Multipole vortex patch equilibria for active scalar equations

Zineb Hassainia, Miles Wheeler

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6 Citations (SciVal)
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We study how a general steady configuration of finitely many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu [Comm. Math. Phys., 350 (2017), pp. 699-747] for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point vortex limit. Provided the point vortex equilibrium is nondegenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons, and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem.

Original languageEnglish
Pages (from-to)6054-6095
Number of pages42
JournalSIAM Journal on Mathematical Analysis
Issue number6
Early online date17 Nov 2022
Publication statusPublished - 31 Dec 2022

Bibliographical note

Funding Information:
\ast Received by the editors April 26, 2021; accepted for publication (in revised form) July 5, 2022; published electronically November 17, 2022. Funding: The work of the first author was supported by the Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. The work of the second author was partially supported by National Science Foundation grant DMS-1400926. \dagger Department of Mathematics, New York University in Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates ( \ddagger Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK (mw2319@bath.

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.


  • Euler equations
  • desingularization
  • generalized SQG equations
  • implicit function theorem
  • vortex dynamics

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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