Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations

[2608.17383] Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2608.17383 (math) [Submitted on 18 Aug 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations Authors: Xuanxuan Zhao View PDF HTML (experimental) Abstract: We construct nonzero compactly supported stationary distributional solutions $u\in L^2(\mathbb{R}^3;\mathbb{R}^3)$ of the unforced incompressible Navier--Stokes equations, together with compactly supported pressures in $L^1$, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized $h$-principle: nonzero stationary solutions are strongly dense in the localized $L^p$ classes for every $1\leq p<2$ and in $H^{-1}$, and weakly dense at $p=2$, while strong $L^2$ density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive $L^2$ norms, periodize the construction to $\mathbb{T}^3$, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations. Comments: 33 pages Subjects: Analysis of PDEs (math.AP) MSC classes: 35Q30, 35A02, 35D30, 76D05 Cite as: arXiv:2608.17383 [math.AP] (or arXiv:2608.17383v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.2608.17383 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xuanxuan Zhao [ view email ] [v1] Tue, 18 Aug 2026 05:18:53 UTC (25 KB) [v2] Thu, 27 Aug 2026 23:31:46 UTC (31 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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