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Potential Singularity of the 3D Euler Equations in the Interior Domain

Hou, Thomas Y. (2022) Potential Singularity of the 3D Euler Equations in the Interior Domain. Foundations of Computational Mathematics . ISSN 1615-3375. doi:10.1007/s10208-022-09585-5. (In Press)

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Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo and Hou (111:12968–12973, 2014) and (12:1722–1776, 2014), which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou and Huang in (arXiv:2102.06663, 2021) and (435:133257, 2022). One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in Hou and Huang (arXiv:2102.06663, 2021) and (435:133257, 2022). More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.

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Hou, Thomas Y.0000-0001-6287-1133
Additional Information:The research was in part supported by NSF Grants DMS-1907977, DMS-1912654, DMS-2205590, and the Choi Family Gift Fund. I would like to thank Dr. De Huang for very helpful discussions regarding the design of the adaptive mesh strategy. I would also like to thank Professor Vladimir Sverak, Jiajie Chen, Dr. De Huang, and the referees for their very constructive comments and suggestions, which significantly improves the quality of this paper. Finally, I have benefited a lot from the AIM SQarRE "Towards a 3D Euler singularity".
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Choi Family Gift FundUNSPECIFIED
Record Number:CaltechAUTHORS:20220919-922490500
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117078
Deposited By: Melissa Ray
Deposited On:23 Sep 2022 19:39
Last Modified:27 Feb 2023 19:30

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