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

physical review. e, statistical, nonlinear and soft matter physics

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