Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

TL;DR

This paper investigates the homogenization of a reaction-diffusion-convection problem in domains connected by a thin composite layer, deriving effective models and transmission conditions relevant for designing impact-resistant materials.

Contribution

It introduces a homogenization framework for nonlinear drift problems in layered domains, focusing on thin-layer limits and effective interface conditions.

Findings

Derived homogenized evolution equations for layered domains.

Established effective transmission conditions across thin interfaces.

Analyzed the impact of layer thickness on homogenization results.

Abstract

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) process for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer. This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces -- a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.