{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:6GNEYFGRTVBXP6H5RNV5CBXDZQ","short_pith_number":"pith:6GNEYFGR","schema_version":"1.0","canonical_sha256":"f19a4c14d19d4377f8fd8b6bd106e3cc27084db7089b703cc251c8530f4529cf","source":{"kind":"arxiv","id":"1410.5929","version":1},"attestation_state":"computed","paper":{"title":"Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Winkler","submitted_at":"2014-10-22T07:39:15Z","abstract_excerpt":"The chemotaxis-Navier-Stokes system linking the chemotaxis equations \\[ n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\chi(c)\\nabla c) \\] and \\[ c_t + u\\cdot\\nabla c = \\Delta c-nf(c) \\] to the incompressible Navier-Stokes equations, \\[ u_t + (u\\cdot\\nabla)u = \\Delta u +\\nabla P + n \\nabla \\Phi, \\qquad \\nabla \\cdot u = 0, \\] is considered under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$, in a bounded convex domain $\\Omega\\subset R^3$ with smooth boundary, where $\\Phi\\in W^{1,\\infty}(\\Omega)$, and where $f\\in C^1([0,\\infty))$ and $\\chi\\in C^"},"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":"1410.5929","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-22T07:39:15Z","cross_cats_sorted":[],"title_canon_sha256":"403b430fb821f5303892f861424943acb707d2d44c717da41c0964027f985d3a","abstract_canon_sha256":"756548ffddf9be80d8053825ebdd1b52f416d23e7aa41656880a54441db83919"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:41.183638Z","signature_b64":"GAdJKzVAyWNyWnm/2dE9S154Wb14EJk+Pz3rxMFVo1iqm1vpShLbN1blAw8GlFjutxefVfNjCVUsg/IjzQ9VCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f19a4c14d19d4377f8fd8b6bd106e3cc27084db7089b703cc251c8530f4529cf","last_reissued_at":"2026-05-18T01:41:41.183033Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:41.183033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Winkler","submitted_at":"2014-10-22T07:39:15Z","abstract_excerpt":"The chemotaxis-Navier-Stokes system linking the chemotaxis equations \\[ n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\chi(c)\\nabla c) \\] and \\[ c_t + u\\cdot\\nabla c = \\Delta c-nf(c) \\] to the incompressible Navier-Stokes equations, \\[ u_t + (u\\cdot\\nabla)u = \\Delta u +\\nabla P + n \\nabla \\Phi, \\qquad \\nabla \\cdot u = 0, \\] is considered under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$, in a bounded convex domain $\\Omega\\subset R^3$ with smooth boundary, where $\\Phi\\in W^{1,\\infty}(\\Omega)$, and where $f\\in C^1([0,\\infty))$ and $\\chi\\in C^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5929","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1410.5929","created_at":"2026-05-18T01:41:41.183120+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.5929v1","created_at":"2026-05-18T01:41:41.183120+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5929","created_at":"2026-05-18T01:41:41.183120+00:00"},{"alias_kind":"pith_short_12","alias_value":"6GNEYFGRTVBX","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_16","alias_value":"6GNEYFGRTVBXP6H5","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_8","alias_value":"6GNEYFGR","created_at":"2026-05-18T12:28:16.859392+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ","json":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ.json","graph_json":"https://pith.science/api/pith-number/6GNEYFGRTVBXP6H5RNV5CBXDZQ/graph.json","events_json":"https://pith.science/api/pith-number/6GNEYFGRTVBXP6H5RNV5CBXDZQ/events.json","paper":"https://pith.science/paper/6GNEYFGR"},"agent_actions":{"view_html":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ","download_json":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ.json","view_paper":"https://pith.science/paper/6GNEYFGR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.5929&json=true","fetch_graph":"https://pith.science/api/pith-number/6GNEYFGRTVBXP6H5RNV5CBXDZQ/graph.json","fetch_events":"https://pith.science/api/pith-number/6GNEYFGRTVBXP6H5RNV5CBXDZQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ/action/storage_attestation","attest_author":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ/action/author_attestation","sign_citation":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ/action/citation_signature","submit_replication":"https://pith.science/pith/6GNEYFGRTVBXP6H5RNV5CBXDZQ/action/replication_record"}},"created_at":"2026-05-18T01:41:41.183120+00:00","updated_at":"2026-05-18T01:41:41.183120+00:00"}