Nonlinear Instability of Vlasov-Maxwell Systems in the Classical and Quasineutral Limits

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 ε→ 0, with ε being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to an homogeneous solution μ of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from μ in arbitrary negative Sobolev norms within time of order | log ε|. Second, we deduce the invalidity of the quasineutral limit in L in arbitrarily short time.