Global solutions for the generalized SQG equation and rearrangements

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-rotati...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2023-03, Vol.376 (3), p.2181-2211
Main Authors: Cao, Daomin, Qin, Guolin, Zhan, Weicheng, Zou, Changjun
Format: Article
Language:English
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8835