A lower bound on blowup rates for the 3D incompressible Euler equation and a single exponential Beale-Kato-Majda type estimate

TL;DR

This paper establishes a new single exponential bound on the growth of solutions to the 3D incompressible Euler equations, providing insights into blowup rates and regularity loss criteria.

Contribution

It introduces a single exponential Beale-Kato-Majda type estimate and derives lower bounds on blowup rates for solutions in high regularity spaces.

Findings

Single exponential bound on solution growth

Lower bounds on blowup rates

New regularity loss criterion

Abstract

We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions.