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 three-dimensional visualizations show the collision of vortex sheets. The BKM criterion on the power-law growth of the supremum of the vorticity, applied on the same time interval, is not inconsistent with the occurrence of a singularity around t 4. These 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 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.

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

“…We note in passing that the limitation in extrapolating in time the temporal evolution of the width of the analyticity strip has been noted, amongst others, in Ref. [36].…”

“…[27] for a recent review of results on finite-time blow-ups via numerical simulations), with the advance of computing power, the search for evidence for or against finite-time blow-up of the three-dimensional Euler equations through numerical simulations have gained ground. As shown by Bustamante and Brachet [36], the temporal measurement of the distance, to the real domain, of the nearest singularity, is limited not only by computing power but also by the onset of thermalisation. Hence an estimate of the time when thermalisation sets in will be have an important bearing on interpreting the accuracy of measurements of complex singularities in time from spectral, and hence Galerkin-truncated, simulations of the Euler equations.…”

“…In recent years (we refer the reader to Ref. [36] for the most recent results and to Ref. [27] for a recent review of results on finite-time blow-ups via numerical simulations), with the advance of computing power, the search for evidence for or against finite-time blow-up of the three-dimensional Euler equations through numerical simulations have gained ground.…”

Asymptotic solution for high-vorticity regions in incompressible three-dimensional Euler equations

“…Comparison of the logarithmic derivatives β 1 = −˙ 1 / 1 and β 2 =ω max /ω max with the velocity gradients −∂v 1 /∂a 1 and ∂v 2 /∂a 2 computed at the global vorticity maximum, see Eq. (9). Prior computing the time derivatives, 1 and ω max are smoothed with the weighted local regression (lowess filter), see [18].…”

Universal mechanism for saturation of vorticity growth in fully developed fluid turbulence

“…Geometrically, this corresponds to a vortex tube getting increasingly thinner and more intense, until its core is annihilated, thus leading to a singular blow-up. A vast amount of work on finite-time singularities, starting from symmetric initial conditions, is available (see Kerr 1993;Pelz 2001;Cichowlas & Brachet 2005;Grafke et al 2008;Hou & Li 2008;Bustamante & Brachet 2012). Several counter-blow-up arguments have also been raised, which usually appeal to more detailed insights into the dynamic morphology of turbulence (Frisch et al 2003;Yang & Pullin 2010).…”