Abstract: The Bahouri-Chemin patch is an important singular steady-state for 2D incompressible Euler equations in the flat torus. In this talk, we consider steady states near the Bahouri-Chemin patch. More precisely, we use two different ways to approximate the semi-linear elliptic equations describing the Bahouri-Chemin patch. In one way, we get a smooth curve consisting of smooth steady states close to the Bahouri-Chemin patch in the sense of the HÃ¶lder norm of the velocity field. It, in particular, shows the importance of non-vanishing conditions for vorticity on the boundary in the example of double exponential growth for the gradient of vorticity. In the other way, we obtain a continuous curve of singular steady states near the Bahouri-Chemin in the sense of the continuous norm of the velocity field. In those examples, an interesting feature is there are multiple particle trajectories across the origin. In the end, if time permits, I will discuss some open questions. This presentation comes from the Joint work of Tarek Elgindi and the Joint work of Chiling Zhang.