Global Bifurcation of Rotating Vortex Patches

Zineb Hassainia, Nader Masmoudi, Miles H. Wheeler

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish
JournalCommunications on Pure and Applied Mathematics
Early online date5 Aug 2019
DOIs
Publication statusE-pub ahead of print - 5 Aug 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Global Bifurcation of Rotating Vortex Patches. / Hassainia, Zineb; Masmoudi, Nader; Wheeler, Miles H.

In: Communications on Pure and Applied Mathematics, 05.08.2019.

Research output: Contribution to journalArticle

@article{772a9eb70d594ee3a4d60733da582648,
title = "Global Bifurcation of Rotating Vortex Patches",
abstract = "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.",
author = "Zineb Hassainia and Nader Masmoudi and Wheeler, {Miles H.}",
year = "2019",
month = "8",
day = "5",
doi = "10.1002/cpa.21855",
language = "English",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",

}

TY - JOUR

T1 - Global Bifurcation of Rotating Vortex Patches

AU - Hassainia, Zineb

AU - Masmoudi, Nader

AU - Wheeler, Miles H.

PY - 2019/8/5

Y1 - 2019/8/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85070509434&partnerID=8YFLogxK

U2 - 10.1002/cpa.21855

DO - 10.1002/cpa.21855

M3 - Article

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -