Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem
|Title:||Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem||Authors:||Bustamante, Miguel
Brachet, Marc E.
|Permanent link:||http://hdl.handle.net/10197/7507||Date:||5-Dec-2012||Abstract:||Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to 40963. 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 40963 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.||Funding Details:||Irish Research Council for Science, Engineering and Technology
University College Dublin
|Type of material:||Journal Article||Publisher:||American Physical Society||Copyright (published version):||2012 American Physical Society||Keywords:||Taylor-Green vortex; Analyticity-strip method; Beale, Kato, and Majda (BKM) theorem||DOI:||10.1103/PhysRevE.86.066302||Language:||en||Status of Item:||Peer reviewed|
|Appears in Collections:||Mathematics and Statistics Research Collection|
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