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 $H^{ \infty}$-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 $H^{ \infty}$-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.

Categories: math.AP