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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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