Asymptotic Analysis of Boundary Layers for Stokes Systems in Periodic Homogenization

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.