Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid

Ebin, D. G. and J. E. Marsden

Bull. Amer. Math. Soc., (N. S.), 75, (1969), 962-967

Abstract:

We announce several results on the structure of the group of diffeomorphisms $\mathcal {D}$ of a compact n -manifold M , possibly with boundary. The group $\mathcal {D}$ has the structure of a differentiable manifold modelled on a Fréchet space and with this structure, the group operations are smooth. See Leslie [5] and Omori [8], for the proof in case M has no boundary. Following Omori, we call $\mathcal {D}$ an ILH Lie group. We shall show that several infinite dimensional subgroups of $\mathcal {D}$ are actually (ILH) submanifolds and hence also have the structure of ILH Lie groups. Also, we construct certain (weak) Riemannian structures on $\mathcal {D}$ (and on certain subgroups) and find the geodesic flows associated to them.