Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

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

Pages: 705 – 738



Vo Anh Khoa (Mathematics and Computer Science Division, Gran Sasso Science Institute, L’Aquila, Italy; and Faculty of Sciences, Hasselt University, Diepenbeek, Belgium)

Adrian Muntean (Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden)


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 [N. Ray, A. Muntean, and P. Knabner, J. Math. Anal. Appl., 390(1):374–393, 2012]. 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.


Stokes–Nernst–Planck–Poisson system, variable scalings, two-scale convergence, perforated domains, homogenization asymptotics, corrector estimates

2010 Mathematics Subject Classification

35B27, 35C20, 35D30, 65M15

The work of the first author was partly supported by a postdoctoral fellowship of the Research Foundation-Flanders (FWO).

Received 25 October 2017

Accepted 21 January 2019

Published 30 August 2019