In this paper, we study the existence of rotating and traveling wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with N-fold symmetry. The traveling-wave solutions take the form of translating vortex pairs. Moreover, these solutions constitute the desingularization of co-rotating N point vortices and counter-rotating pairs. Some other quantitative properties are also established. 

　　Publication: 

　　Transactions of the American Mathematical Society 

　　DOI: 10.1090/tran/8835 

　　Author: 

　　Daomin Cao 

　　Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190 

　　University of Chinese Academy of Sciences, Beijing 100049, P.R. China 

　　Email address: dmcao@amt.ac.cn 

　　Guolin Qin 

　　Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190 

　　University of Chinese Academy of Sciences, Beijing 100049, P.R. China 

　　Weicheng Zhan 

　　Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190 

　　University of Chinese Academy of Sciences, Beijing 100049, P.R. China 

　　Changjun Zou 

　　Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190 

　　University of Chinese Academy of Sciences, Beijing 100049, P.R. China