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

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.