Toward the Finite-Time Blowup of the 3D Incompressible Euler Equations: a Numerical Investigation

Guo Luo California Institute of Technology Thomas Y. Hou California Institute of Technology

Numerical Analysis and Scientific Computing mathscidoc:1705.25002

SIAM Multiscale Modeling and Simulation, 12, (4), 1722-1776, 2014
Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 × 10^{12})^2$ near the point of the singularity, we are able to advance the solution up to $\tau_2 = 0.003505$ and predict a singularity time of $t_s ≈ 0.0035056$, while achieving a pointwise relative error of $O(10^{−4})$ in the vorticity vector ω and observing a $(3 × 10^8)$-fold increase in the maximum vorticity $\|ω\|_{\infty}$. The numerical data are checked against all major blowup/non-blowup criteria, including Beale–Kato–Majda, Constantin–Fefferman–Majda, and Deng–Hou–Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.
3D axisymmetric Euler equations, finite-time blowup
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@inproceedings{guo2014toward,
  title={Toward the Finite-Time Blowup of the 3D Incompressible Euler Equations: a Numerical Investigation},
  author={Guo Luo, and Thomas Y. Hou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530161111691754771},
  booktitle={SIAM Multiscale Modeling and Simulation},
  volume={12},
  number={4},
  pages={1722-1776},
  year={2014},
}
Guo Luo, and Thomas Y. Hou. Toward the Finite-Time Blowup of the 3D Incompressible Euler Equations: a Numerical Investigation. 2014. Vol. 12. In SIAM Multiscale Modeling and Simulation. pp.1722-1776. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530161111691754771.
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