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

TL;DR

This paper establishes a new blow-up criterion for the incompressible Euler equations based on the Hessian of the pressure, providing conditions under which solutions remain regular or blow up in finite time.

Contribution

It introduces a novel pressure Hessian-based blow-up criterion and derives conditions preventing singularity formation in Euler solutions.

Findings

Blow-up occurs only if a specific integral involving pressure Hessian diverges.

No blow-up if pressure Hessian norm is bounded by c/(T-t)^2 with c<1.

Localized regularity results under additional integrability assumptions.

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.