Homogenization of a reaction-diffusion-advection problem in an evolving micro-domain and including nonlinear boundary conditions

TL;DR

This paper rigorously derives a homogenized macroscopic model for a reaction-diffusion-advection system in a time-evolving perforated medium with nonlinear boundary conditions, accounting for microstructural changes.

Contribution

It introduces a novel homogenization approach for microstructures that evolve over time, incorporating nonlinear reactions and boundary conditions in a rigorous mathematical framework.

Findings

Established strong two-scale compactness results for nonlinear terms.

Derived a macroscopic model depending on micro- and macro-variables.

Accounted for microstructural evolution via time- and space-dependent reference elements.

Abstract

We consider a reaction-diffusion-advection problem in a perforated medium, with nonlinear reactions in the bulk and at the microscopic boundary, and low diffusion scaling. The microstructure changes in time; the microstructural evolution is known a priori. The aim of the paper is the rigorous derivation of a homogenized model. We use appropriately scaled function spaces, which allow us to show compactness results, especially regarding the time-derivative and we prove strong two-scale compactness results of Kolmogorov-Simon-type, which allow to pass to the limit in the nonlinear terms. The derived macroscopic model depends on the micro- and the macro-variable, and the evolution of the underlying microstructure is approximated by time- and space-dependent reference elements.