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Potentially Singular Solutions of the 3D Incompressible Euler Equations

Luo, Guo and Hou, Thomas Y. (2013) Potentially Singular Solutions of the 3D Incompressible Euler Equations. . (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20160315-140244755

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Abstract

Whether the 3D 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 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 τ_2 = 0.003505 and predict a singularity time of τ_s ≈ 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector ω and observing a (3 x 10^8)-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.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1310.0497arXivDiscussion Paper
Additional Information:December 10, 2013. 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 us to use his computing resources. This research was supported in part by an NSF FRG Grant DMS-1159138 and a DOE Grant DE-FG02-06ER25727.
Funders:
Funding AgencyGrant Number
NSFDMS-1159138
Department of Energy (DOE)DE-FG02-06ER25727
Record Number:CaltechAUTHORS:20160315-140244755
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20160315-140244755
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:65371
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:15 Mar 2016 22:55
Last Modified:15 Mar 2016 22:55

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