Asymptotic Analysis of Boundary Layers for Stokes Systems in Periodic Homogenization

TL;DR

This paper analyzes the boundary layer behavior of Stokes systems with periodic coefficients in homogenization, proving velocity convergence and developing an asymptotic expansion of Poisson's kernel using physical space methods.

Contribution

It introduces a novel framework for boundary layer analysis in homogenization that avoids quasiperiodic techniques, applicable to arbitrary normals.

Findings

Proves convergence of velocity in boundary layers for all normals.

Develops an asymptotic expansion of Poisson's kernel for the Stokes operator.

Provides a new physical space approach for boundary layer analysis.

Abstract

We investigate the asymptotics of boundary layers in periodic homogenization. The analysis is focused on a Stokes system with periodic coefficients and periodic Dirichlet data posed in the half-space $\{y\in \mathbb{R}^d: y\cdot n -s>0\}$. In particular, we establish the convergence of the velocity as $y\cdot n \rightarrow \infty$. We obtain this convergence for arbitrary normals $n\in \mathbb{S}^{d-1}$. Moreover, we build an asymptotic expansion of Poisson's kernel for the periodically oscillating Stokes operator in the half-space. The presence of the pressure and the incompressibility condition impose certain innovations. In particular, we provide a framework for the analysis of the boundary layers in homogenization that relies only on physical space techniques and not on techniques that rely on the quasiperiodic structure of the problem.