Limits of the Stokes and Navier–Stokes equations in a punctured periodic domain

Abstract

We treat three problems on a two-dimensional “punctured periodic domain”: we take [Formula: see text], where [Formula: see text] and [Formula: see text] is the closure of an open connected set that is star-shaped with respect to [Formula: see text] and has a [Formula: see text] boundary. We impose periodic boundary conditions on the boundary of [Formula: see text], and Dirichlet boundary conditions on [Formula: see text]. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function [Formula: see text], and examine the behavior of solutions as [Formula: see text]. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of [Formula: see text] with periodic boundary conditions.