  We prove a blow up criterion in terms of the Hessian of the pressure of
smooth solutions $u\in C([0, T); W^{2,q} (\mathbb R^3))$, $q>3$ of the
incompressible Euler equations. We show that a blow up at $t=T$ happens only if
$$\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (\tau)\|_{L^\infty} d\tau \exp
\left( \int_{s} ^t \int_0 ^{\s} \|D^2 p (\tau)\|_{L^\infty} d\tau d\s \right)
\right\}dsdt \, = +\infty.$$ As consequences of this criterion we show that
there is no blow up at $t=T$ if $ \|D^2 p(t)\|_{L^\infty} \le \frac
{c}{(T-t)^2}$ with $c<1$ as $t\nearrow T$. Under the additional assumption of
$\int_0 ^T \|u(t)\|_{L^\infty (B(x_0, \rho))} dt <+\infty$, we obtain localized
versions of these results.
