Ill-posedness of the two-dimensional stationary Navier--Stokes equations on the whole plane

We consider the two-dimensional stationary Navier–Stokes equations on the whole plane ${\mathbb{R}}^2$. In the higher-dimensional cases ${\mathbb{R}}^n$ with $n⩾3$, the well-posedness and ill-posedness in scaling critical spaces are well-investigated by numerous papers. However, despite the attention of many researchers, the corresponding problem in the two-dimensional whole plane case was a long-standing open problem due to inherent difficulties of two-dimensional analysis. The aim of this paper is to address this issue and prove the ill-posedness in the scaling critical Besov spaces based on $L^p({\mathbb{R}}^2)$ for all $1⩽p⩽2$ in the sense of the discontinuity of the solution map and the non-existence of small solutions. To overcome the difficulty, we propose a new method based on the contradictory argument that reduces the problem to the analysis of the corresponding nonstationary Navier–Stokes equations and shows the existence of nonstationary solutions with strange large time behavior, if we suppose to contrary that the stationary problem is well-posed.