Corrector estimates and homogenization errors of unsteady flow ruled by Darcy's law

TL;DR

This paper investigates homogenization errors in unsteady flow governed by Darcy's law with memory effects, providing sharp estimates for velocity and pressure, and developing boundary-layer correctors to handle boundary layer challenges.

Contribution

It introduces a novel approach to estimate homogenization errors in non-stationary Darcy flow with memory effects, including the construction of boundary-layer correctors using Bogovskii's operator.

Findings

Established sharp homogenization error estimates for velocity and pressure.

Developed boundary-layer correctors with detailed regularity estimates.

Addressed boundary layer challenges due to incompressibility and initial-boundary incompatibility.

Abstract

Focusing on Darcy's law incorporating memory effects, this paper studies non-stationary Stokes equations on perforated domains. We establish a sharp homogenization error for both velocity and pressure in terms of the energy norm. The main challenge lies in gauging the boundary layers induced by the incompressibility condition. To address this, we construct boundary-layer correctors using Bogovskii's operator. Also, the present work provides detailed regularity estimates for these correctors, where a significant difficulty arises from the incompatibility between initial and boundary values. The methodologies developed herein hold great potential for tackling the same issue in other evolutionary models beyond a homogenization setting.