J. Burbea, Motions of vortex patches, Letters in Mathematical Physics, vol.25, issue.1, pp.1-16, 1982.
DOI : 10.1007/BF02281165

A. Castro, D. Córdoba, and J. Gomez-serrano, Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations. Duke Math, J, vol.165, issue.5, pp.935-984, 2016.

A. Castro, D. Córdoba, and J. Gomez-serrano, Uniformly Rotating Analytic Global Patch Solutions for Active Scalars, Annals of PDE, vol.3, issue.1, pp.34-2016
DOI : 10.1016/0021-9991(84)90051-2

C. Cerretelli and C. H. Williamson, A new family of uniform vortices related to vortex configurations before merging, Journal of Fluid Mechanics, vol.493, pp.219-229, 2003.
DOI : 10.1017/S0022112003005536

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues J. of Func, Analysis, vol.8, pp.321-340, 1971.

G. S. Deem and N. J. Zabusky, States," Interactions, Recurrence, and Breaking, Interactions, Recurrence, and Breaking, pp.859-862, 1978.
DOI : 10.1006/jcph.1997.5703

M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Communications on Pure and Applied Mathematics, vol.135, issue.1, pp.21-98, 1979.
DOI : 10.1111/j.1749-6632.1979.tb29498.x

Z. Hassainia and T. Hmidi, On the V-states for the Generalized Quasi-Geostrophic Equations, Communications in Mathematical Physics, vol.30, issue.1, pp.321-377, 2015.
DOI : 10.1016/0021-9991(79)90089-5
URL : https://hal.archives-ouvertes.fr/hal-00991981

Z. Hassainia, N. Masmoudi, and M. H. Wheeler, Global bifurcation of rotating vortex patches

T. Hmidi, J. Mateu, and J. Verdera, Boundary Regularity of Rotating Vortex Patches, Archive for Rational Mechanics and Analysis, vol.53, issue.2, pp.171-208, 2013.
DOI : 10.1016/0021-9991(84)90051-2
URL : https://hal.archives-ouvertes.fr/hal-00727718

T. Hmidi and J. Mateu, Bifurcation of rotating patches from Kirchhoff vortices, Discrete and Continuous Dynamical Systems, vol.36, issue.10, pp.5401-5422, 2016.
DOI : 10.3934/dcds.2016038
URL : https://hal.archives-ouvertes.fr/hal-01365294

J. R. Kamm, Shape and stability of two-dimensional uniform vorticity regions, 1987.

G. R. Kirchhoff, Vorlesungen¨berVorlesungen¨ Vorlesungen¨ber mathematische Physik, Mechanik. Teubner, 1876.

P. Liu, J. Shi, and Y. Wang, Imperfect transcritical and pitchfork bifurcations, Journal of Functional Analysis, vol.251, issue.2, pp.573-600, 2007.
DOI : 10.1016/j.jfa.2007.06.015

A. E. Love, On the Stability of certain Vortex Motions, Proc. Lond, p.18, 1893.
DOI : 10.1112/plms/s1-25.1.18

P. Luzzatto-fegiz and C. H. Williamson, Stability of elliptical vortices from " Imperfect-Velocity-Impulse " diagrams, Theor. Comput. Fluid Dyn, vol.24, pp.1-4, 2010.

P. Luzzatto-fegiz and C. H. Williamson, An efficient and general numerical method to compute steady uniform vortices, Journal of Computational Physics, vol.230, issue.17, pp.6495-6511, 2011.
DOI : 10.1016/j.jcp.2011.04.035

H. Pp-lotka and D. G. , Quasi-geostrophic shallow-water vortex???patch equilibria and their stability, Geophysical & Astrophysical Fluid Dynamics, vol.106, issue.6, pp.574-595, 2012.
DOI : 10.1007/BF02236522

L. M. Polvani, Geostrophic vortex dynamics, 1988.
DOI : 10.1575/1912/4961
URL : http://darchive.mblwhoilibrary.org/bitstream/1912/4961/1/Polvani_thesis.pdf

L. M. Polvani, N. J. Zabusky, and G. R. , Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger, Journal of Fluid Mechanics, vol.33, issue.-1, pp.215-242, 1989.
DOI : 10.1063/1.857485
URL : http://darchive.mblwhoilibrary.org/bitstream/1912/4961/1/Polvani_thesis.pdf

J. Segura, Bounds for ratios of modified Bessel functions and associated Tur??n-type inequalities, Journal of Mathematical Analysis and Applications, vol.374, issue.2, pp.516-528, 2011.
DOI : 10.1016/j.jmaa.2010.09.030

J. Shi, Persistence and Bifurcation of Degenerate Solutions, Journal of Functional Analysis, vol.169, issue.2, pp.494-531, 1999.
DOI : 10.1006/jfan.1999.3483

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, 2008.
DOI : 10.1017/9781107588417

G. N. Watson, A Treatise on the Theory of Bessel Functions, The Mathematical Gazette, vol.18, issue.231, 1944.
DOI : 10.2307/3605513

U. Rennes and C. Irmar-umr, F-35000 Rennes, France E-mail address: thmidi@univ-rennes1, E-mail address: Coralie.Renault@ens-rennes.fr