# Existence and uniqueness of steady weak solutions to the Navier-Stokes equations in $\mathbb{R}^2$

## TL;DR

This paper proves the existence of weak solutions to the steady Navier-Stokes equations in the entire plane, overcoming previous limitations by constructing approximate solutions with prescribed mean velocities, and establishes a weak-strong uniqueness result for small data.

## Contribution

It demonstrates the existence and parameterization of weak solutions in , a case unresolved since 1933, and introduces a weak-strong uniqueness theorem for small data.

## Key findings

- Existence of infinitely many weak solutions in 
- Construction of solutions with prescribed mean velocity
- Weak-strong uniqueness for small data

## 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.