Coadjoint orbits of vortex sheets in ideal fluids

Abstract

We describe coadjoint orbits associated to the motion of codimension one singular vorticities in ideal fluids, e.g. vortex sheets in 3D. We show that these coadjoint orbits can be identified with a certain class of decorated nonlinear Grassmannians, that consist of codimension one submanifolds belonging to an isodrast (i.e., enclosing a given volume, provided their homology class vanishes), endowed with a closed 1-form of a given type. Such isodrasts are defined via an integrable distribution associated to a manifold endowed with a volume form, similar to the isodrastic distribution [35] in symplectic setting. With the choice of a Riemannian metric, the expression of the orbit symplectic form is shown to take a particularly simple expression reminiscent of Darboux coordinates. In general we get coadjoint orbits of the universal central extension of the group of exact volume preserving diffeomorphisms and we give a necessary and sufficient condition under which the central extension is not needed. We also focus on the case in which we get coadjoint orbits of the group of volume preserving diffeomorphisms, which is relevant for the ideal fluid, and discuss how our results extend the class of coadjoint orbits of exact vortex sheets considered in [25].