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Boundary layer analysis and quasi-neutral limits in the drift-diffusion equations. (English) Zbl 0994.35020
Let $$\Omega$$ be an open and bounded domain in $$\mathbb{R} ^d, d\geq 1$$, whose boundary consists of two disjoint sets $$\Gamma_D$$ and $$\Gamma_N$$. Set $\Omega_T=(0,T)\times\Omega, \quad \Sigma_D=(0,T)\times\Gamma_D, \quad \Sigma_N=(0,T)\times\Gamma_N, T>0.$ The author considers an unmagnetized plasma consisting of electrons with density $$n ^\varepsilon_e$$ and a single species of positively charged ions with density $$n ^\varepsilon_i$$ . These particle densities and the electric potential $$\phi ^\varepsilon$$ satisfy the usual drift-diffusion equations in $$\Omega_T$$, complemented by initial conditions, and Dirichlet [resp. Neumann] boundary conditions on $$\Sigma_D$$ [resp.$$\Sigma_N$$]. He studies asymptotic properties of $$n ^\varepsilon_e$$, $$n ^\varepsilon_i$$ and $$\phi ^\varepsilon$$, when the Debye-length $$\varepsilon$$ goes to zero, under assumptions which are closely related to those of a previous paper by himself and A. Jüngel [A hierarchy of hydrodynamic models for plasmas; quasi-neutral limits in the drift-diffusion, Asymptotic Anal. 28, No. 1, 49-73 (2001; Zbl 1045.76058)]. He shows that the quasi-neutral limit of the sequence $$(n ^\varepsilon_e, n ^\varepsilon_i, \phi ^\varepsilon),$$ obtained in [loc. cit.], is unique and characterizes the limit problem. In the case $$\Omega=(0,1)$$, he determines the boundary layers; this yields a sharp estimate for $$\|n ^\varepsilon_i-n ^\varepsilon_e\|_{L ^2(\Omega_T)}$$, and the strong convergence of $$n ^\varepsilon_i$$ and $$n ^\varepsilon_e$$ in $$L ^2(\Omega_T)$$. Possible extensions of these results to multi-dimensional domains are pointed out. The proofs use variational methods in vector-valued $$L ^p$$ or, more generally, Sobolev spaces. In the last section of the paper, similar results, for the non-linear drift-diffusion equation, are given without proof.
MSC:
 35B25 Singular perturbations in context of PDEs 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35Q60 PDEs in connection with optics and electromagnetic theory
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