Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem

Abstract

Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to $4096^3$. The results are analyzed in terms of the classical analyticity strip method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of the time-decay of the width of the analyticity strip $δ(t)$ is observed at the highest resolution for $3.7<t<3.85$ while preliminary 3D visualizations show the collision of vortex sheets. The BKM criterium on the power-law growth of supremum of the vorticity, applied on the same time-interval, is not inconsistent with the occurrence of a singularity around $t \simeq 4$. These new findings lead us to investigate how fast the analyticity strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A new simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if $δ(t)$ vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for $δ(t)$ then its exponent must be greater than some critical value, thus providing a new test that is applied to our $4096^3$ Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time-interval on which a well-resolved power-law behavior of $δ(t)$ takes place, and check whether the new regime is genuine and not simply a crossover to a faster exponential decay.