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## On the non-relativistic limit of Vlasov-Maxwell

In this note, I briefly explain my recent joint work with D. Han-Kwan (CNRS, Ecole polytechnique) and F. Rousset (Paris-Sud) on the non-relativistic limit of Vlassov-Maxwell. Precisely, we consider the relativistic Vlasov-Maxwell system, modeling the dynamics of electrons with electron density distribution ${f(t,x,v)}$, which reads

$\displaystyle \partial_t f + \hat v \cdot \nabla_x f + (E + \epsilon \hat v \times B)\cdot \nabla_v f = 0$

on ${\mathbb{T}^3\times \mathbb{R}^3}$, with the relativistic velocity ${\hat v = v/\sqrt{1+ \epsilon^2 |v|^2}}$.

What is a plasma? A plasma is an ionized gas that consists of charged particles: positive ions and negative electrons. To describe the dynamics of a plasma, let ${f^\pm(t,x,v)}$ be the (nonnegative) density distribution of ions and electrons, respectively, at time ${t\ge 0}$, position ${x\in \Omega \subset \mathbb{R}^3}$, and particle velocity (or momentum) ${v\in \mathbb{R}^3}$. The dynamics of a plasma is commonly modeled by the Vlasov equations
$\displaystyle \frac{d}{dt} f^\pm (t,X(t), V(t)) = 0 \ \ \ \ \ (1)$