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Vortex Patches Choreography for Active Scalar Equations

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Abstract

This paper deals with the existence of N vortex patches located at the vertex of a regular polygon with N sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and \(\text {(SQG)}_\beta \) equations, with \(\beta \in (0,1)\), but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the N point vortex system, that is, N point vortices located at the vertex of a regular polygon with N sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite-dimensional implicit function theorem and the well-chosen of the function spaces.

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Authors and Affiliations

  1. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Claudia García

  2. Research Unit “Modeling Nature” (MNat), Universidad de Granada, 18071, Granada, Spain

    Claudia García

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  1. Claudia García
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Correspondence to Claudia García.

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Communicated by Rustum Choksi.

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This work has been partially supported by the MINECO-Feder (Spain) research Grant number RTI2018–098850-B–I00, the Junta de Andalucía (Spain) project FQM 954, the Junta de Andaluciía (Spain) research Grant P18–RT–2422 and the MECD (Spain) research Grant FPU15/04094 (C.G), European Research Council through Grant ERC-StG-852741 (CAPA).

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García, C. Vortex Patches Choreography for Active Scalar Equations. J Nonlinear Sci 31, 75 (2021). https://doi.org/10.1007/s00332-021-09729-x

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