English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study Sobolev-type metrics of fractional order $s\geq0$ on the group$\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold $M$. We showthat for the important special case $M=S^1$ the geodesic distance on$\Diff_c(S^1)$ vanishes if and only if $s\leq\frac12$. For other manifolds weobtain a partial characterization: the geodesic distance on $\Diff_c(M)$vanishes for $M=\R\times N, s<\frac12$ and for $M=S^1\times N, s\leq\frac12$,with $N$ being a compact Riemannian manifold. On the other hand the geodesicdistance on $\Diff_c(M)$ is positive for $\dim(M)=1, s>\frac12$ and$\dim(M)\geq2, s\geq1$.  For $M=\R^n$ we discuss the geodesic equations for these metrics. For $n=1$we obtain some well known PDEs of hydrodynamics: Burgers' equation for $s=0$,the modified Constantin-Lax-Majda equation for $s=\frac 12$ and theCamassa-Holm equation for $s=1$.

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group