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

TL;DR

This paper investigates the well-posedness of fractional order Sobolev metrics on diffeomorphism groups of Euclidean space, establishing global solutions for higher regularity and local solutions for lower regularity metrics.

Contribution

It proves global well-posedness of geodesic equations for Sobolev metrics with regularity s > 1 + d/2 and local existence for s between 1/2 and 1 + d/2 on diffeomorphism groups.

Findings

Global well-posedness for s > 1 + d/2

Local existence for 1/2 ≤ s < 1 + d/2

Smooth Riemannian structure on diffeomorphism groups

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 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.