Formation of Finite-Time Singularities in the 3D Axisymmetric Euler Equations: A Numerics Guided Study
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, by first describing a class of potentially singular solutions to the Euler equations numerically discovered in axisymmetric geometries, and then by presenting evidence from rigorous analysis that strongly supports the existence of such singular solutions. The initial data leading to these singular solutions possess certain special symmetry and monotonicity properties, and the subsequent flows are assumed to satisfy a periodic boundary condition along the axial direction and a no-flow, free-slip boundary condition on the solid wall. The numerical study employs a hybrid 6th-order Galerkin/finite difference discretization of the governing equations in space and a 4th-order Runge--Kutta discretization in time, where the emerging singularity is captured on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10¹²)² near the point of the singularity, the simulations are able to advance the solution to a point that is asymptotically close to the predicted singularity time, while achieving a pointwise relative error of O(10⁻⁴) in the vorticity vector and obtaining a 3 x 10⁸-fold increase in the maximum vorticity. The numerical data are checked against all major blowup/nonblowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A close scrutiny of the data near the point of the singularity also reveals a self-similar structure in the blowup, as well as a one-dimensional model which is seen to capture the essential features of the singular solutions along the solid wall, and for which existence of finite-time singularities can be established rigorously.
© 2019 Society for Industrial and Applied Mathematics. Published electronically November 6, 2019. This paper originally appeared in Multiscale Modeling and Simulation, Volume 12, Number 4, 2014, pages 1722–1776, under the title "Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation." https://doi.org/10.1137/19M1288061. This research was supported in part by NSF FRG grant DMS-1159138 and NSF grants DMS-1613861, DMS-1907977, and DMS-1912654. The authors would like to gratefully acknowledge the computing resources provided by the SHC cluster at Caltech Center for Advanced Computing Research (CACR) and the Brutus cluster at ETH Zürich (ETHZ). The authors gratefully acknowledge the excellent support provided by the staff members at SHC, especially Sharon Brunett, and the support provided by Prof. Petros Koumoutsakos at ETHZ, who kindly allowed them to use his computing resources. The authors also thank the anonymous referees for their helpful comments. The first author gratefully acknowledges the travel support provided by NSF FRG grant DMS-1159133, made available to him by Prof. Alexander Kiselev, for his trip to 2013 Stanford summer school, and by the Department of Computing and Mathematical Sciences at Caltech, for his trip to 2013 AMS Fall central sectional meeting at Washington University in St. Louis.
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