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The condition which vorticity fields need to satisfy is\n  $\\lim\\limits_{|x|\\rightarrow +\\infty}|w(x,t)|=0$ and\n  $\\lim\\limits_{r\\rig"},"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":"1701.00868","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-01-03T23:38:09Z","cross_cats_sorted":[],"title_canon_sha256":"47f0c3ccbf7381606fe9563fcc47a499235b24aaa5100b450e2bb9d1d4c51ef3","abstract_canon_sha256":"4a8aacb5c1759cbed654b93b0cf3fa41990a649419d3d883bdb4c8bdc8373e14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:22.021622Z","signature_b64":"95bMUSSONqpHp331GsqfOOfGCUe5XNrQFPNkdGF4sbvHtVIK24uR1BWFE+OiPR3JksuJpgjA9u0LkCJNTEpABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"acaed3f1eb3750f04815215a425d27b7e1139577eebe9c50db62ebb495437a3c","last_reissued_at":"2026-05-18T00:53:22.021137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:22.021137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved Liouville theorems for axially symmetric Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Na Zhao, Qi S Zhang, Zhen Lei","submitted_at":"2017-01-03T23:38:09Z","abstract_excerpt":"In this paper, we consider the Liouville property for ancient solutions of the incompressible Navier-Stokes equations. 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