{"paper":{"title":"Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chiun-Chuan Chen, Horng-Tzer Yau, Robert M. Strain, Tai-Peng Tsai","submitted_at":"2007-01-27T19:55:00Z","abstract_excerpt":"Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \\cite{MR673830} that they could only blow up on the axis of symmetry.\n  Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \\le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\\le t < 0$ and $0<C_*<\\infty$ allowed to be large, we then prove that $v$ is regular at time zero."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701796","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0701796/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}