Local and Global Well-posedness of the fractional order EPDiff equation on ${\mathbb{R}}^d$

Of concern is the study of fractional order Sobolev–type metrics on the group of $H^{\mathrm{\infty }}$-diffeomorphism of ${\mathbb{R}}^d$ and on its Sobolev completions ${\mathcal{D}}^q({\mathbb{R}}^d)$. It is shown that the $H^s$-Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds ${\mathcal{D}}^s({\mathbb{R}}^d)$ for $s>1+\frac{d}{2}$. As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold ${\mathcal{D}}^s({\mathbb{R}}^d)$ and on the smooth regular Fr'echet-Lie group of all $H^{\mathrm{\infty }}$-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order $\frac{1}{2}\le s<1+d\mathrm{/}2$ is derived.