On a Type I Singularity Condition in Terms of the Pressure for the Euler Equations in ℝ3.

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 $$\begin{align*} &\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \exp \left(\int_{s} ^t \int_0 ^{\sigma} \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \textrm{d}\sigma \right) \right\} \textrm{d}s \textrm{d}t \, = +\infty.\end{align*}$$ 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))} \textrm{d}t <+\infty $|⁠ , we obtain localized versions of these results. [ABSTRACT FROM AUTHOR]