The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity

TL;DR

This paper demonstrates the propagation of velocity moments and regularity in the 3D Vlasov-Poisson system with a uniform magnetic field, addressing singularities at cyclotron periods and extending uniqueness results.

Contribution

It adapts existing methods to include magnetic fields, showing moment propagation, regularity, and extending uniqueness classes for solutions.

Findings

Propagation of velocity moments established

Regularity of solutions propagated over time

Uniqueness extended to solutions with bounded macroscopic density

Abstract

We show propagation of moments in velocity for the 3-dimensional Vlasov-Poisson system with a uniform magnetic field $B = (0, 0, {\omega})$ by adapting the work of Lions, Perthame. The added magnetic field also produces singularities at times which are the multiples of the cyclotron period $t = \dfrac{2\pi k}{\omega}, k \in \mathbb{N}$ . This result also allows to show propagation of regularity for the solution. For uniqueness, we extend Loeper's result by showing that the set of solutions with bounded macroscopic density is a uniqueness class.