Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Amp\`ere systems

This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Amp`ere system, where the fully non-linear Monge-Amp`ere equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.