A Caltech Library Service

Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data

Chen, Jiajie and Hou, Thomas Y. (2022) Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data. . (Unpublished)

[img] PDF - Submitted Version
See Usage Policy.


Use this Persistent URL to link to this item:


Inspired by the numerical evidence of a potential 3D Euler singularity [Luo-Hou-14a, Luo-Hou-14b], we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. There are several essential difficulties in proving finite time blowup of 3D Euler with smooth initial data. One of the essential difficulties is to control a number of nonlocal terms that do not seem to offer any damping effect. Another essential difficulty is that the strong advection normal to the boundary introduces a large growth factor for the perturbation if we use weighted L² estimates. We overcome this difficulty by using a combination of a weighted L∞ norm and a weighted C^(1/2) norm, and develop sharp functional inequalities using the symmetry properties of the kernels and some techniques from optimal transport. Moreover we decompose the linearized operator into a leading order operator plus a finite rank operator. The leading order operator is designed in such a way that we can obtain sharp stability estimates. The contribution from the finite rank operator can be captured by an auxiliary variable and its contribution to linear stability can be estimated by constructing approximate solution in space-time. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper InCaltech News
Chen, Jiajie0000-0002-0194-1975
Hou, Thomas Y.0000-0001-6287-1133
Alternate Title:Stable nearly self-similar blowup of the 2D Boussinesq equations with smooth data
Additional Information:The research was in part supported by NSF Grants DMS-1907977 and 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. We would also like to thank Drs. Pengfei Liu and De Huang for a number of stimulating discussions in the early stage of this project, and Dr. Tarek Elgindi for his suggestion to include the 3D Euler blowup result in this paper. Part of the computation in this paper was performed using the Caltech IMSS High Performance Computing Service. The support from its staff is greatly appreciated.
Funding AgencyGrant Number
Choi Family Gift FundUNSPECIFIED
Record Number:CaltechAUTHORS:20230227-191437852
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:119517
Deposited By: George Porter
Deposited On:28 Feb 2023 03:09
Last Modified:28 Feb 2023 03:09

Repository Staff Only: item control page