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

*(English)*Zbl 1045.76058Summary: 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.

##### MSC:

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 |