Homogenization of a nonlinear drift–diffusion system for multiple charged species in a porous medium. (December 2022)

• Record Type: Journal Article
• Authors: Bhattacharya, Apratim; Gahn, Markus; Neuss-Radu, Maria
• Is Part Of: Nonlinear analysis. Volume 68(2022)
• Publication Date: 2022-12
• Subjects: Drift–diffusion model -- Nonlinear diffusion -- Multiple charged species -- Porous media -- Homogenization -- Two-scale convergence; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2022.103651
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We consider a nonlinear drift–diffusion system for multiple charged species in a porous medium in 2D and 3D with periodic microstructure. The system consists of a transport equation for the concentration of the species and Poisson's equation for the electric potential. The diffusion terms depend nonlinearly on the concentrations. We consider non-homogeneous Neumann boundary condition for the electric potential. The aim is the rigorous derivation of an effective (homogenized) model in the limit when the scale parameter ϵ tends to zero. This is based on uniform a priori estimates for the solutions of the microscopic model. The crucial result is the uniform L ∞ -estimate for the concentration in space and time. This result exploits the fact that the system admits a nonnegative energy functional which decreases in time along the solutions of the system. By using weak and strong (two-scale) convergence properties of the microscopic solutions, effective models are derived in the limit ϵ → 0 for different scalings of the microscopic model.