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

Abstract

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 $$\begin{align*} &\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \exp \left( \int_{s} ^t \int_0 ^{\sigma} \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \textrm{d}\sigma \right) \right\} \textrm{d}s \textrm{d}t \, = +\infty.\end{align*}$$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.