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

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 $${\int }_0^T{\int }_0^t\{{\int }_0^s\mathrm{\parallel }{D}^{2}p(\tau ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}d\tau \exp ({\int }_s^t{\int }_0^{\text{s}}\mathrm{\parallel }{D}^{2}p(\tau ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}d\tau d\text{s})\}dsdt=+\mathrm{\infty }\mathrm{.}$$ 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$\mathrm{.}Undertheadditionalassumptionof$\int_0 ^T |u(t)|{L^\infty (B(x_0, \rho))} dt <+\infty$, we obtain localized versions of these results.