Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows

TL;DR

This study investigates the late-time behavior of 3D Euler flow solutions near singularities, revealing a Gaussian spectral regime and demonstrating that mapped variables enable unprecedented proximity to blowup times in simulations.

Contribution

The paper introduces a novel analysis of late-time singular regimes in 3D Euler flows using mapping techniques, uncovering Gaussian spectra and extending the accessible simulation proximity to singularity.

Findings

Late-time regime exhibits Gaussian spectrum rather than exponential.

Analyticity-strip width approaches zero slowly, remaining above collocation scale.

Mapped variables enable simulation proximity to singularity at 10^{-140} time difference.

Abstract

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems, presented in Bustamante (2011) and extended to the symmetry-plane case by Mulungye et al. (2015), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^2.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t < T^* - 10^{-9000}$, where $T^*$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating point double precision ($\approx 10^{-16}$) creates a `machine-epsilon' barrier. Due to this limitation on the \emph{original} independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^*- t \approx 10^{-140}.$