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Article Dans Une Revue Journal of Differential Equations Année : 2015

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

Joachim Escher
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Martin Bauer
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Boris Kolev

Résumé

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.

Dates et versions

hal-01111245 , version 1 (29-01-2015)

Identifiants

Citer

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