Title: Existence and uniqueness of steady weak solutions to the Navier-Stokes equations in $\mathbb{R}^2$,
Abstract: The existence of weak solutions to the stationary Navier-Stokes equations in
the whole plane $\mathbb{R}^2$ is proven. This particular geometry was the only
case left open since the work of Leray in 1933. The reason is that due to the
absence of boundaries the local behavior of the solutions cannot be controlled
by the enstrophy in two dimensions. We overcome this difficulty by constructing
approximate weak solutions having a prescribed mean velocity on some given
bounded set. As a corollary, we obtain infinitely many weak solutions in
$\mathbb{R}^2$ parameterized by this mean velocity, which is reminiscent of the
expected convergence of the velocity field at large distances to any prescribed
constant vector field. This explicit parameterization of the weak solutions
allows us to prove a weak-strong uniqueness theorem for small data. The
question of the asymptotic behavior of the weak solutions remains however open,
when the uniqueness theorem doesn't apply.