Coadjoint orbits of vortex sheets in ideal fluids (Q6146401)

The authors deal with vortex sheets in ideal fluids focusing on the case of codimension one closed support. Their vorticity distribution is described by a closed one-form, forming vortex sheets. They handle the general case, which requires to use central extensions of the group of volume preserving diffeomorphisms, giving rise to a cohomological characterization for vortex sheets that yield coadjoint orbits of the volume preserving diffeomorphisms, thus enlarging the class of exact vortex sheets. The authors also described the coadjoint orbits in terms of a class of extended nonlinear Grassmannians, called decorated nonlinear Grassmannians. The related vorticity supports in a given orbit made possible for authors to use the volume form variant of the isodrastic distribution considered before by Weinstein in the symplectic context. The authors illustrate the obtained results with several examples of vortex sheets in 3D, in particular they show that the orbit symplectic form descends from a closed 2-form on the manifold of parametrized hypersurfaces, naturally induced by the volume form and by the vorticity density.