On the interaction of strain and vorticity for solutions of the Navier--Stokes equation

[2407.02691] On the interaction of strain and vorticity for solutions of the Navier--Stokes equation Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2407.02691 (math) [Submitted on 2 Jul 2024 ( v1 ), last revised 13 Apr 2026 (this version, v2)] Title: On the interaction of strain and vorticity for solutions of the Navier--Stokes equation Authors: Evan Miller View a PDF of the paper titled On the interaction of strain and vorticity for solutions of the Navier--Stokes equation, by Evan Miller View PDF HTML (experimental) Abstract: In this paper, we prove a new identity for divergence free vector fields, showing that \begin{equation*} \left<-\Delta S,\omega\otimes\omega\right>=0, \end{equation*} where $S_{ij}=\frac{1}{2}\left(\partial_iu_j+\partial_ju_i\right)$ is the symmetric part of the velocity gradient, and $\omega=\nabla\times u$ is the vorticity. This identity will allow us to understand the interaction of different aspects of the nonlinearity in the Navier--Stokes equation from the strain and vorticity perspective, particularly as they relate to the depletion of the nonlinearity by advection. We will prove global regularity for the strain-vorticity interaction model equation, a model equation for studying the impact of the vorticity on the evolution of strain which has the same identity for enstrophy growth as the full Navier--Stokes equation. We will also use this identity to obtain several new regularity criteria for the Navier--Stokes equation, one of which will help to clarify the circumstances in which advection can work to deplete the nonlinearity, preventing finite-time blowup. Subjects: Analysis of PDEs (math.AP) MSC classes: 35Q30 Cite as: arXiv:2407.02691 [math.AP] (or arXiv:2407.02691v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.2407.02691 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Pure Appl. Analysis 8 (2026) 247-270 Related DOI : https://doi.org/10.2140/paa.2026.8.247 Focus to learn more DOI(s) linking to related resources Submission history From: Evan Miller [ view email ] [v1] Tue, 2 Jul 2024 22:16:25 UTC (18 KB) [v2] Mon, 13 Apr 2026 19:33:38 UTC (18 KB) Full-text links: Access Paper: View a PDF of the paper titled On the interaction of strain and vorticity for solutions of the Navier--Stokes equation, by Evan Miller View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2024-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

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