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Assuming $m>\\frac{4}{3}$ and sufficiently regular nonnegative initial d"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1712.00262","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-01T10:22:52Z","cross_cats_sorted":[],"title_canon_sha256":"462fe9e2327799dbee2bbffb5da9f6dca1bd963b1cf4910f0f512a8627afcd48","abstract_canon_sha256":"12c5fcc58cfbe1550f0ea2091f4a98768da59f148cf217f4a9b336e139dee886"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:00.296045Z","signature_b64":"+GdZMEjLOF4Fn/1cp5yAQAkIwFKCln2jYMLPzj5tBxQQWGj1hXAjK/hG0Z/1ceAYOKs5D/s7Wky39zxXcEwCAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d243bcb9e8eddfc099712db27c46abbc15cc104c6390cfbba1c8d37d0d310387","last_reissued_at":"2026-05-18T00:09:00.295345Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:00.295345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tobias Black","submitted_at":"2017-12-01T10:22:52Z","abstract_excerpt":"We consider the chemotaxis-fluid system \\begin{align}\\label{star}\\tag{$\\diamondsuit$} \\left\\{ \\begin{array}{r@{\\,}c@{\\,}c@{\\ }l@{\\quad}l@{\\quad}l@{\\,}c} n_{t}&+&u\\cdot\\!\\nabla n&=\\Delta n^m-\\nabla\\!\\cdot(n\\nabla c),\\ &x\\in\\Omega,& t>0,\\\\ c_{t}&+&u\\cdot\\!\\nabla c&=\\Delta c-c+n,\\ &x\\in\\Omega,& t>0,\\\\ u_{t}&+&(u\\cdot\\nabla)u&=\\Delta u+\\nabla P+n\\nabla\\phi,\\ &x\\in\\Omega,& t>0,\\\\ &&\\nabla\\cdot u&=0,\\ &x\\in\\Omega,& t>0, \\end{array}\\right. \\end{align} in a bounded domain $\\Omega\\subset\\mathbb{R}^3$ with smooth boundary and $m>1$. 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