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Suppose the solution satisfies either $|v (x,t)| \\le C_*{|t|^{-1/2}} $ or, for some $\\e > 0$, $|v (x,t)| \\le C_* r^{-1+\\epsilon} |t|^{-\\epsilon /2}$ for $-T_0\\le t < 0$ and $0<C_*<\\infty$ allowed to be large. We prove that $v$ is regular at time zero."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0709.4230","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2007-09-26T18:14:12Z","cross_cats_sorted":[],"title_canon_sha256":"d5b0e9fb9960e8360898c5ee76a41d0cbba77bce4a4d034bb18dc47b23740b16","abstract_canon_sha256":"5983b95b72bcfea76e44e2d0db2db293146a0238b0463ed17a70c89f3ad32cde"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T17:24:29.971308Z","signature_b64":"Xa3vVZnZ+PXQOWavHR4YhwF3Esv5qhQSC9z4LkndCYYUVoKYXdluAwbHpuxEhiQUiCl7mZple1RuFtnUx3J4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"010fc3ab9eb8ebc4a5541b5ad9dbdd995272006151f30311198610a06a0cc0f1","last_reissued_at":"2026-07-04T17:24:29.970887Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T17:24:29.970887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chiun-Chuan Chen, Horng-Tzer Yau, Robert M. Strain, Tai-Peng Tsai","submitted_at":"2007-09-26T18:14:12Z","abstract_excerpt":"Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \\le C_*{|t|^{-1/2}} $ or, for some $\\e > 0$, $|v (x,t)| \\le C_* r^{-1+\\epsilon} |t|^{-\\epsilon /2}$ for $-T_0\\le t < 0$ and $0<C_*<\\infty$ allowed to be large. 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