TY - JOUR
T1 - Toward the finite-time blowup of the 3D axisymmetric Euler equations
T2 - A numerical investigation
AU - Luo, Guo
AU - Hou, Thomas Y.
PY - 2014
Y1 - 2014
N2 - 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 × 1012)2 near the point of the singularity, we are able to advance the solution up to τ2 = 0.003505 and predict a singularity time of ts ≈ 0.0035056, while achieving a pointwise relative error of O(10-4) in the vorticity vector ω and observing a (3 × 108)-fold increase in the maximum vorticity ||ω||∞. 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.
AB - 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 × 1012)2 near the point of the singularity, we are able to advance the solution up to τ2 = 0.003505 and predict a singularity time of ts ≈ 0.0035056, while achieving a pointwise relative error of O(10-4) in the vorticity vector ω and observing a (3 × 108)-fold increase in the maximum vorticity ||ω||∞. 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.
KW - 3D axisymmetric Euler equations
KW - Finite-time blowup
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U2 - 10.1137/140966411
DO - 10.1137/140966411
M3 - 21_Publication in refereed journal
VL - 12
SP - 1722
EP - 1776
JO - Multiscale Modeling and Simulation
JF - Multiscale Modeling and Simulation
SN - 1540-3459
IS - 4
ER -