  Of concern is the study of the long-time existence of solutions to the
Euler--Arnold equation of the right-invariant $H^{3/2}$-metric on the
diffeomorphism group of the circle. In previous work by Escher and Kolev it has
been shown that this equation admits long-time solutions if the order $s$ of
the 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 the
analogous result also for the boundary case. The behaviour of the
$H^{3/2}$-metric is, however, still different from its higher order counter
parts, as it does not induce a complete Riemannian metric on any group of
Sobolev diffeomorphisms.
