In this paper, we study the existence and qualitative properties of multiscale steady vortex patches for Euler equations in a 2D bounded domain. By considering certain maximization problems for the vorticity, we obtain the existence of double vortex patches which are trapped in a neighborhood of two points. Limiting localizations of these two points are determined by the Robin function and the boundary of the domain, rather than critical points of the Kirchhoff--Routh function $H_2$, which is quite different from all the known results. Moreover, the strengths of two components of vorticity are of different order. Multiscale vortex patches concentrating near $k$ points are also constructed for any integer $k\geq 2.$.

　　Publication:

　　SIAM Journal on Mathematical Analysis, Vol. 54, Iss. 2, pp: 1488-1514.

　　Author:

　　Daomin Cao

　　Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, and University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China

　　Email: dmcao@amt.ac.cn

　　Jie Wan

　　School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People's Republic of China

　　Email: wanjie@bit.edu.cn