Homogenization of the discrete diffusive coagulation-fragmentation equations in perforated domains

TL;DR

This paper studies the homogenization of discrete coagulation-fragmentation equations in perforated domains, deriving effective macroscopic models that incorporate microscale boundary effects and geometric corrections.

Contribution

It introduces a homogenization approach using two-scale convergence for coagulation-fragmentation equations in perforated domains, linking microscale boundary conditions to macroscopic source terms.

Findings

Effective macroscopic equations with source terms derived from microscale boundary conditions.

Perforation geometry influences the diffusion coefficients at the macroscale.

Homogenization provides a rigorous link between microscopic boundary effects and macroscopic behavior.

Abstract

The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation-fragmentation-diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng-Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a global source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in the diffusion coefficients.