Asymptotic analysis of a semi-linear elliptic system in perforated domains: well-posedness and correctors for the homogenization limit

TL;DR

This paper rigorously analyzes a semi-linear elliptic system in perforated domains, establishing well-posedness, deriving homogenization limits, and providing corrector estimates for the convergence to macroscopic models involving diffusion and chemical reactions.

Contribution

It introduces a rigorous homogenization framework for semi-linear elliptic systems in perforated media, including well-posedness and explicit corrector estimates.

Findings

Proved weak solvability and positivity of solutions.

Derived homogenization limits for the system.

Established convergence rates with corrector estimates.

Abstract

In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations. Our techniques include Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as energy-like estimates.