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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 ${\hat v = v/\sqrt{1+ \epsilon^2 |v|^2}}$.

## Stability of a collisionless plasma

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

$\displaystyle \frac{d}{dt} f^\pm (t,X(t), V(t)) = 0 \ \ \ \ \ (1)$

Daniel Han-Kwan and I have just submitted a paper entitled: “Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits”, which is also available on arxiv: arXiv:1506.08537. In this paper, we study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit ${\varepsilon \rightarrow 0}$, with ${\varepsilon}$ being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution ${\mu}$ of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from ${\mu}$ in arbitrary negative Sobolev norms within time of order ${|\log \varepsilon|}$. Second, we deduce the invalidity of the quasineutral limit in ${L^2}$ in arbitrarily short time.