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 a homogeneous solution μ of Vlasov-Poisson systems 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 L² in arbitrarily short time.