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A hierachy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations. (English) Zbl 0956.35010
The paper under review is concerned with the systems of PDE’s: \[ \begin{aligned} \delta_{\alpha} \frac{\partial n_{\alpha}}{\partial t} &= \text{div} ( \nabla p_{\alpha}(n_{\alpha})+ q_{\alpha}n_{\alpha}\nabla\Phi)\;(\alpha=e,i),\quad -\lambda^{2}\Delta \Phi = n_{i}-n_{e},\tag{I}\\ \delta_{i}\frac{\partial n_{i}}{\partial t}&= \text{div} ( \nabla p_{i}(n_{i})+ n_{i}\nabla \Phi),\qquad -\lambda^{2}\Delta \Phi= n_{i}- f_{e}(\Phi). \tag{II}\end{aligned} \] These systems are obtained as zero-relaxation-time limit of a Euler-Poisson system modelling a plasma consisting of electrons and ions (\(n_{e}, n_{i}\) density of electrons or ions, respectively, \(q_{e}=-1, q_{i}=1\) charges, \(p_{\alpha}\) pressure functions, \(\Phi\) electrostatic potential, \(f_{e}^{-1}\) enthalpy function). Systems (I) and (II) are considered in a cylinder \(Q=\Omega \times (0,T)\) \((\Omega \subset \mathbb{R}^{d}\) bounded domain) and completed by mixed boundary conditions upon \(n_{e},n_{i}\) and \(\Phi\) on \(\partial\Omega\times (0,T)\) and initial conditions upon \(n_{e}\) and \(n_{i}\) on \(\Omega \times\{0\}\). The authors prove the convergence of weak solutions of (I) to a weak solution of (II) when \(\delta_{e} \rightarrow 0\), \(\delta_{i}> 0\) fixed, and the convergence of weak solutions of (I) or (II) respectively, to a weak solution of \[ -\lambda^{2}\Delta\Phi = f_{i}(c-\Phi)-f_{e}(\Phi) \] when \(\delta_{e}=\delta_{i}=\delta \rightarrow 0\) or \(\delta_{i}\rightarrow 0\), respectively.
Independently of the first result, the authors prove the existence of a weak solution to \[ \frac{\partial n}{\partial t}- \text{div} ( \nabla p(n)+ n \nabla \Phi) = 0,\quad -\Delta \Phi =n-f(\Phi) . \]

MSC:
35B25 Singular perturbations in context of PDEs
78A35 Motion of charged particles
35B40 Asymptotic behavior of solutions to PDEs
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