%0 Journal Article %T Local and Global Well-posedness of the fractional order EPDiff equation on $R^d$ %+ Institute for Applied Mathematics (IFAM) %+ University of Vienna [Vienna] %+ Institut de Mathématiques de Marseille (I2M) %A Escher, Joachim %A Bauer, Martin %A Kolev, Boris %< avec comité de lecture %@ 0022-0396 %J Journal of Differential Equations %I Elsevier %V 258 %N 6 %P 2010–2053 %8 2015-03-15 %D 2015 %Z 1411.4081 %R 10.1016/j.jde.2014.11.021 %K Sobolev metrics of fractional order %K Diffeomorphism groups %K EPDiff equation %Z MSC 2010: 58D05, 35Q35 %Z Mathematics [math]/Mathematical Physics [math-ph]Journal articles %X 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 $\frac{1}{2} \leq s < 1 + d/2$ is derived. %G English %L hal-01111245 %U https://hal.science/hal-01111245 %~ CNRS %~ UNIV-AMU %~ EC-MARSEILLE %~ INSMI %~ I2M %~ I2M-2014- %~ TDS-MACS