Corrector homogenization estimates for a non-stationary Stokes-Nernst-Planck-Poisson system in perforated domains

TL;DR

This paper rigorously derives homogenization estimates for a complex non-stationary Stokes-Nernst-Planck-Poisson system in perforated domains, quantifying the error between microscopic and macroscopic models.

Contribution

It provides the first corrector homogenization estimates for the coupled nonlinear system in perforated domains, addressing microstructure effects and flux interactions.

Findings

Established error bounds between micro- and macro-concentrations

Analyzed effects of different scalings and boundary conditions

Controlled nonlinear drift terms in homogenization process

Abstract

We consider a non-stationary Stokes-Nernst-Planck-Poisson system posed in perforated domains. Our aim is to justify rigorously the homogenization limit for the upscaled system derived by means of two-scale convergence in \cite{RMK12}. In other words, we wish to obtain the so-called corrector homogenization estimates that specify the error obtained when upscaling the microscopic equations. Essentially, we control in terms of suitable norms differences between the micro- and macro-concentrations and between the corresponding micro- and macro-concentration gradients. The major challenges that we face are the coupled flux structure of the system, the nonlinear drift terms and the presence of the microstructures. Employing various energy-like estimates, we discuss several scalings choices and boundary conditions.