On the blowup rate of vorticity for the Euler equations in a bounded domain

Abstract

Given that a solution to the 3D incompressible Euler equations on a bounded domain blows up at a time $T_{\ast }$ and that $T_{\ast }$ is the first such time, we provide pointwise-in-time lower bounds on $\Vert D^k\omega {\Vert }_{L^{\infty }(\Omega )}$ for $k\ge 1$. We also show that the Gronwall-type inequality satisfied by $\Vert \omega (t){\Vert }_{L^{\infty }}$, in the cases that $\Omega ={\mathbb{R}}^3$, ${\mathbb{T}}^3$, or a bounded domain, exhibits wildly oscillating solutions.