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 ${\sup }_{s\in [0,t]}\Vert \omega {\Vert }_{L^{\infty }}$ 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~$\Vert D^k\omega {\Vert }_{L^{\infty }}$. Finally, we state some consequences on the blow-up rate of the derivative of the deformation tensor.