Limits of the Stokes and Navier–Stokes equa- tions in a punctured periodic domain

In this paper we treat three problems on a two-dimensional ‘punctured periodic domain’: we take Ωr = (−L,L) \Dr, where Dr = B(0, r) is the disc of radius r centred at the origin. We impose periodic boundary conditions on the boundary of the box Ω = (−L,L), and Dirichlet boundary conditions on the circumference of the disc. 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 (which must satisfy ́ Ω f = 0 for the stationary problems), and examine the behaviour 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. Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00.