Convergence of the Vlasov-Poisson system to the incompressible Euler equations

We consider the displacement of an electronic cloud generated by the local difference of charge with a uniform neutralizing background of non-moving ions. The equations are given by the Vlasov-Poisson system, with a coupling constant epsilon = (tau/2 pi)(2) where tau is the (constant) oscillation period of the electrons. In the so-called quasi-neutral regime, namely as epsilon --> 0, the current is expected to converge to a solution of the incompressible Euler equations, at least in the case of a vanishing initial temperature. This result is proved by adapting an argument used by P.-L. Lions [Li] to prove the convergence of the Leray solutions of the 3d Navier-Stokes equation to the so-called dissipative solutions of the Euler equations. For this purpose, the total energy of the system is modulated by a test-function. An alternative proof is given, based on the concept of measure-valued (mv) solutions introduced by DiPerna and Majda [DM] and already used by Brenier and Grenier [BG], [Gr2] for the asymptotic analysis of the Vlasov-Poisson system in the quasi-neutral regime. Through this analysis, a link is established between Lions' dissipative solutions and Diperna-Majda's my solutions of the Euler equations. A second interesting asymptotic regime, still leading to the Euler equations, known as the gyrokinetic limit of the Vlasov-Poisson system, is obtained when the electrons are forced by a strong constant external magnetic field and has been investigated in [Gr3], [GSR]. As for the quasi-neutral limit, we justify the gyrokinetic limit by using the concepts of dissipative solutions and modulated total energy.