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

[2210.07191] Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2210.07191 (math) [Submitted on 13 Oct 2022 ( v1 ), last revised 16 Aug 2026 (this version, v4)] Title: Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis Authors: Jiajie Chen , Thomas Y. Hou View a PDF of the paper titled Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis, by Jiajie Chen and 1 other authors View PDF HTML (experimental) Abstract: Inspired by numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite-time, nearly self-similar blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. The proof encounters several essential difficulties. 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 when using weighted $L^2$ or $H^k$ estimates. We overcome these difficulties by combining weighted $L^\infty$ estimates and weighted $C^{1/2}$ estimates, and by developing 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 and a finite-rank operator. We design the leading order operator to obtain sharp stability estimates. The contribution from the finite-rank operator to linear stability is estimated by constructing approximate space-time solutions. These ingredients enable us to establish the nonlinear stability of the approximate self-similar profile and to prove stable nearly self-similar blowup. Comments: Revised Sec 2.3 on proof outline. Added discussion of ideas for coercivity estimates in Sec 2.7. Added Sec 2.8 summarizing analytic low-rank correction in Part II, Sec 4.4 on qualitative regularity of perturbation, outline of energy estimates in Sec 5.1, and App C.4 on ideas for rigorous weighted estimates in Part II. Added explanatory discussion and minor edits. 159 pages. Supplement: 25 pages Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:2210.07191 [math.AP] (or arXiv:2210.07191v4 [math.AP] for this version) https://doi.org/10.48550/arXiv.2210.07191 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jiajie Chen [ view email ] [v1] Thu, 13 Oct 2022 17:13:08 UTC (2,951 KB) [v2] Wed, 19 Oct 2022 17:54:53 UTC (2,980 KB) [v3] Tue, 9 May 2023 17:50:13 UTC (3,497 KB) [v4] Sun, 16 Aug 2026 08:46:30 UTC (6,107 KB) Full-text links: Access Paper: View a PDF of the paper titled Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis, by Jiajie Chen and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2022-10 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar 1 blog link ( what is this? ) export BibTeX citation Loading... BibTeX formatted citation × loading... 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