Homogenization of evolutionary incompressible Navier-Stokes system in perforated domains

TL;DR

This paper investigates the homogenization of the evolutionary incompressible Navier-Stokes equations in three-dimensional perforated domains with many small, periodically arranged holes, using uniform estimates and generalized cell problems.

Contribution

It introduces a novel approach to handle the time derivative by integrating in time and extends weak solutions to analyze the homogenization limit in perforated domains.

Findings

Established uniform estimates for weak solutions.

Applied generalized cell problems to study the limit process.

Provided a framework for homogenization in complex perforated structures.

Abstract

In this paper, we consider the homogenization problems for evolutionary incompressible Navier-Stokes system in three dimensional domains perforated with a large number of small holes which are periodically located. We first establish certain uniform estimates for the weak solutions. To overcome the extra difficulties coming from the time derivative, we use the idea of Temam and consider the equations by integrating in time variable. After suitably extending the weak solutions to the whole domain, we employ the generalized cell problem to study the limit process.