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 ωr = (-L,L)2rK, where r > 0 and K is the closure of an open connected set that is star-shaped with respect to 0 and has a C1 boundary. We impose periodic boundary conditions on the boundary of ω = (-L,L)2, and Dirichlet boundary conditions on (rK). In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier-Stokes equations, all with a fixed forcing function f, and examine the behavior of solutions as r → 0. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of ω with periodic boundary conditions