Skip to main content Accessibility help

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

  • Rachel M. Mulungye (a1), Dan Lucas (a2) and Miguel D. Bustamante (a1)


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. (Physica D, vol. 132, 1999, pp. 497–510). By employing the method of mapping to regular systems, presented by Bustamante (Physica D, vol. 240 (13), 2011, pp. 1092–1099) and extended to the symmetry-plane case by Mulungye et al. (J. Fluid Mech., vol. 771, 2015, pp. 468–502), 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 $t<T^{\ast }-10^{-9000}$ , where $T^{\ast }$ 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 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: $T^{\ast }-t\approx 10^{-140}$ .


Corresponding author

Email address for correspondence:


Hide All
Bardos, C. & Titi, E. 2007 Euler equations for incompressible ideal fluids. Russ. Math. Surveys 62 (3), 409451.
Beale, J., Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94, 6166.
Bustamante, M. D. 2011 3D Euler equations and ideal MHD mapped to regular systems: probing the finite-time blowup hypothesis. Physica D 240 (13), 10921099.
Bustamante, M. D. & Brachet, M. 2012 Interplay between the Beale–Kato–Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem. Phys. Rev. E 86 (6), 066302.
Constantin, P. 2000 The Euler equations and nonlocal conservative Riccati equations. Intl Math. Res. Not. 2000 (9), 455465.
Fefferman, C. L. 2000 Existence and smoothness of the Navier–Stokes equation. In The Millennium Prize Problems, pp. 5767. Clay Mathematics Institute.
Gibbon, J. 2008 The three-dimensional Euler equations: where do we stand? Physica D 237, 18941904.
Gibbon, J. D., Fokas, A. S. & Doering, C. R. 1999 Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations. Physica D 132, 497510.
Gibbon, J. D. & Ohkitani, K. 2001 Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics. Nonlinearity 14 (5), 12391264.
Hou, T. Y. & Li, R. 2006 Dynamic depletion of vortex stretching and non-blowup of the 3D incompressible Euler equations. J. Nonlinear Sci. 16 (6), 639664.
Mulungye, R. M., Lucas, D. & Bustamante, M. D. 2015 Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions. J. Fluid Mech. 771, 468502.
Ohkitani, K. & Gibbon, J. D. 2000 Numerical study of singularity formation in a class of Euler and Navier–Stokes flows. Phys. Fluids 12, 31813194.
Perlin, M. & Bustamante, M. D. 2015 A robust quantitative comparison criterion of two signals based on the Sobolev norm of their difference. J. Engng Math. (in press).
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed