A hierarchy of hydrodynamic models for plasmas.: Quasi-neutral limits in the drift-diffusion equations

This paper is a continuation of a series of papers in which (quasi-) hydrodynamic models for plasmas are rigorously derived by means of asymptotic analysis. Here, the quasi-neutral limit (zero-Debye-length limit) in the drift-diffusion equations is performed in the two cases: weakly ionized plasmas and not weakly ionized plasmas. The model consists of the continuity equations for the electrons and ions, the constitutive relations for the particle current densities, and the Poisson equation for the electrostatic potential in a bounded domain. In the case of a weakly ionized plasma, the continuity equation for the electrons is replaced by a relation between the electrostatic potential and the electron density such that the Poisson equation becomes nonlinear. The equations are complemented by mixed Dirichlet-Neumann boundary conditions and initial conditions. The quasi-neutral limits are shown without assuming compatibility conditions on the boundary densities. The proofs rely on the use of the so-called entropy functional which yields appropriate uniform estimates, and compensated compactness methods.