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 defined by $\{y\in {ℝ}^d:y\cdot n-s>0\}$. In particular, we establish the convergence of the velocity as $y\cdot n\to \infty$. 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 pressure and incompressibility conditions 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.

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