# American Institute of Mathematical Sciences

• Previous Article
Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type
• DCDS Home
• This Issue
• Next Article
On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation
October  2016, 36(10): 5401-5422. doi: 10.3934/dcds.2016038

## Bifurcation of rotating patches from Kirchhoff vortices

 1 IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Received  September 2015 Revised  April 2016 Published  July 2016

In this paper we investigate the existence of a new family of rotating patches for the planar Euler equations. We shall prove the existence of countable branches bifurcating from the ellipses at some implicit angular velocities. The proof uses bifurcation tools combined with the explicit parametrization of the ellipse through the exterior conformal mappings. The boundary is shown to belong to Hölderian class.
Citation: Taoufik Hmidi, Joan Mateu. Bifurcation of rotating patches from Kirchhoff vortices. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5401-5422. doi: 10.3934/dcds.2016038
##### References:

show all references

##### References:
 [1] Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020263 [2] Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure & Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002 [3] Quan Wang. Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 543-563. doi: 10.3934/dcdsb.2014.19.543 [4] Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391 [5] Sepideh Mirrahimi. Adaptation and migration of a population between patches. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 753-768. doi: 10.3934/dcdsb.2013.18.753 [6] Shikun Wang. Dynamics of a chemostat system with two patches. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6261-6278. doi: 10.3934/dcdsb.2019138 [7] Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349 [8] B. Emamizadeh, F. Bahrami, M. H. Mehrabi. Steiner symmetric vortices attached to seamounts. Communications on Pure & Applied Analysis, 2004, 3 (4) : 663-674. doi: 10.3934/cpaa.2004.3.663 [9] Chang-Yuan Cheng, Xingfu Zou. On predation effort allocation strategy over two patches. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020281 [10] Peter Constantin. Transport in rotating fluids. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 165-176. doi: 10.3934/dcds.2004.10.165 [11] D. Bresch, B. Desjardins, D. Gérard-Varet. Rotating fluids in a cylinder. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 47-82. doi: 10.3934/dcds.2004.11.47 [12] Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439 [13] Takashi Suzuki. Brownian point vortices and dd-model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 161-176. doi: 10.3934/dcdss.2014.7.161 [14] A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100 [15] Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223 [16] Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121 [17] A. V. Borisov, I. S. Mamaev, S. M. Ramodanov. Dynamics of a circular cylinder interacting with point vortices. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 35-50. doi: 10.3934/dcdsb.2005.5.35 [18] Mathieu Desbrun, Evan S. Gawlik, François Gay-Balmaz, Vladimir Zeitlin. Variational discretization for rotating stratified fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 477-509. doi: 10.3934/dcds.2014.34.477 [19] Kyungwoo Song, Yuxi Zheng. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1365-1380. doi: 10.3934/dcds.2009.24.1365 [20] Linxiang Wang, Roderick Melnik. Dynamics of shape memory alloys patches with mechanically induced transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1237-1252. doi: 10.3934/dcds.2006.15.1237

2019 Impact Factor: 1.338