Global Bifurcation of Rotating Vortex Patches

We rigorously construct continuous curves of rotating vortex patch solutions to the two‐dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This i...

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Bibliographic Details
Published inCommunications on pure and applied mathematics Vol. 73; no. 9; pp. 1933 - 1980
Main Authors Hassainia, Zineb, Masmoudi, Nader, Wheeler, Miles H.
Format Journal Article
LanguageEnglish
Published New York John Wiley and Sons, Limited 01.09.2020
Subjects
Angular velocity
Bifurcations
Computational fluid dynamics
Corners
Euler-Lagrange equation
Rotating fluids
Rotation
Vortices
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Summary:We rigorously construct continuous curves of rotating vortex patch solutions to the two‐dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [30, 38] of singular limiting patches with 90 corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are “cat's‐eyes”‐type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular. © 2019 Wiley Periodicals, Inc.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.21855