Corotating and counter-rotating vortex pairs for Euler equations
Journées équations aux dérivées partielles (2016), Talk no. 6, 16 p.

We study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations. From the numerical experiments implemented in [7, 16, 17] it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle [14, 18], however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments. The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.

Published online:
DOI: 10.5802/jedp.647
Classification: 35Q35, 76B03, 76C05
Keywords: Euler equations, steady vortex pairs, desingularization.
Hmidi, Taoufik 1; Mateu, Joan 2

1 IRMAR, Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France
2 Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Catalonia Spain
@article{JEDP_2016____A6_0,
     author = {Hmidi, Taoufik and Mateu, Joan},
     title = {Corotating and counter-rotating vortex pairs for {Euler} equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:6},
     pages = {1--16},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2016},
     doi = {10.5802/jedp.647},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.647/}
}
TY  - JOUR
AU  - Hmidi, Taoufik
AU  - Mateu, Joan
TI  - Corotating and counter-rotating vortex pairs for Euler equations
JO  - Journées équations aux dérivées partielles
N1  - talk:6
PY  - 2016
SP  - 1
EP  - 16
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.647/
DO  - 10.5802/jedp.647
LA  - en
ID  - JEDP_2016____A6_0
ER  - 
%0 Journal Article
%A Hmidi, Taoufik
%A Mateu, Joan
%T Corotating and counter-rotating vortex pairs for Euler equations
%J Journées équations aux dérivées partielles
%Z talk:6
%D 2016
%P 1-16
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.647/
%R 10.5802/jedp.647
%G en
%F JEDP_2016____A6_0
Hmidi, Taoufik; Mateu, Joan. Corotating and counter-rotating vortex pairs for Euler equations. Journées équations aux dérivées partielles (2016), Talk no. 6, 16 p. doi : 10.5802/jedp.647. http://www.numdam.org/articles/10.5802/jedp.647/

[1] Bertozzi, A. L.; Constantin, P. Global regularity for vortex patches, Comm. Math. Phys., Volume 152 (1993) no. 1, pp. 19-28 http://projecteuclid.org/euclid.cmp/1104252307 | MR

[2] Burbea, Jacob Motions of vortex patches, Lett. Math. Phys., Volume 6 (1982) no. 1, pp. 1-16 | DOI | MR

[3] Castro, Angel; Córdoba, Diego; Gómez-Serrano, Javier Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations, Duke Math. J., Volume 165 (2016) no. 5, pp. 935-984 | DOI | MR

[4] Castro, Angel; Córdoba, Diego; Gómez-Serrano, Javier Uniformly rotating analytic global patch solutions for active scalars, Ann. PDE, Volume 2 (2016) no. 1, Art. 1, 34 pages | DOI | MR

[5] Chemin, Jean-Yves Fluides parfaits incompressibles, Astérisque, Volume 230 (1995), 177 pages | MR

[6] de la Hoz, Francisco; Hmidi, Taoufik; Mateu, Joan; Verdera, Joan Doubly connected V-states for the planar Euler equations, SIAM J. Math. Anal., Volume 48 (2016) no. 3, pp. 1892-1928 | DOI | MR

[7] Deem, Gary S; Zabusky, Norman J Vortex waves: Stationary" V states," interactions, recurrence, and breaking, Physical Review Letters, Volume 40 (1978) no. 13, 859 pages

[8] Denisov, Sergey A. The centrally symmetric V-states for active scalar equations. Two-dimensional Euler with cut-off, Comm. Math. Phys., Volume 337 (2015) no. 2, pp. 955-1009 | DOI | MR

[9] Dritschel, David G. A general theory for two-dimensional vortex interactions, J. Fluid Mech., Volume 293 (1995), pp. 269-303 | DOI | MR

[10] Hmidi, Taoufik; Mateu, Joan Bifurcation of rotating patches from Kirchhoff vortices, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 10, pp. 5401-5422 | DOI | MR

[11] Hmidi, Taoufik; Mateu, Joan Degenerate bifurcation of the rotating patches, Adv. Math., Volume 302 (2016), pp. 799-850 | DOI | MR

[12] Hmidi, Taoufik; Mateu, Joan; Verdera, Joan Boundary regularity of rotating vortex patches, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 1, pp. 171-208 | DOI | MR

[13] Kamm, James Russell SHAPE AND STABILITY OF TWO-DIMENSIONAL UNIFORM VORTICITY REGIONS, ProQuest LLC, Ann Arbor, MI, 1987, 145 pages Thesis (Ph.D.)–California Institute of Technology | MR

[14] Keady, G. Asymptotic estimates for symmetric vortex streets, J. Austral. Math. Soc. Ser. B, Volume 26 (1985) no. 4, pp. 487-502 | DOI | MR

[15] Lamb, Horace Hydrodynamics, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993, xxvi+738 pages (With a foreword by R. A. Caflisch [Russel E. Caflisch]) | MR

[16] Pierrehumbert, RT A family of steady, translating vortex pairs with distributed vorticity, Journal of Fluid Mechanics, Volume 99 (1980) no. 01, pp. 129-144

[17] Saffman, P. G.; Szeto, R. Equilibrium shapes of a pair of equal uniform vortices, Phys. Fluids, Volume 23 (1980) no. 12, pp. 2339-2342 | DOI | MR

[18] Turkington, Bruce Corotating steady vortex flows with N-fold symmetry, Nonlinear Anal., Volume 9 (1985) no. 4, pp. 351-369 | DOI | MR

[19] Wu, H. M.; Overman, E. A. II; Zabusky, N. J. Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases. I. Numerical algorithms and results, J. Comput. Phys., Volume 53 (1984) no. 1, pp. 42-71 | DOI | MR

[20] Yudovič, V. I. Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., Volume 3 (1963), pp. 1032-1066 | MR

Cited by Sources: