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 $$\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (τ)\|_{L^\infty} dτ\exp \left( \int_{s} ^t \int_0 ^{\s} \|D^2 p (τ)\|_{L^\infty} dτd\s \right) \right\}dsdt \, = +\infty.$$ As consequences of this criterion we show that there is no blow up at $t=T$ if $\Vert D^{2}p(t){\Vert }_{L^{\infty }}\le \frac{c}{(T-t{)}^2}$ with $c<1$ as $t\nearrow T$. Under the additional assumption of ${\int }_0^T\Vert u(t){\Vert }_{L^{\infty }(B(x_0,\rho ))}dt<+\infty$, we obtain localized versions of these results.

10 pages