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

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. We prove the uniqueness of the solution under the enhanced Navier boundary conditions for a class of incompressible flow with constant vorticity. Meanwhile, 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 establish various sufficient conditions to guarantee the uniqueness of the solution.