Well-posedness of the two-dimensional stationary Navier--Stokes equations around a uniform flow

Abstract

In this paper, we consider the solvability of the two-dimensional stationary Navier--Stokes equations on the whole plane ${\mathbb{R}}^2$. In [6], it was proved that the stationary Navier--Stokes equations on ${\mathbb{R}}^2$ is ill-posed for solutions around zero. In contrast, considering solutions around the non-zero constant flow, the perturbed system has a better regularity in the linear part, which enables us to prove the unique existence of solutions in the scaling critical spaces of the Besov type.