Some results on the steady solution for the incompressible Euler equation

• 主讲人：Daomin Cao (Academy of Mathematics and Systems Science, CAS)
• 举办方： Beijing-Saint Petersburg Mathematics Colloquium
• 时间： 2025-01-09 20:00 - 2025-01-09 21:00
• 地点： Online

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Abstract: 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.

 

Bio: Cao Daomin is currently a professor at the Academy of Mathematics and Systems Science, China Academy of Sciences. He received his doctorate from China Academy of Sciences in 1989. He is mainly engaged in the study of nonlinear partial differential equations and nonlinear variational problems. In recent years, he mainly studies the existence and stability of steady solutions of incompressible Euler equations. He is currently associated editor of “Acta Mathematicae Applicatae Sinica” and “Acta Mathematica Scientia”, and is a member of the editorial board of “Applied Analysis, Annales Academiae Scientiarum Fennicae, Mathematica”, etc.