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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."},"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":"math/0701796","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-01-27T19:55:00Z","cross_cats_sorted":[],"title_canon_sha256":"8ce69f09fb04d93e6b39d4b9272ebfc6db7ea9c95da5bac7b2df16cd85c4aac4","abstract_canon_sha256":"123dca31775c51f547c43aeca33eafb9591f080c2a322892f9d92e64c4bb5c26"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T17:24:32.267793Z","signature_b64":"TN749xgQ56oRUsssubDzG33fZH4cb0P5TIhVhqu+Djt88SWQ0q31emYFGkTKOf1Xf5p6lxDNAwall5ga+WbmAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31b8652c467075fcb40e9dc91c0d26257d97a3996f76a2245888d547997c6fd8","last_reissued_at":"2026-07-04T17:24:32.267375Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T17:24:32.267375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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