Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit

TL;DR

This paper analyzes the Vlasov-Maxwell system in the non-relativistic limit, providing stability estimates over long times and developing higher-order approximations as the speed of light becomes very large.

Contribution

It introduces a higher-order Vlasov-Darwin approximation and establishes polynomial-time stability estimates in the non-relativistic regime.

Findings

Sobolev stability estimates valid for polynomial times in the speed of light

Construction of higher-order Vlasov-Darwin approximation

Validation near stable homogeneous equilibria

Abstract

In this paper, we study the Vlasov-Maxwell system in the non-relativistic limit, that is in the regime where the speed of light is a very large parameter. We consider data lying in the vicinity of homogeneous equilibria that are stable in the sense of Penrose (for the Vlasov-Poisson system), and prove Sobolev stability estimates that are valid for times which are polynomial in terms of the speed of light and of the inverse of size of initial perturbations. We build a kind of higher-order Vlasov-Darwin approximation which allows us to reach arbitrarily large powers of the speed of light.