, Volume 314, Issue 1, pp 265-280
Date: 18 Jul 2012

A Lower Bound on Blowup Rates for the 3D Incompressible Euler Equation and a Single Exponential Beale-Kato-Majda Type Estimate

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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 \({s>\frac{5}{2}}\) . 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 Constantin in (SIAM Rev. 36(1):73–98, 1994). In particular, we derive lower bounds on the blowup rate of such solutions.

Communicated by P. Constantin