On a Type I singularity condition in terms of the pressure for the Euler equations in $\mathbb R^3$

Abstract: 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.