Centre de diffusion de revues académiques mathématiques


Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Taoufik Hmidi; Joan Mateu
Corotating and counter-rotating vortex pairs for Euler equations
Journées équations aux dérivées partielles (2016), Exp. No. 6, 16 p., doi: 10.5802/jedp.647
Article PDF
Class. Math.: 35Q35, 76B03, 76C05
Mots clés: Euler equations, steady vortex pairs, desingularization.

Résumé - Abstract

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.


[1] A. L. Bertozzi & P. Constantin, “Global regularity for vortex patches”, Comm. Math. Phys. 152 (1993) no. 1, p. 19-28
[2] Jacob Burbea, “Motions of vortex patches”, Lett. Math. Phys. 6 (1982) no. 1, p. 1-16 Article
[3] Angel Castro, Diego Córdoba & Javier Gómez-Serrano, “Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations”, Duke Math. J. 165 (2016) no. 5, p. 935-984 Article
[4] Angel Castro, Diego Córdoba & Javier Gómez-Serrano, “Uniformly rotating analytic global patch solutions for active scalars”, Ann. PDE 2 (2016) no. 1 Article
[5] Jean-Yves Chemin, “Fluides parfaits incompressibles”, Astérisque 230 (1995)
[6] Francisco de la Hoz, Taoufik Hmidi, Joan Mateu & Joan Verdera, “Doubly connected $V$-states for the planar Euler equations”, SIAM J. Math. Anal. 48 (2016) no. 3, p. 1892-1928 Article
[7] Gary S Deem & Norman J Zabusky, “Vortex waves: Stationary" V states," interactions, recurrence, and breaking”, Physical Review Letters 40 (1978) no. 13
[8] Sergey A. Denisov, “The centrally symmetric $V$-states for active scalar equations. Two-dimensional Euler with cut-off”, Comm. Math. Phys. 337 (2015) no. 2, p. 955-1009 Article
[9] David G. Dritschel, “A general theory for two-dimensional vortex interactions”, J. Fluid Mech. 293 (1995), p. 269-303 Article
[10] Taoufik Hmidi & Joan Mateu, “Bifurcation of rotating patches from Kirchhoff vortices”, Discrete Contin. Dyn. Syst. 36 (2016) no. 10, p. 5401-5422 Article
[11] Taoufik Hmidi & Joan Mateu, “Degenerate bifurcation of the rotating patches”, Adv. Math. 302 (2016), p. 799-850 Article
[12] Taoufik Hmidi, Joan Mateu & Joan Verdera, “Boundary regularity of rotating vortex patches”, Arch. Ration. Mech. Anal. 209 (2013) no. 1, p. 171-208 Article
[13] James Russell Kamm, SHAPE AND STABILITY OF TWO-DIMENSIONAL UNIFORM VORTICITY REGIONS, ProQuest LLC, Ann Arbor, MI, 1987, Thesis (Ph.D.)–California Institute of Technology
[14] G. Keady, “Asymptotic estimates for symmetric vortex streets”, J. Austral. Math. Soc. Ser. B 26 (1985) no. 4, p. 487-502 Article
[15] Horace Lamb, Hydrodynamics, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993, With a foreword by R. A. Caflisch [Russel E. Caflisch]
[16] RT Pierrehumbert, “A family of steady, translating vortex pairs with distributed vorticity”, Journal of Fluid Mechanics 99 (1980) no. 01, p. 129-144
[17] P. G. Saffman & R. Szeto, “Equilibrium shapes of a pair of equal uniform vortices”, Phys. Fluids 23 (1980) no. 12, p. 2339-2342 Article
[18] Bruce Turkington, “Corotating steady vortex flows with $N$-fold symmetry”, Nonlinear Anal. 9 (1985) no. 4, p. 351-369 Article
[19] H. M. Wu, E. A. Overman & N. J. Zabusky, “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. 53 (1984) no. 1, p. 42-71 Article
[20] V. I. Yudovič, “Non-stationary flows of an ideal incompressible fluid”, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), p. 1032-1066
Copyright Cellule MathDoc 2018 | Crédit | Plan du site