Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

TL;DR

This paper rigorously derives effective equations for a reaction-diffusion system with large nonlinear drift and Robin boundary conditions in a perforated domain, using advanced two-scale compactness techniques.

Contribution

It introduces a novel two-scale compactness with drift method to handle strong nonlinearities in homogenization of reaction-diffusion problems.

Findings

Established two-scale compactness with drift for nonlinear problems

Proved strong convergence of the corrector function

Derived effective coefficients for the homogenized model

Abstract

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.