On uniqueness of symmetric Navier-Stokes flows around a body in the plane

TL;DR

This paper proves the uniqueness of symmetric weak solutions to the stationary Navier-Stokes equations in a 2D exterior domain under certain symmetry and smallness conditions, using Hardy inequalities and density properties.

Contribution

It establishes a new uniqueness result for symmetric solutions of the 2D stationary Navier-Stokes equations with specific energy and decay conditions.

Findings

Uniqueness holds if the solution satisfies the energy inequality.

Smallness of the solution's supremum norm ensures uniqueness.

Uses Hardy inequality and density property in the proof.

Abstract

We investigate the uniqueness of symmetric weak solutions to the stationary Navier-Stokes equation in a two-dimensional exterior domain $\Omega$. It is known that, under suitable symmetry condition on the domain and the data, the problem admits at least one symmetric weak solution tending to zero at infinity. Given two symmetric weak solutions $u$ and $v$, we show that if $u$ satisfies the energy inequality $\| \nabla u \|_{L^2 (\Omega)}^2 \le (f,u)$ and $\sup_{x \in \Omega} (|x|+1)|v(x)|$ is sufficiently small, then $u=v$. The proof relies upon a density property for the solenoidal vector field and the Hardy inequality for symmetric functions.