Hou, Thomas Y. and Zhang, Shumao (2022) Potential Singularity of the Axisymmetric Euler Equations with C^α Initial Vorticity for A Large Range of α. Part I: the 3-Dimensional Case. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20230227-194424192
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Abstract
n Part I of our sequence of 2 papers, we provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with C^α initial vorticity for a large range of α. We employ an adaptive mesh method using a highly effective mesh to resolve the potential singularity sufficiently close to the potential blow-up time. Resolution study shows that our numerical method is at least second-order accurate. Scaling analysis and the dynamic rescaling formulation are presented to quantitatively study the scaling properties of the potential singularity. We demonstrate that this potential blow-up is stable with respect to the perturbation of initial data. Our study shows that the 3D Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α is smaller than some critical value α^∗. By properly rescaling the initial data in the z-axis, this upper bound for potential blow-up α^∗ can asymptotically approach 1/3. Compared with Elgindi's blow-up result in a similar setting [15], our potential blow-up scenario has a different Hölder continuity property in the initial data and the scaling properties of the two initial data are also quite different.
Item Type: | Report or Paper (Discussion Paper) | ||||||
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Additional Information: | The research was in part supported by DMS-2205590. We would like to acknowledge the generous support from Mr. K. C. Choi through the Choi Family Gift Fund and the Choi Family Postdoc Gift Fund. | ||||||
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DOI: | 10.48550/arXiv.2212.11912 | ||||||
Record Number: | CaltechAUTHORS:20230227-194424192 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20230227-194424192 | ||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
ID Code: | 119522 | ||||||
Collection: | CaltechAUTHORS | ||||||
Deposited By: | George Porter | ||||||
Deposited On: | 28 Feb 2023 03:40 | ||||||
Last Modified: | 02 Jun 2023 01:30 |
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