The zero-electron-mass limit in the Euler-Poisson system for both well- and ill-prepared initial data

The Euler-Poisson system consists of the balance laws for the electron density and current density coupled to the Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well and ill prepared initial data on the whole space case is discussed in this paper. Although it has some relations to the incompressible limit of the Euler equations, i.e., the limit velocity satisfies the incompressible Euler equations with damping, things are more complicated due to the linear singular perturbation including the coupling with the Poisson equation. A careful analysis on the structure of the linear perturbation has been done so that we are able to show the convergence for well prepared initial data and ill prepared initial data where the convergence occurs away from time $t=0$.