Some results on the steady solution for the incompressible Euler equation

For the 2D incompressible Euler equation, there is a lot of results on the existence of vortex solutions, such as solutions with vorticity compactly supported or vorticity concentrated in several small sets. For the 3D case, the situation is much more difficult. A well-known solution is the vortex ring with small cross-section, traveling with a constant speed along its symmetric-axis, which was observed by Helmohtz in 1858. To get existence of solutions, people usually consider axi-symmetric case and a lot of axi-symmetric solutions have been established by different methods. But very little is known about the uniqueness. The stability of such vortex ring with small cross-section is a long-time open problem. In this talk the speaker will first talk about the uniqueness of such vortex ring. Then the speaker will introduce results on the nonlinear stability of such vortex ring. Lastly the speaker will talk about solutions with helical symmetry. The results presented in this talk are from joint papers with Guolin Qin, Weilin Yu, Weicheng Zhan and Changjun Zou.