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.
Mots clés : Euler equations, steady vortex pairs, desingularization.
@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), Exposé no. 6, 16 p. doi : 10.5802/jedp.647. http://www.numdam.org/articles/10.5802/jedp.647/
[1] Global regularity for vortex patches, Comm. Math. Phys., Volume 152 (1993) no. 1, pp. 19-28 http://projecteuclid.org/euclid.cmp/1104252307 | MR
[2] Motions of vortex patches, Lett. Math. Phys., Volume 6 (1982) no. 1, pp. 1-16 | DOI | MR
[3] 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] Uniformly rotating analytic global patch solutions for active scalars, Ann. PDE, Volume 2 (2016) no. 1, Art. 1, 34 pages | DOI | MR
[5] Fluides parfaits incompressibles, Astérisque, Volume 230 (1995), 177 pages | MR
[6] Doubly connected -states for the planar Euler equations, SIAM J. Math. Anal., Volume 48 (2016) no. 3, pp. 1892-1928 | DOI | MR
[7] Vortex waves: Stationary" V states," interactions, recurrence, and breaking, Physical Review Letters, Volume 40 (1978) no. 13, 859 pages
[8] The centrally symmetric -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] A general theory for two-dimensional vortex interactions, J. Fluid Mech., Volume 293 (1995), pp. 269-303 | DOI | MR
[10] Bifurcation of rotating patches from Kirchhoff vortices, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 10, pp. 5401-5422 | DOI | MR
[11] Degenerate bifurcation of the rotating patches, Adv. Math., Volume 302 (2016), pp. 799-850 | DOI | MR
[12] Boundary regularity of rotating vortex patches, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 1, pp. 171-208 | DOI | MR
[13] 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] Asymptotic estimates for symmetric vortex streets, J. Austral. Math. Soc. Ser. B, Volume 26 (1985) no. 4, pp. 487-502 | DOI | MR
[15] Hydrodynamics, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993, xxvi+738 pages (With a foreword by R. A. Caflisch [Russel E. Caflisch]) | MR
[16] A family of steady, translating vortex pairs with distributed vorticity, Journal of Fluid Mechanics, Volume 99 (1980) no. 01, pp. 129-144
[17] Equilibrium shapes of a pair of equal uniform vortices, Phys. Fluids, Volume 23 (1980) no. 12, pp. 2339-2342 | DOI | MR
[18] Corotating steady vortex flows with -fold symmetry, Nonlinear Anal., Volume 9 (1985) no. 4, pp. 351-369 | DOI | MR
[19] Steady-state solutions of the Euler equations in two dimensions: rotating and translating -states with limiting cases. I. Numerical algorithms and results, J. Comput. Phys., Volume 53 (1984) no. 1, pp. 42-71 | DOI | MR
[20] Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., Volume 3 (1963), pp. 1032-1066 | MR
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