Global quasi-neutral limit for a two-fluid Euler-Poisson system in one space dimension

Abstract

The quasi-neutral limit of one-fluid Euler-Poisson systems leads to incompressible Euler equations. It was widely studied in previous works. In this paper, we deal with the quasi-neutral limit in a two-fluid Euler-Poisson system. This limit presents a different feature since it leads to compressible Euler equations. We justify this limit for global smooth solutions near constant equilibrium states in one space dimension. Specifically, we prove a global existence of smooth solutions by establishing uniform energy estimates with respect to the Debye length and the time. This allows to pass to the limit in the system for all time. Moreover, we establish global error estimates between the solution of the two-fluid Euler-Poisson system and that of the compressible Euler equations. The proof is based on classical uniform energy estimates together with various dissipation estimates. In order to control the quasi-neutrality of the velocities of two-fluids, similar conditions on the initial data are needed.