{"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). We develop a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06040","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.06040/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}