Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation
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 x 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) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). 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.
© 2014 by SIAM. Received by the editors April 24, 2014; accepted for publication (in revised form) August 15, 2014; published electronically November 18, 2014. This research was supported in part by NSF FRG grant DMS-1159138 and DOE grant DE-FG02-06ER25727. 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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