{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"3b88c794268a4066908cc7ec5b43225b1949faa89f0e6578a793f5f87cbf237b","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["54960aeaedc16638e40aaed7a08c54d12b3bbbeb5b124174117517217805a9bc","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"b971731c2d8cf3b8fac3eecaad2e7f1b4ea3002ddad35c1d022a361e6d1534cd","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.fluid-singularities.axisymmetric-swirl-ns","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"Flows that are symmetric around an axis but also spin around it are the leading candidates for viscous blowup. Without spin they are known to stay smooth; with spin the question is open even in this restricted class.","posed_since":"","precise":"3D Navier-Stokes on $R^{3}$ with $f = 0$ and smooth, finite-energy, axisymmetric initial data with nonzero swirl $u_{\\theta}$. Prove global regularity, or give smooth axisymmetric data whose solution has $\\operatorname{limsup}_{t \\to T} \\mid\\mid u(t)\\mid\\mid_{\\infty} = \\infty$. Answer: yes or no with proof; this is a special case of mathph.fluid-singularities.ns-unforced.","problem_ref":"54960aeaedc16638e40aaed7a08c54d12b3bbbeb5b124174117517217805a9bc","references":"","settled_by":"A global regularity proof for all smooth axisymmetric data, or a verified (e.g. computer-assisted) blowup construction in this class without forcing.","status_note":"Hou (arXiv 2107.06509, Foundations of Computational Mathematics) reported numerical evidence of potentially singular axisymmetric flow with maximum vorticity growing by a factor of about $10^{7}$; regularity is proven for zero swirl (Ukhovskii and Yudovich; Ladyzhenskaya, 1968).","title":"Regularity or blowup of axisymmetric Navier-Stokes flows with 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