocity moments for uniform magnetic fields

Extension of uniqueness results to general magnetic fields

Method combining classical moments with induction based on cyclotron period

Abstract

We present two results regarding the three-dimensional Vlasov--Poisson system in the full space with an external magnetic field. First, we investigate the propagation of velocity moments for solutions to the system when the magnetic field is uniform and time-dependent. We combine the classical moment approach with an induction procedure depending on the cyclotron period $T_c=||B||_{\infty}^{-1}$. This allows us to obtain, like in the unmagnetized case, the propagation of velocity moments of order $k>2$ in the full space case and of order $k>3$ in the periodic case. Second, this time taking a general magnetic field that depends on both time and position, we manage to extend a result by Miot arXiv:1409.6988v1 [math.AP] regarding uniqueness for Vlasov--Poisson to the magnetized framework.