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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. (2019) Finite time blowup of 2D Boussinesq and 3D Euler equations with C^(1,α) velocity and boundary. . (Unpublished)

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Inspired by the recent numerical evidence of a potential 3D Euler singularity [28, 29], 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 [28,29] 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 the method of analysis proposed in our recent joint work with Huang in [5] and the simplification of the Biot-Savart law derived by Elgindi in [11] for C^(1,α) velocity to establish the 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:Report or Paper (Discussion Paper)
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Additional Information: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 contribution in proving Lemma 9.1. We are grateful to Dongyi Wei for telling us the estimate of the mixed derivative terms related to Proposition 7.13. 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.
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Record Number:CaltechAUTHORS:20200122-142818756
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100851
Deposited By: Tony Diaz
Deposited On:22 Jan 2020 23:03
Last Modified:22 Jan 2020 23:03

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