Toward the finite-time blowup of the 3D axisymmetric Euler equations: a numerical investigation. (English) Zbl 1316.35235
Summary: 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 \((3\times 10^{12})^2\) 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 \(t_s\approx 0.0035056\), while achieving a pointwise relative error of \(O(10^{-4})\) in the vorticity vector \(\omega\) and observing a \((3\times 10^8)\)-fold increase in the maximum vorticity \(\|\omega\|_\infty\). 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.
MSC:
| 35Q31 | Euler equations |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 35B44 | Blow-up in context of PDEs |
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