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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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