AMaSiS 2021 - Abstract
Bhattacharya, Apratim
Coauthors: Markus Gahn (University of Heidelberg), and Maria Neuss-Radu (FAU Erlangen-Nürnberg)
Friedrich--Alexander--Universität Erlangen, Germany
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 the transport equation for the concentration of the species and the Poisson equation for the electric potential. The diffusion terms depend nonlinearly on the concentrations. We consider zero flux boundary conditions for the species concentrations and 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 is based on a maximum principle, and exploits the fact that there exists a nonnegative free energy functional which is monotonically decreasing 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.
Acknowledgments: AB acknowledges the support by the RTG 2339 IntComSin of the German Science Foundation (DFG).