Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits

TL;DR

This paper investigates the nonlinear instability of relativistic Vlasov-Maxwell systems in the classical and quasineutral limits, revealing rapid divergence from equilibrium solutions and invalidity of the quasineutral approximation.

Contribution

It constructs solutions demonstrating instability in both limits and shows the quasineutral limit fails in short time, advancing understanding of plasma model limitations.

Findings

Solutions diverge from homogeneous states rapidly in classical limit

Quasineutral limit is invalid in short time in L^2 norm

Instability occurs even with initial polynomial convergence

Abstract

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 $\varepsilon \to 0$, 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.