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Potential Singularity of the Axisymmetric Euler Equations with C^α Initial Vorticity for A Large Range of α. Part II: the N-Dimensional Case

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 (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20230227-194427740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)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2212.11924arXivDiscussion Paper
ORCID:
AuthorORCID
Hou, Thomas Y.0000-0001-6287-1133
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.
Funders:
Funding AgencyGrant Number
NSFDMS-2205590
Choi Family Gift FundUNSPECIFIED
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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