Geometry of fluid motion
Khesin, Boris1
1 Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2002-2003), Exposé 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].

Khesin, Boris 1

1 Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada 
@article{SEDP_2002-2003____A10_0,
     author = {Khesin, Boris},
     title = {Geometry of fluid motion},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:10},
     pages = {1--10},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2002-2003},
     zbl = {1056.37096},
     mrnumber = {2030705},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2002-2003____A10_0/}
}
TY  - JOUR
AU  - Khesin, Boris
TI  - Geometry of fluid motion
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:10
PY  - 2002-2003
SP  - 1
EP  - 10
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_2002-2003____A10_0/
LA  - en
ID  - SEDP_2002-2003____A10_0
ER  - 
%0 Journal Article
%A Khesin, Boris
%T Geometry of fluid motion
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:10
%D 2002-2003
%P 1-10
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org/item/SEDP_2002-2003____A10_0/
%G en
%F SEDP_2002-2003____A10_0
Khesin, Boris. Geometry of fluid motion. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2002-2003), Exposé 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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

[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

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

[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 | Zbl

[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

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