A lower bound on blowup rates for the 3D incompressible Euler equation and a single exponential Beale-Kato-Majda type estimate

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 $s>\frac{5}{2}$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions.