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Published February 28, 2023 | Accepted Version
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Potential singularity of the 3D Euler equations in the interior domain


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-Hou in [31, 32], 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-Huang in [20, 21]. 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 \cite{Hou-Huang-2021,Hou-Huang-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.

Additional Information

The research was in part supported by NSF Grants DMS-1907977 and DMS-1912654, 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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Additional details

August 20, 2023
October 25, 2023