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Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with C^(1,α) Velocity and Boundary

Chen, Jiajie and Hou, Thomas Y. (2021) Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with C^(1,α) Velocity and Boundary. Communications in Mathematical Physics, 383 (3). pp. 1559-1667. ISSN 0010-3616. doi:10.1007/s00220-021-04067-1. https://resolver.caltech.edu/CaltechAUTHORS:20200122-142818756

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Abstract

Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30, 31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with C^(1,α) initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with C^(1,α) initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30, 31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use C^(1,α) initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with C^(1,α) initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00220-021-04067-1DOIArticle
https://rdcu.be/cibQKPublisherFree ReadCube access
https://arxiv.org/abs/1910.00173arXivDiscussion Paper
ORCID:
AuthorORCID
Chen, Jiajie0000-0002-0194-1975
Alternate Title:Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with C1,α Velocity and Boundary
Additional Information:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021. Received 19 November 2019; Accepted 12 March 2021; Published 02 April 2021. The research was in part supported by NSF Grants DMS-1613861, DMS-1907977 and DMS-1912654. We would like to thank De Huang for his stimulating discussion on and contribution to Lemma 9.1. We are grateful to Dongyi Wei for telling us the estimate of the mixed derivative terms related to Proposition 7.12. We would also like to thank Tarek Elgindi, Dongyi Wei and Zhifei Zhang for their valuable comments and suggestions on our earlier version of the manuscript. We are also grateful to the two referees for their constructive comments on the original manuscript, which improve the quality of our paper.
Funders:
Funding AgencyGrant Number
NSFDMS-1613861
NSFDMS-1907977
NSFDMS-1912654
Issue or Number:3
DOI:10.1007/s00220-021-04067-1
Record Number:CaltechAUTHORS:20200122-142818756
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200122-142818756
Official Citation:Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with C^(1,α) Velocity and Boundary. Commun. Math. Phys. 383, 1559–1667 (2021). https://doi.org/10.1007/s00220-021-04067-1
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100851
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:22 Jan 2020 23:03
Last Modified:16 Apr 2021 19:34

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