The gyrokinetic limit for the two dimensional Vlasov-Poisson system with multi-point charges

TL;DR

This paper studies the gyrokinetic limit of the 2D Vlasov-Poisson system with multiple point charges, showing convergence to a measure-valued Euler solution without smallness restrictions, despite frequent charge intersections.

Contribution

It extends previous results to multi-point charges and removes the smallness condition, introducing new techniques to handle rapid charge intersections.

Findings

Convergence to measure-valued Euler solutions with defect measure.

Extension of prior work to multi-point charges.

Removal of the smallness condition on initial data.

Abstract

In this article, we investigate the gyrokinetic limit for the two dimensional Vlasov-Poisson system with multi-point charges. We show that the solution converges to a measure-valued solution of the Euler equation with a defect measure, which extends the results of Miot (Nonlinearity 32(2):654-677, 2019) to the case of multi-point charges and removes the smallness condition $\sup_{0<\varepsilon<1}\|f_{\varepsilon}^0\|_{L^1}<1$. The main difficulty arises from the fact that the orbits of point charges will intersect frequently and rapidly as the magnetic field intensity becomes large. To overcome the problem, we adopt the techniques recently developed by Wu and Zhang (Journal of Statistical Physics 190:183, 2023) combined with a new technique introduced here.