Extension of Hamel paradox for the 2D exterior Navier-Stokes problem

TL;DR

This paper extends the Hamel paradox to the 2D exterior Navier-Stokes problem, demonstrating non-uniqueness with certain boundary data and proposing conditions for uniqueness based on decay at infinity.

Contribution

It generalizes the Hamel paradox to include non-trivial flux boundary data and identifies conditions for solution uniqueness in exterior Navier-Stokes flows.

Findings

Non-uniqueness extends to an open set of boundary data.

Decay conditions at infinity can restore uniqueness.

Analysis of linearized system around potential flows.

Abstract

In this paper, we continue the analysis of the stationary exterior Navier-Stokes problem with interior boundary data and vanishing condition at infinity. We first show an existence result that extends a previous contribution of the second author by considering boundary data prescribing a non-trivial flux on the internal boundary. We obtain in particular that the non-uniqueness result of G. Hamel extends to an open set of internal boundary data. We then show that one way to recover uniqueness of a solution is to complement the perturbation of velocity field with a decay condition at infinity for small circulation through the interior boundary. Our method is based on a fine analysis of the linearized Navier-Stokes system around potential flows in the exterior domain.