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

TL;DR

This paper investigates how the eigenvalues and solutions of the Stokes and Navier-Stokes equations behave as a small hole in a planar domain shrinks to a point, establishing convergence results for eigenvalues, eigenspaces, and vorticity.

Contribution

It provides the first rigorous analysis of the asymptotic behavior of Stokes and Navier-Stokes solutions in domains with vanishing holes, connecting perforated and punctured domain problems.

Findings

Eigenvalues of the Stokes operator converge to those in the whole domain as the hole shrinks.

Eigenspaces and Stokes semigroup also converge in the limit.

Vorticity solutions of Navier-Stokes in perforated domains converge to those in punctured domains.

Abstract

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.