On the uniqueness and non-uniqueness of the steady planar Navier-Stokes equations in an exterior domain

TL;DR

This paper explores conditions for the uniqueness and non-uniqueness of steady planar Navier-Stokes solutions in exterior domains, highlighting the impact of boundary conditions on solution behavior.

Contribution

It provides new proofs of uniqueness under enhanced boundary conditions and counterexamples under standard Navier conditions, advancing understanding of solution behavior.

Findings

Uniqueness holds for incompressible flow with constant vorticity under enhanced Navier boundary conditions.

Counterexamples show non-uniqueness under standard Navier boundary conditions.

Sufficient conditions for uniqueness are established for flows with Dirichlet boundary conditions.

Abstract

In this paper we investigate the uniqueness of solutions of the steady planar Navier-Stokes equations with different boundary conditions in the exterior domain. For a class of incompressible flow with constant vorticity, we prove the uniqueness of the solution under the enhanced Navier boundary conditions. At the same time, some counterexamples are given to show that the uniqueness of the solution fails under the Navier boundary conditions. For the general incompressible flow with Dirichlet boundary condition, we prove various sufficient conditions for the uniqueness of the solution.