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

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 $\frac{1}{2} \leq s < 1 + d/2$ is derived.
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Submitted on : Thursday, January 29, 2015 - 11:05:14 PM
Last modification on : Monday, March 4, 2019 - 2:04:19 PM

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Joachim Escher, Martin Bauer, Boris Kolev. Local and Global Well-posedness of the fractional order EPDiff equation on $R^d$. Journal of Differential Equations, Elsevier, 2015, 258 (6), pp.2010-2053. ⟨10.1016/j.jde.2014.11.021⟩. ⟨hal-01111245⟩

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