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Published September 9, 2014 | public
Journal Article Open

Potentially singular solutions of the 3D axisymmetric Euler equations


The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(t_s − t)^(−2.46), where t_s ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ_2 = 0.003505 at which time the vorticity amplifies by more than (3 × 10^8)-fold and the maximum mesh resolution exceeds (3 × 10^(12))^2. The vorticity vector is observed to maintain four significant digits throughout the computations.

Additional Information

© 2014 National Academy of Sciences. Edited by Alexandre J. Chorin, University of California, Berkeley, CA, and approved July 30, 2014 (received for review March 20, 2014). Published online before print August 25, 2014, doi: 10.1073/pnas.1405238111. We gratefully acknowledge the computing resources provided by the Shared Heterogeneous Cluster (SHC) at Caltech Center for Advanced Computing Research and the Brutus Cluster at Eidgenössische Technische Hochschule Zürich (ETHZ). We 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 us to use his computing resources. We also thank the anonymous referees for their helpful comments. This research was supported in part by National Science Foundation (NSF) Focused Research Group (FRG) Grant DMS-1159138 and Department of Energy Grant DE-FG02-06ER25727. G.L. gratefully acknowledges the travel support provided by NSF FRG Grant DMS-1159133, made available to him by Prof. Alexander Kiselev, for his trip to the 2013 Stanford Summer School, and by the Department of Computing and Mathematical Sciences at Caltech for his trip to the 2013 American Mathematical Society Fall Central Sectional Meeting at Washington University in St. Louis. Author contributions: G.L. and T.Y.H. designed research, performed research, analyzed data, and wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1405238111/-/DCSupplemental.

Attached Files

Published - 12968.full.pdf

Submitted - 1310.0497.pdf

Supplemental Material - pnas.201405238SI.pdf


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August 20, 2023
August 20, 2023