  We prove that the geodesic equations of all Sobolev metrics of fractional
order one and higher on spaces of diffeomorphisms and, more generally,
immersions are locally well posed. This result builds on the recently
established real analytic dependence of fractional Laplacians on the underlying
Riemannian metric. It extends several previous results and applies to a wide
range of variational partial differential equations, including the well-known
Euler-Arnold equations on diffeomorphism groups as well as the geodesic
equations on spaces of manifold-valued curves and surfaces.
