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A hierarchy of hydrodynamic models for plasmas. Quasi-neutral limits in the drift-diffusion equations. (English) Zbl 1045.76058
Summary: We rigorously derive (quasi-) hydrodynamic models for plasmas by means of asymptotic analysis. The quasi-neutral limit (zero-Debye-length limit) in the drift-diffusion equations is performed in two cases: weakly ionized plasmas, and not-weakly ionized plasmas. The model consists of continuity equations for electrons and ions, the constitutive relations for particle current densities, and Poisson equation for electrostatic potential in a bounded domain. In the case of a weakly ionized plasma, the continuity equation for electrons is replaced by a relation between the electrostatic potential and 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.

76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
82D10 Statistical mechanical studies of plasmas
35Q60 PDEs in connection with optics and electromagnetic theory
35Q35 PDEs in connection with fluid mechanics