scholarly journals On the Finite-Time Blowup of a One-Dimensional Model for the Three-Dimensional Axisymmetric Euler Equations

2017 ◽  
Vol 70 (11) ◽  
pp. 2218-2243 ◽  
Author(s):  
Kyudong Choi ◽  
Thomas Y. Hou ◽  
Alexander Kiselev ◽  
Guo Luo ◽  
Vladimir Sverak ◽  
...  
Keyword(s):  
Euler Equations ◽  
Finite Time ◽  
Three Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Finite Time Blowup ◽  
Time Blowup

scholarly journals Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions

2015 ◽  
Vol 771 ◽  
pp. 468-502 ◽  
Author(s):  
Rachel M. Mulungye ◽  
Dan Lucas ◽  
Miguel D. Bustamante
Keyword(s):  
Numerical Solution ◽  
Euler Equations ◽  
Finite Time ◽  
Symmetry Plane ◽  
Original Model ◽  
Finite Time Blowup ◽  
Fluid Equations ◽  
Model Equations ◽  
Singularity Time ◽  
Time Blowup

Motivated by the work on stagnation-point-type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. (Physica D, vol. 132 (4), 1999, pp. 497–510) and the subsequent demonstration of finite-time blowup by Constantin (Int. Math. Res. Not. IMRN, vol. 9, 2000, pp. 455–465) we introduce a one-parameter family of models of the 3D Euler fluid equations on a 2D symmetry plane. Our models are seen as a deformation of the 3D Euler equations which respects the variational structure of the original equations so that explicit solutions can be found for the supremum norms of the basic fields: vorticity and stretching rate of vorticity. In particular, the value of the model’s parameter determines whether or not there is finite-time blowup, and the singularity time can be computed explicitly in terms of the initial conditions and the model’s parameter. We use a representative of this family of models, whose solution blows up at a finite time, as a benchmark for the systematic study of errors in numerical simulations. Using a high-order pseudospectral method, we compare the numerical integration of our ‘original’ model equations against a ‘mapped’ version of these equations. The mapped version is a globally regular (in time) system of equations, obtained via a bijective nonlinear mapping of time and fields from the original model equations. The mapping can be constructed explicitly whenever a Beale–Kato–Majda type of theorem is available therefore it is applicable to the 3D Euler equations (Bustamante, Physica D, vol. 240 (13), 2011, pp. 1092–1099). We show that the mapped system’s numerical solution leads to more accurate (by three orders of magnitude) estimates of supremum norms and singularity time compared with the original system. The numerical integration of the mapped equations is demonstrated to entail only a small extra computational cost. We study the Fourier spectrum of the model’s numerical solution and find that the analyticity strip width (a measure of the solution’s analyticity) tends to zero as a power law in a finite time. This is in agreement with the finite-time blowup of the fields’ supremum norms, in the light of rigorous bounds stemming from the bridge (Bustamante & Brachet, Phys. Rev. E, vol. 86 (6), 2012, 066302) between the analyticity-strip method and the Beale–Kato–Majda type of theorems. We conclude by discussing the implications of this research on the analysis of numerical solutions to the 3D Euler fluid equations.

scholarly journals On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations

2021 ◽  
Vol 74 (6) ◽  
pp. 1282-1350
Author(s):  
Jiajie Chen ◽  
Thomas Y. Hou ◽  
De Huang
Keyword(s):  
Euler Equations ◽  
Finite Time ◽  
Finite Time Blowup ◽  
Time Blowup
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