In this paper, we consider the solvability of the two-dimensional stationary Navier--Stokes equations on the whole plane R². In 6, it was proved that the stationary Navier--Stokes equations on R² 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.
Fujii et al. (Sat,) studied this question.