{"paper":{"title":"$\\varepsilon$-regularity criteria in anisotropic Lebesgue spaces and Leray's self-similar solutions to the 3D Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daoguo Zhou, Gang Wu, Yanqing Wang","submitted_at":"2019-01-06T07:11:19Z","abstract_excerpt":"In this paper, we establish some $\\varepsilon$-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier-Stokes equations as follows: $$ \\limsup\\limits_{\\varrho\\rightarrow0}\n  \\varrho^{1-\\frac{2}{p}-\\sum\\limits^{3}_{j=1}\\frac{1}{q_{j}}} \\|u\\|_{L_{t}^{p}L^{\\overrightarrow{q}}_{x}(Q(\\varrho))} \\leq\\varepsilon, ~~\\frac{2}{p}+\\sum\\limits^{3}_{j=1}\\frac{1}{q_{j}} \\leq2~~~~~\\text{with}~q_{j} > 1;\\\\$$$$ \\sup_{-1\\leq t\\leq0}\\|u\\|_{L^{\\overrightarrow{q}}(B(1))} < \\varepsilon,~~\\frac{1}{q_{1}}+\\frac{1}{q_{2}}+\\frac{1}{q_{3}} <2\\quad \\text{with}\\, 1<q_{j}<\\infty;$$ $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01510","kind":"arxiv","version":3},"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"}