Beyond Arnold’s geodesic framework of an ideal hydrodynamics

In 1966 V.I. Arnold developed a group-theoretic approach to ideal hydrodynamics in which the Euler equation for an inviscid incompressible fluid is described as the geodesic flow equation for a right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. In the talk we discuss two ramifications of this approach.
First of all, it was observed that problems of optimal mass transport are in a sense dual to the Euler hydrodynamics. By regarding volume-preserving diffeomorphisms as a subgroup of all diffeomorphisms, we describe $L^2$ and $H^1$ versions of the Kantorovich–Wasserstein and Fisher–Rao metrics on the spaces of densities. It turns out that for the homogeneous $H^1$ metric the Wasserstein space is isometric to (a piece of) an infinite-dimensional sphere and it leads to an integrable generalization of the Hunter–Saxton equation.
The second generalization is an Arnold-like geodesic and Hamiltonian description for fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface and equipped with a one-sided invariant metric.