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

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 (2019), 654-677) to the case of multi-point charges and removes the smallness condition 0<ε<1\|fε⁰\|L¹<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, which complicates the proof of time continuity of the orbits. To overcome this problem, we adopt the estimates on the singular moment recently developed by Wu and Zhang (J. Stat. Phys., 190 (2023), Paper No.183), combined with a new technique of iterating with non-uniform time periods introduced here.