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Of concern is the study of the long-time existence of solutions to theEuler--Arnold equation of the right-invariant $H^{3/2}$-metric on thediffeomorphism group of the circle. In previous work by Escher and Kolev it hasbeen shown that this equation admits long-time solutions if the order $s$ ofthe metric is greater than $3/2$, the behaviour for the critical Sobolev index$s=3/2$ has been left open. In this article we fill this gap by proving theanalogous result also for the boundary case. The behaviour of the$H^{3/2}$-metric is, however, still different from its higher order counterparts, as it does not induce a complete Riemannian metric on any group ofSobolev diffeomorphisms.

Geodesic completeness of the $H^{3/2}$ metric on $\mathrm{Diff}(S^{1})$