{"paper":{"title":"Criticality of 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":"2015-05-11T14:06:47Z","abstract_excerpt":"Smooth solutions to the axi-symmetric Navier-Stokes equations obey the following maximum principle: $$\\sup_{t\\geq 0}\\|rv^\\theta(t, \\cdot)\\|_{L^\\infty} \\leq \\|rv^\\theta(0, \\cdot)\\|_{L^\\infty}.$$ We prove that all solutions with initial data in $H^{\\frac{1}{2}}$ is smooth globally in time if $rv^\\theta$ satisfies a kind of Form Boundedness Condition (FBC) which is invariant under the natural scaling of the Navier-Stokes equations. In particular, if $rv^\\theta$ satisfies \\begin{equation}\\nonumber \\sup_{t \\geq 0}|rv^\\theta(t, r, z)| \\leq C_\\ast|\\ln r|^{- 2},\\ \\ r \\leq \\delta_0 \\in (0, \\frac{1}{2})"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.02628","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}