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Using a new pointwise Calder\\'on--Zygmund estimate adapted to cylindrical geometry, we improve the decay estimates of Carrillo--Pan--Zhang (2020, JFA) and prove\n  \\[\n  |\\nabla u_r|+|\\nabla u_z|\n  \\lesssim r^{-5/4}[\\log(\\mathrm e+r)]^{5/4},\n  \\quad\n  |\\omega_r|+|\\omega_z|\n  \\lesssim r^{-9/8}[\\log(\\mathrm e+r)]^{9/8}, \\quad r\\gg1.\n  \\]\n  (ii). We develop a "},"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":"2608.06040","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T13:49:30Z","cross_cats_sorted":[],"title_canon_sha256":"155132d7694528aa4834d656a85f6163e8edb4b69fefbef2aead48f9aefaf50e","abstract_canon_sha256":"77536df6f01868a27c946b7344ac62318d71e8e30d9b3e1243bc49a9706b8cd5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-07T00:55:09.761608Z","signature_b64":"+ohg1rGph3TH4IuymAx+fJqI3C7q7VEl9RkI/HmXVQh16tVG22z9EBpXkV2fTICPCnnb20jk8xrG5+HqK1rCDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"857d816a8e16d6f99ac42c7608a1f1f96b2ea7db8c313add0d113a389bce0618","last_reissued_at":"2026-08-07T00:55:09.760196Z","signature_status":"signed_v1","first_computed_at":"2026-08-07T00:55:09.760196Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New decay estimates and Liouville type theorems for the 3D axisymmetric stationary Navier-Stokes equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guoxu Yang, Wendong Wang","submitted_at":"2026-08-06T13:49:30Z","abstract_excerpt":"The Liouville problem for the three-dimensional stationary Navier--Stokes equations remains open, even for axisymmetric \\(D\\)-solutions. In this paper, we obtain two results based on decay in the cylindrical radial variable \\(r=|x'|\\).\n  (i). Using a new pointwise Calder\\'on--Zygmund estimate adapted to cylindrical geometry, we improve the decay estimates of Carrillo--Pan--Zhang (2020, JFA) and prove\n  \\[\n  |\\nabla u_r|+|\\nabla u_z|\n  \\lesssim r^{-5/4}[\\log(\\mathrm e+r)]^{5/4},\n  \\quad\n  |\\omega_r|+|\\omega_z|\n  \\lesssim r^{-9/8}[\\log(\\mathrm e+r)]^{9/8}, \\quad r\\gg1.\n  \\]\n  (ii). 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