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This result improves the one in Zhen Lei and Qi S. Zhang \\cite{1}. As a corollary, we also prove the global regularity under the assumption that $|ru_\\theta(r,z,t)|\\leq\\ |\\ln r|^{-3/2},\\ \\ \\forall\\ 0<r\\leq\\delta_0\\in"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.03318","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-13T19:38:13Z","cross_cats_sorted":[],"title_canon_sha256":"ab8d8f5a2757712e0780f179bd51720c552a46ac447b29402036409e38abca91","abstract_canon_sha256":"69c0f1f63c4697488b07ce1567efcb2483af2af8d16619c4cbb8de6c719c7853"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:21.108039Z","signature_b64":"nGEKpJt9NFeuWimyZEaEQDB8QuUeliwoM/BJLW4H8+WMh9e2hkmgqqjQgl6dh3V+vRBQ3Xx30GS0w5eEc/WEDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd40878c2da3a7b8f3f07dbfebe3c8e5ab15222af24a1b217780218c7ebb18a5","last_reissued_at":"2026-05-18T01:35:21.107297Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:21.107297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity Criterion to the axially symmetric Navier-Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongyi Wei","submitted_at":"2015-08-13T19:38:13Z","abstract_excerpt":"Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\\|ru_\\theta(r,z,t)\\|_{L^\\infty}\\leq\\|ru_\\theta(r,z,0)\\|_{L^\\infty}.$ We first prove the global regularity of solutions if $\\|ru_\\theta(r,z,0)\\|_{L^\\infty}$ or $ \\|ru_\\theta(r,z,t)\\|_{L^\\infty(r\\leq r_0)}$ is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang \\cite{1}. 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