{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QV6YC2UOC3LPTGWEFR3ARIPR7F","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"77536df6f01868a27c946b7344ac62318d71e8e30d9b3e1243bc49a9706b8cd5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T13:49:30Z","title_canon_sha256":"155132d7694528aa4834d656a85f6163e8edb4b69fefbef2aead48f9aefaf50e"},"schema_version":"1.0","source":{"id":"2608.06040","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.06040","created_at":"2026-08-07T00:55:09Z"},{"alias_kind":"arxiv_version","alias_value":"2608.06040v1","created_at":"2026-08-07T00:55:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.06040","created_at":"2026-08-07T00:55:09Z"},{"alias_kind":"pith_short_12","alias_value":"QV6YC2UOC3LP","created_at":"2026-08-07T00:55:09Z"},{"alias_kind":"pith_short_16","alias_value":"QV6YC2UOC3LPTGWE","created_at":"2026-08-07T00:55:09Z"},{"alias_kind":"pith_short_8","alias_value":"QV6YC2UO","created_at":"2026-08-07T00:55:09Z"}],"graph_snapshots":[{"event_id":"sha256:891886685b6631fe010839ae782ff8472afeeabaf37847e59a1225640202c0e5","target":"graph","created_at":"2026-08-07T00:55:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.06040/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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). We develop a ","authors_text":"Guoxu Yang, Wendong Wang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T13:49:30Z","title":"New decay estimates and Liouville type theorems for the 3D axisymmetric stationary Navier-Stokes equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06040","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a822a49a434f953c21675458b21bd40e731c5d9d37cba7b13a175b6d3e035239","target":"record","created_at":"2026-08-07T00:55:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"77536df6f01868a27c946b7344ac62318d71e8e30d9b3e1243bc49a9706b8cd5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T13:49:30Z","title_canon_sha256":"155132d7694528aa4834d656a85f6163e8edb4b69fefbef2aead48f9aefaf50e"},"schema_version":"1.0","source":{"id":"2608.06040","kind":"arxiv","version":1}},"canonical_sha256":"857d816a8e16d6f99ac42c7608a1f1f96b2ea7db8c313add0d113a389bce0618","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"857d816a8e16d6f99ac42c7608a1f1f96b2ea7db8c313add0d113a389bce0618","first_computed_at":"2026-08-07T00:55:09.760196Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-07T00:55:09.760196Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+ohg1rGph3TH4IuymAx+fJqI3C7q7VEl9RkI/HmXVQh16tVG22z9EBpXkV2fTICPCnnb20jk8xrG5+HqK1rCDg==","signature_status":"signed_v1","signed_at":"2026-08-07T00:55:09.761608Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.06040","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a822a49a434f953c21675458b21bd40e731c5d9d37cba7b13a175b6d3e035239","sha256:891886685b6631fe010839ae782ff8472afeeabaf37847e59a1225640202c0e5"],"state_sha256":"f9d01cc46109b213458fd179bac9f40a5faed1a7deb821de06067c594729ea6c"}