Global solutions for the generalized SQG equation and rearrangements

Cao, Daomin; Qin, Guolin; Zhan, Weicheng; Zou, Changjun

doi:10.1090/tran/8835

Global solutions for the generalized SQG equation and rearrangements
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by Daomin Cao, Guolin Qin, Weicheng Zhan and Changjun Zou;

Trans. Amer. Math. Soc. 376 (2023), 2181-2211

DOI: https://doi.org/10.1090/tran/8835

Published electronically: January 4, 2023

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Abstract:

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.

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