{"paper":{"title":"Logarithmical Blow-up Criteria for the Nematic Liquid Crystal Flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jihong Zhao, Qiao Liu","submitted_at":"2012-09-25T14:36:32Z","abstract_excerpt":"We investigate the blow-up criterion for the local in time classical solution of the nematic liquid crystal flows in dimension two and three. More precisely, $0<T_{*}<+\\infty$ is the maximal time interval if and only if (i) for $n=3$, {align*} \\int_{0}^{T_{*}}\\frac{\\|\\omega\\|_{\\dot{B}^{0}_{\\infty,\\infty}}+\\|\\nabla d\\|_{\\dot{B}^{0}_{\\infty,\\infty}}^{2}}{\\sqrt{1+\\text{ln}(e+\\|\\omega\\|_{\\dot{B}^{0}_{\\infty,\\infty}} +\\|\\nabla d\\|_{\\dot{B}^{0}_{\\infty,\\infty}})}}\\text{d}t=\\infty, {align*} or {align*} \\int_{0}^{T_{*}}\\frac{\\|\\nabla u\\|_{\\dot{B}^{-1}_{\\infty,\\infty}}^{2}+\\|\\nabla d\\|_{\\dot{B}^{0}_{\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5623","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"}