Geometry of fluid motion

Boris Khesin[1]

  • [1] Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada

Séminaire Équations aux dérivées partielles (2002-2003)

  • Volume: 2002-2003, page 1-10

Abstract

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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].

How to cite

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Khesin, Boris. "Geometry of fluid motion." Séminaire Équations aux dérivées partielles 2002-2003 (2002-2003): 1-10. <http://eudml.org/doc/11052>.

@article{Khesin2002-2003,
abstract = {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].},
affiliation = {Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada},
author = {Khesin, Boris},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {geodesic flow; Euler equations; energy estimates; geodesics on Lie groups; helicity; topological obstructions to energy relaxation; magnetic field in a perfectly conducting medium; Hamiltonian methods; conservation laws},
language = {eng},
pages = {1-10},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Geometry of fluid motion},
url = {http://eudml.org/doc/11052},
volume = {2002-2003},
year = {2002-2003},
}

TY - JOUR
AU - Khesin, Boris
TI - Geometry of fluid motion
JO - Séminaire Équations aux dérivées partielles
PY - 2002-2003
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2002-2003
SP - 1
EP - 10
AB - 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].
LA - eng
KW - geodesic flow; Euler equations; energy estimates; geodesics on Lie groups; helicity; topological obstructions to energy relaxation; magnetic field in a perfectly conducting medium; Hamiltonian methods; conservation laws
UR - http://eudml.org/doc/11052
ER -

References

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  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. Zbl0148.45301
  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. Zbl0623.57016MR891881
  3. Arnold, V.I. & Khesin, B.A. (1998) Topological methods in hydrodynamics. Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, pp. xv+374. Zbl0902.76001MR1612569
  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. Zbl0820.58019MR1028280
  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. Zbl0973.76097
  6. Freedman, M.H. & He, Z.-X. (1991) Divergence-free fields: energy and asymptotic crossing number. Annals of Math.  134:1, 189–229. Zbl0746.57011MR1114611
  7. Khesin, B. & Misiołek, G. (2002) Euler equations on homogeneous spaces and Virasoro orbits. Preprint arXiv: math.SG/0210397, to appear Adv. Math., 26pp. Zbl1017.37039
  8. Moffatt, H.K. (1969) The degree of knottedness of tangled vortex lines. J. Fluid. Mech. 35, 117–129. Zbl0159.57903
  9. Moffatt, H.K. & Tsinober, A. (1992) Helicity in laminar and turbulent flow. Annual Review of Fluid Mechanics  24, 281–312. Zbl0751.76018MR1145012
  10. Ovsienko, V.Yu. & Khesin, B.A. (1987) Korteweg-de Vries super-equation as an Euler equation. Funct. Anal. Appl.  21:4, 329–331. Zbl0655.58018MR925082
  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. Zbl0598.76006
  12. Vogel, T. (2000) On the asymptotic linking number. Preprint arXiv: math.DS/0011159, to appear in Proc. AMS, 9pp. Zbl1015.57018MR1963779

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