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 the accumulation of $\|ω\|_{L^\infty}$ up to time $t$ and the supremum over $[0,t]$ of $\|ω\|_{L^\infty}$ for $t$ sufficiently close to~$T_\ast$. Notably, this gives a quantitative description of the BKM blow-up criterion. Moreover, we provide lower bounds on the supremum over $[0,t]$ of $\|D^k ω\|_{L^\infty}$. When the domain is $\mathbb{T}^3$, we establish pointwise-in-time lower bounds on $\|D^kω\|_{L^\infty}$ for $t$ sufficiently close to~$T_\ast$.

Theorems & Definitions (16)

• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.4
• proof : Proof of Theorem \ref{['T01']}
• Lemma 4.1