  We prove a Beale-Kato-Majda type criterion for the loss of regularity for
solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for
$s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type,
we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length
parameter introduced by P. Constantin in \cite{co1}. In particular, we derive
lower bounds on the blowup rate of such solutions.
