Geometry of fluid motion
Séminaire Équations aux dérivées partielles (Polytechnique) (2002-2003), Talk no. 10, 10 p.

We survey two problems illustrating geometric-topological and Hamiltonian methods in fluid mechanics: energy relaxation of a magnetic field and conservation laws for ideal fluid motion. More details and results, as well as a guide to the literature on these topics can be found in [3].

@article{SEDP_2002-2003____A10_0,
     author = {Khesin, Boris},
     title = {Geometry of fluid motion},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2002-2003},
     note = {talk:10},
     mrnumber = {2030705},
     zbl = {1056.37096},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2002-2003____A10_0}
}
Khesin, Boris. Geometry of fluid motion. Séminaire Équations aux dérivées partielles (Polytechnique) (2002-2003), Talk no. 10, 10 p. http://www.numdam.org/item/SEDP_2002-2003____A10_0/

[1] Arnold, V.I. (1966) Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361. | Numdam | Zbl 0148.45301

[2] Arnold, V.I. (1973) The asymptotic Hopf invariant and its applications. Proc. Summer School in Diff. Equations at Dilizhan, Erevan (in Russian); English transl.: Sel. Math. Sov.  5 (1986), 327–345. | MR 891881 | Zbl 0623.57016

[3] Arnold, V.I. & Khesin, B.A. (1998) Topological methods in hydrodynamics. Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, pp. xv+374. | MR 1612569 | Zbl 0902.76001

[4] Khesin, B.A. & Chekanov, Yu.V. (1989) Invariants of the Euler equation for ideal or barotropic hydrodynamics and superconductivity in D dimensions. Physica D 40:1, 119–131. | MR 1028280 | Zbl 0820.58019

[5] Freedman, M.H. (1999) Zeldovich’s neutron star and the prediction of magnetic froth. Proceedings of the Arnoldfest, Fields Institute Communications 24 (ed. E.Bierstone, et al.), pp. 165–172. | Zbl 0973.76097

[6] Freedman, M.H. & He, Z.-X. (1991) Divergence-free fields: energy and asymptotic crossing number. Annals of Math.  134:1, 189–229. | MR 1114611 | Zbl 0746.57011

[7] Khesin, B. & Misiołek, G. (2002) Euler equations on homogeneous spaces and Virasoro orbits. Preprint arXiv: math.SG/0210397, to appear Adv. Math., 26pp.

[8] Moffatt, H.K. (1969) The degree of knottedness of tangled vortex lines. J. Fluid. Mech. 35, 117–129. | Zbl 0159.57903

[9] Moffatt, H.K. & Tsinober, A. (1992) Helicity in laminar and turbulent flow. Annual Review of Fluid Mechanics  24, 281–312. | MR 1145012 | Zbl 0751.76018

[10] Ovsienko, V.Yu. & Khesin, B.A. (1987) Korteweg-de Vries super-equation as an Euler equation. Funct. Anal. Appl.  21:4, 329–331. | MR 925082 | Zbl 0655.58018

[11] Serre, D. (1984) Invariants et dégénérescence symplectique de l’équation d’Euler des fluids parfaits incompressibles. C.R. Acad. Sci. Paris, Sér. A 298:14, 349–352; also personal communication of L. Tartar. | Zbl 0598.76006

[12] Vogel, T. (2000) On the asymptotic linking number. Preprint arXiv: math.DS/0011159, to appear in Proc. AMS, 9pp. | MR 1963779 | Zbl 1015.57018