Vol. 9, No. 7, 2016

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An analytical and numerical study of steady patches in the disc

Francisco de la Hoz, Zineb Hassainia, Taoufik Hmidi and Joan Mateu

Vol. 9 (2016), No. 7, 1609–1670
DOI: 10.2140/apde.2016.9.1609

We prove the existence of m-fold rotating patches for the Euler equations in the disc, for the simply connected and doubly connected cases. Compared to the planar case, the rigid boundary introduces rich dynamics for the lowest symmetries m = 1 and m = 2. We also discuss some numerical experiments highlighting the interaction between the boundary of the patch and the rigid one.

Euler equations, $V$-states, bifurcation
Mathematical Subject Classification 2010
Primary: 35Q35, 37G40, 35Q31
Secondary: 76B47
Received: 20 October 2015
Revised: 27 April 2016
Accepted: 28 May 2016
Published: 7 November 2016
Francisco de la Hoz
Department of Applied Mathematics and Statistics and Operations Research
Faculty of Science and Technology
University of the Basque Country UPV/EHU
Barrio Sarriena S/N
48940 Leioa
Zineb Hassainia
Courant Institute for Mathematical Sciences
New York University
New York, NY 10012-1185
United States
Taoufik Hmidi
Institut de Recherche Mathématique de Rennes
Université de Rennes 1
Campus de Beaulieu
35 042 Rennes CEDEX
Joan Mateu
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Barcelona