Local and global well-posedness of the fractional order EPDiff equation on RdRd

Of concern is the study of fractional order Sobolev-type metrics on the group of H∞H∞-diffeomorphism of RdRd and on its Sobolev completions Dq(Rd)Dq(Rd). It is shown that the HsHs-Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds Ds(Rd)Ds(Rd) for s>1+d2. As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold Ds(Rd)Ds(Rd) and on the smooth regular Fréchet–Lie group of all H∞H∞-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order 12≤s<1+d/2 is derived.