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Mosc. Math. J., 2012, Volume 12, Number 2, Pages 413–434 (Mi mmj473)  

This article is cited in 7 scientific papers (total in 7 papers)

Symplectic structures and dynamics on vortex membranes

Boris Khesin

Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Abstract: We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes of codimension 2 in any $\mathbb{R}^n$, which generalizes to any dimension the classical binormal, or vortex filament, equation in $\mathbb{R}^3$.
This framework also allows one to define the symplectic structures on the spaces of vortex sheets, which interpolate between the corresponding structures on vortex filaments and smooth vorticities.

Key words and phrases: vortex filament equation, Euler equation, vortex sheet, mean curvature flow, localized induction approximation, symplectic structure, vortex membrane.

DOI: https://doi.org/10.17323/1609-4514-2012-12-2-413-434

Full text: http://www.ams.org/.../abst12-2-2012.html
References: PDF file   HTML file

Bibliographic databases:

MSC: Primary 35Q35; Secondary 53C44, 58E40
Received: November 2, 2011; in revised form January 19, 2012
Language:

Citation: Boris Khesin, “Symplectic structures and dynamics on vortex membranes”, Mosc. Math. J., 12:2 (2012), 413–434

Citation in format AMSBIB
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\by Boris~Khesin
\paper Symplectic structures and dynamics on vortex membranes
\jour Mosc. Math.~J.
\yr 2012
\vol 12
\issue 2
\pages 413--434
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\crossref{https://doi.org/10.17323/1609-4514-2012-12-2-413-434}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2978763}
\zmath{https://zbmath.org/?q=an:1258.35162}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309365900011}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Khesin B., “Dynamics of Symplectic Fluids and Point Vortices”, Geom. Funct. Anal., 22:5 (2012), 1444–1459  crossref  zmath  isi  scopus
    2. Khesin B., “The Vortex Filament Equation in Any Dimension”, Iutam Symposium on Topological Fluid Dynamics: Theory and Applications, Procedia Iutam, 7, eds. Moffatt H., Bajer K., Kimura Y., Elsevier Science BV, 2013, 135–140  crossref  isi  scopus
    3. Vankerschaver J., Leok M., “A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects”, J. Nonlinear Sci., 24:1 (2014), 1–37  crossref  mathscinet  zmath  isi  elib  scopus
    4. R. L. Jerrard, Ch. Seis, “On the vortex filament conjecture for Euler flows”, Arch. Ration. Mech. Anal., 224:1 (2017), 135–172  crossref  mathscinet  zmath  isi  scopus
    5. Ch. Song, “Gauss map of the skew mean curvature flow”, Proc. Amer. Math. Soc., 145:11 (2017), 4963–4970  crossref  mathscinet  zmath  isi  scopus
    6. Izosimov A., Khesin B., “Vortex Sheets and Diffeomorphism Groupoids”, Adv. Math., 338 (2018), 447–501  crossref  mathscinet  zmath  isi  scopus
    7. Hernandez-Garduno A., Shashikanth B.N., “Reconstruction Phases in the Planar Three-and Four-Vortex Problems”, Nonlinearity, 31:3 (2018), 783–814  crossref  mathscinet  zmath  isi  scopus
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