Lower bounds on the blowup rate of vorticity in the Euler equations

Abstract

Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In particular, when the domain is $\mathbb{R}^3$ or $\mathbb{T}^3$, we provide lower bounds on $\int_{0}^{t}\Vert \omega\Vert_{L^\infty}\,ds$ and $\sup_{s\in[0,t]}\|\omega\|_{L^\infty}$ for $t$ sufficiently close to~$T_\ast$. Notably, this gives a quantitative description of the BKM blow-up criterion. Moreover, we provide pointwise-in-time lower bounds on~$\|D^k \omega\|_{L^\infty}$. Finally, we state some consequences on the blow-up rate of the derivative of the deformation tensor.