{"paper":{"title":"On the integrability properties of Leray-Hopf solutions of the Navier-Stokes equations on $\\mathbb{R}^3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Sauli Lindberg","submitted_at":"2024-12-17T16:31:09Z","abstract_excerpt":"Let $r,s \\in [2,\\infty]$ and consider the Navier-Stokes equations on $\\mathbb{R}^3$. We study the following two questions for suitable $s$-homogeneous Banach spaces $X \\subset \\mathcal{S}'$: does every $u_0 \\in L^2_\\sigma$ have a weak solution that belongs to $L^r(0,\\infty;X)$, and are the $L^r(0,\\infty;X)$ norms of the solutions bounded uniformly in viscosity? We show that if $\\frac{2}{r} + \\frac{3}{s} < \\frac{3}{2}-\\frac{1}{2r}$, then for a Baire generic datum $u_0 \\in L^2_\\sigma$, no weak solution $u^\\nu$ belongs to $L^r(0,\\infty;X)$. If $\\frac{3}{2}-\\frac{1}{2r} \\leq \\frac{2}{r} + \\frac{3}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.13066","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.13066/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"}