Thin vortex tubes in the stationary Euler equation
[Tubes de vorticité étroits dans l’équation d’Euler stationnaire]
Journées équations aux dérivées partielles (2013), article no. 4, 13 p.

On expose quelques nouveaux résultats sur l’existence de solutions stationnaires à l’équation d’Euler sur 3 avec un ensemble de tubes de vorticité étroits (qui peuvent être noués et entrelacés) qu’on peut prescrire a priori.

In this paper we outline some recent results concerning the existence of steady solutions to the Euler equation in 3 with a prescribed set of (possibly knotted and linked) thin vortex tubes.

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     author = {Enciso, Alberto and Peralta-Salas, Daniel},
     title = {Thin vortex tubes in the stationary {Euler} equation},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {4},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2013},
     doi = {10.5802/jedp.100},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.100/}
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Enciso, Alberto; Peralta-Salas, Daniel. Thin vortex tubes in the stationary Euler equation. Journées équations aux dérivées partielles (2013), article  no. 4, 13 p. doi : 10.5802/jedp.100. http://www.numdam.org/articles/10.5802/jedp.100/

[1] V.I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966) 319–361. | Numdam | MR 202082 | Zbl 0148.45301

[2] V.I. Arnold, B. Khesin, Topological methods in hydrodynamics, Springer, New York, 1999. | MR 1612569 | Zbl 0902.76001

[3] D. Córdoba, C. Fefferman, On the collapse of tubes carried by 3D incompressible flows, Comm. Math. Phys. 222 (2001) 293–298. | MR 1859600 | Zbl 0999.76020

[4] J. Deng, T.Y. Hou, X. Yu, Geometric properties and nonblowup of 3D incompressible Euler flow. Comm. PDE 30 (2005) 225–243. | MR 2131052 | Zbl 1142.35549

[5] A. Enciso, D. Peralta-Salas, Knots and links in steady solutions of the Euler equation, Ann. of Math. 175 (2012) 345–367. | MR 2874645 | Zbl 1238.35092

[6] A. Enciso, D. Peralta-Salas, Existence of knotted vortex tubes in steady Euler flows, arXiv:1210.6271.

[7] A. Enciso, D. Peralta-Salas, Non-existence and structure for Beltrami fields with nonconstant proportionality factor, arXiv:1402.6825.

[8] A. González-Enríquez, R. de la Llave, Analytic smoothing of geometric maps with applications to KAM theory, J. Differential Equations 245 (2008) 1243–1298. | MR 2436830 | Zbl 1160.37024

[9] H. von Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math. 55 (1858) 25–55. | Zbl 055.1448cj

[10] D. Kleckner, W.T.M. Irvine, Creation and dynamics of knotted vortices, Nature Phys. 9 (2013) 253–258.

[11] R.B. Pelz, Symmetry and the hydrodynamic blow-up problem, J. Fluid Mech. 444 (2001) 299–320. | MR 1856973 | Zbl 1002.76095

[12] W. Thomson (Lord Kelvin), Vortex Statics, Proc. R. Soc. Edinburgh 9 (1875) 59–73 (reprinted in: Mathematical and physical papers IV, Cambridge University Press, Cambridge, 2011, pp. 115–128).

[13] N. Nadirashvili, Liouville theorem for Beltrami flow, Geom. Funct. Anal. 24 (2014) 916–921.

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