The study of self-similar solutions plays an important role in understanding the regularity theory or the asymptotic behaviors of a solution to the Navier-Stokes equations. We have been studying it for the stationary Navier-Stokes equations in various dimensions $n=2$ or $n\geq 4$ in domains with or without boundary. When $n\geq 4$, we proved that any steady solution satisfying $|u(x)|\leq C/|x|$ in $\mathbb{R}^n\setminus \{0\}$ must be trivial. Neither smallness assumptions on $C$ nor self-similarity assumptions on $u$ are required. When $n=2$, we studied this topic in a sector with the no-slip boundary condition and established necessary and sufficient conditions in terms of the angle of the sector and the flux to guarantee the existence of self-similar solutions of a given type. In addition, we investigated uniqueness and non-uniqueness of flows with a given type. Our main idea for the higher dimensional result is to use weighted energy estimates and the equation of the head pressure. For the two-dimensional result, we used detailed properties of complete and incomplete elliptic functions. These are joint works with Changfeng Gui and Hao Liu (U. of Macau), Chunjing Xie (Shanghai Jiao Tong University), and Yun Wang (Soochow University).
