Abstract
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.
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Accepted January 25, 2001¶Published online June 28, 2001
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Dolbeault, J., Markowich, P. & Unterreiter, A. On Singular Limits of Mean-Field Equations. Arch. Rational Mech. Anal. 158, 319–351 (2001). https://doi.org/10.1007/s002050100148
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DOI: https://doi.org/10.1007/s002050100148