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Under a general scaling invariant condition on $b$, we prove that the quantity $\\Gamma = r v^\\theta$ is H\\\"older continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \\cite{KNSS} and Seregin-Sverak in \\cite{SS}. As another application, we prove that if $b \\in L^\\infty([0, T], BMO^{-1})$"},"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":"1011.5066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-11-23T11:38:15Z","cross_cats_sorted":[],"title_canon_sha256":"a6b32f43f4d31659f3d6a3853aad46e2363a1641f11b17ef668774463f357bac","abstract_canon_sha256":"54e4110da71193f6e717dd0146247602d260b0725bdf0be04ebf2cff1e9eadc0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:37.982209Z","signature_b64":"mW9qE9Ul136YBGFZ4fdLixykZqdFoxUC6Lq77DPgWmohV/K3Y3E5bgd4mG7VzqfRkKEzePi0U7lZDV71S34VAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28ec2504584a77813b15c0620f4bd729fe39869b7186a282d799df84ef55edf3","last_reissued_at":"2026-05-18T04:34:37.981547Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:37.981547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qi S. Zhang, Zhen Lei","submitted_at":"2010-11-23T11:38:15Z","abstract_excerpt":"Let $v(x, t)= v^r e_r + v^\\theta e_\\theta + v^z e_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b = v^r e_r + v^z e_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $\\Gamma = r v^\\theta$ is H\\\"older continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \\cite{KNSS} and Seregin-Sverak in \\cite{SS}. 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