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Symplectic structures and dynamics on vortex membranes. (English) Zbl 1258.35162
Summary: 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.

MSC:
 35Q31 Euler equations 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 58E40 Variational aspects of group actions in infinite-dimensional spaces
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