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This article is cited in 22 scientific papers (total in 22 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.
Received: November 2, 2011; in revised form January 19, 2012
Citation:
Boris Khesin, “Symplectic structures and dynamics on vortex membranes”, Mosc. Math. J., 12:2 (2012), 413–434
Linking options:
https://www.mathnet.ru/eng/mmj473 https://www.mathnet.ru/eng/mmj/v12/i2/p413
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Abstract page: | 235 | References: | 35 |
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