{"paper":{"title":"On the Critical One Components Regularity for the $3-D$ Navier-Stokes System in $L^p_T(\\dot{B}^{\\frac 1 2+\\frac 2 p}_{2,\\infty})$ spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Pei Zheng, Wei Luo, Zhanyang Yin","submitted_at":"2026-07-02T01:39:26Z","abstract_excerpt":"We consider the conditional regularity of the mild solution $v$ of the $3-D$ incompressible Navier-Stokes equations with initial data $v_0\\in \\dot{H}^{\\frac 1 2}$ and vorticity $\\Omega_0\\in L^{r_0}$ for some $r_0\\in (1,2)$. We prove that if the solution associated with initial data $v_0$ blows up at a finite time $T^\\ast$, then for any $2<p<\\infty$, and any unit vectors $e$ in $\\mathbb{R}^3$, the integral\n  $$\\int_0^{T^\\ast}\\left\\Vert (v(t)|e)_{\\mathbb{R}^3}\\right\\Vert_{\\dot{B}^{\\frac 1 2+\\frac 2 p}_{2,\\infty}}^p{\\rm d}t$$ blows up at $T^\\ast$. The conclusion improves the recent results in Che"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01587","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/2607.01587/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"}