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 II: the N-Dimensional Case. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20230227-194427740
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Abstract
In Part II of this sequence to our previous paper for the 3-dimensional Euler equations [8], we investigate potential singularity of the n-diemnsional axisymmetric Euler equations with C^α initial vorticity for a large range of α. We use the adaptive mesh method to solve the n-dimensional axisymmetric Euler equations and use the scaling analysis and dynamic rescaling method to examine the potential blow-up and capture its self-similar profile. Our study shows that the n-dimensional axisymmetric Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α < α^∗, and this upper bound α∗ can asymptotically approach 1 − 2/n. Moreover, we introduce a stretching parameter δ along the z-direction. Based on a few assumptions inspired by our numerical experiments, we obtain α^∗ = 1 − 2/n by studying the limiting case of δ→0. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the n-dimensional Euler equations. This one-dimensional model sheds useful light to our understanding of the blowup mechanism for the n-dimensional Euler equations. As shown in [8], the scaling behavior and regularity properties of our initial data are quite different from those of the initial data considered by Elgindi in [6].
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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Record Number: | CaltechAUTHORS:20230227-194427740 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20230227-194427740 | ||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
ID Code: | 119523 | ||||||
Collection: | CaltechAUTHORS | ||||||
Deposited By: | George Porter | ||||||
Deposited On: | 28 Feb 2023 15:01 | ||||||
Last Modified: | 28 Feb 2023 15:01 |
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