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

Abstract

Of concern is the study of fractional order Sobolev--type metrics on the group of $H^{\infty }$-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 Fréchet-Lie group of all $H^{\infty }$-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order $\frac{1}{2}\le s<1+d/2$ is derived.

37 pages

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