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On the Slightly Perturbed De Gregorio Model on S¹

Chen, Jiajie (2021) On the Slightly Perturbed De Gregorio Model on S¹. Archive for Rational Mechanics and Analysis, 241 (3). pp. 1843-1869. ISSN 0003-9527. doi:10.1007/s00205-021-01685-w.

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It is conjectured that the generalization of the Constantin–Lax–Majda model (gCLM) ω_t + auω_x = u_xω, due to Okamoto, Sakajo and Wunsch, can develop a finite time singularity from smooth initial data for a < 1. For the endpoint case where a is close to and less than 1, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection a is slightly larger than 1, we prove that the solution exists globally with ||ω(t)||_(H¹) decaying in a rate of O(t⁻¹) for large time. The transition threshold between two different behaviors is a = 1, which corresponds to the De Gregorio model.

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Chen, Jiajie0000-0002-0194-1975
Alternate Title:On the Slightly Perturbed De Gregorio Model on S1
Additional Information:© 2021 The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature. Received 07 November 2020; Accepted 28 May 2021; Published 09 June 2021. The author would like to thank Thomas Hou for helpful comments on an earlier version of this work. We would also like to thank the referee for the constructive comments on the original manuscript, which improve the quality of our paper. This research was supported in part by Grants DMS-1907977 and DMS-1912654 from the National Science Foundation.
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Issue or Number:3
Record Number:CaltechAUTHORS:20210611-153013840
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Official Citation:Chen, J. On the Slightly Perturbed De Gregorio Model on S1. Arch Rational Mech Anal 241, 1843–1869 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109490
Deposited By: Tony Diaz
Deposited On:11 Jun 2021 22:53
Last Modified:14 Jul 2021 18:03

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