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On Singular Limits of Mean-Field Equations

Unterreiter, Andreas; Markowich, Peter; Dolbeault, Jean (2001), On Singular Limits of Mean-Field Equations, Archive for Rational Mechanics and Analysis, 158, 4, p. 319-351. http://dx.doi.org/10.1007/s002050100148

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Type
Article accepté pour publication ou publié
Date
2001
Journal name
Archive for Rational Mechanics and Analysis
Volume
158
Number
4
Publisher
Springer
Pages
319-351
Publication identifier
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Author(s)
Unterreiter, Andreas
Markowich, Peter
Dolbeault, Jean cc
Abstract (EN)
Mean-field equations arise as steady state versions of convection-diffusion systems where the convective field is determined by solution of a Poisson equation whose right-hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of two convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean-field equation by a variational analysis of a saddle point problem (usually without coercivity). Also we analyze the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.
Subjects / Keywords
Mean-field equations; convection-diffusion systems; Poisson equation; small-Debye-length limit