Asymptotic limit for the Stokes and Navier-Stokes problems in a planar domain with a vanishing hole

We show that the eigenvalues of the Stokes operator in a domain with a small hole converge to the eigenvalues of the Stokes operator in the whole domain, when the diameter of the hole tends to 0. The convergence of the eigenspaces and the convergence of the Stokes semigroup are also established. Concerning the Navier--Stokes equations, we prove that the vorticity of the solution in the perforated domain converges as the hole shrinks to a point $r$ to the vorticity of the solution in the punctured domain (i.e. the whole domain with the point $r$ removed). The main ingredients of the analysis are a suitable decomposition of the vorticity space, the formalism elaborated in [7] and some basics of potential theory.