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

TL;DR

This paper establishes quantitative lower bounds on the blow-up rate of vorticity in solutions to the 3D Euler equations, providing insights into the conditions leading to finite-time singularities.

Contribution

It introduces new lower bounds on vorticity growth near blow-up time, advancing understanding of the blow-up mechanism in Euler equations.

Findings

Lower bounds on the integral of vorticity norm before blow-up.

Lower bounds on the maximum vorticity at blow-up time.

Implications for the blow-up rate of derivatives of vorticity.

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.