{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:KA3L3L5UBKDIGNR26BC7L3DK6X","short_pith_number":"pith:KA3L3L5U","schema_version":"1.0","canonical_sha256":"5036bdafb40a8683363af045f5ec6af5e6c94b9001cdb0f2d312040b5b4d6247","source":{"kind":"arxiv","id":"1209.3718","version":1},"attestation_state":"computed","paper":{"title":"On some Liouville Type Theorems for the Compressible Navier-Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dong Li, Xinwei Yu","submitted_at":"2012-09-17T16:54:49Z","abstract_excerpt":"We prove several Liouville type results for stationary solutions of the $d$-dimensional compressible Navier-Stokes equations. In particular, we show that when the dimension $d \\geqslant 4$, the natural requirements $\\rho \\in L^{\\infty} (\\mathbbm{R}^d)$, $v \\in \\dot{H}^1 (\\mathbbm{R}^d)$ suffice to guarantee that the solution is trivial. For dimensions $d=2,3$, we assume the extra condition $v \\in L^{\\frac{3d}{d-1}}(\\mathbb R^d)$. This improves a recent result of Chae (2012)."},"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":"1209.3718","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-17T16:54:49Z","cross_cats_sorted":[],"title_canon_sha256":"a81c24174f046f898ed394ca3c1c17b27afb05d3193b9d6593e63616d86abe0f","abstract_canon_sha256":"cd36b4aa554ad146b0776f45de27d19cdafc37546185ee1def92cd34c3127012"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:23.646856Z","signature_b64":"U1ljX7xb90YReJKkJ542dsfrjxz9eLDI8Rcau+G/Bkf05W8jz5P8iYFX8lbF5xsESYCxqTMpgAOXlcM2iLrpCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5036bdafb40a8683363af045f5ec6af5e6c94b9001cdb0f2d312040b5b4d6247","last_reissued_at":"2026-05-18T03:45:23.646349Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:23.646349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some Liouville Type Theorems for the Compressible Navier-Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dong Li, Xinwei Yu","submitted_at":"2012-09-17T16:54:49Z","abstract_excerpt":"We prove several Liouville type results for stationary solutions of the $d$-dimensional compressible Navier-Stokes equations. In particular, we show that when the dimension $d \\geqslant 4$, the natural requirements $\\rho \\in L^{\\infty} (\\mathbbm{R}^d)$, $v \\in \\dot{H}^1 (\\mathbbm{R}^d)$ suffice to guarantee that the solution is trivial. For dimensions $d=2,3$, we assume the extra condition $v \\in L^{\\frac{3d}{d-1}}(\\mathbb R^d)$. This improves a recent result of Chae (2012)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3718","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":"1209.3718","created_at":"2026-05-18T03:45:23.646432+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.3718v1","created_at":"2026-05-18T03:45:23.646432+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3718","created_at":"2026-05-18T03:45:23.646432+00:00"},{"alias_kind":"pith_short_12","alias_value":"KA3L3L5UBKDI","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"KA3L3L5UBKDIGNR2","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"KA3L3L5U","created_at":"2026-05-18T12:27:11.947152+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/KA3L3L5UBKDIGNR26BC7L3DK6X","json":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X.json","graph_json":"https://pith.science/api/pith-number/KA3L3L5UBKDIGNR26BC7L3DK6X/graph.json","events_json":"https://pith.science/api/pith-number/KA3L3L5UBKDIGNR26BC7L3DK6X/events.json","paper":"https://pith.science/paper/KA3L3L5U"},"agent_actions":{"view_html":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X","download_json":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X.json","view_paper":"https://pith.science/paper/KA3L3L5U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.3718&json=true","fetch_graph":"https://pith.science/api/pith-number/KA3L3L5UBKDIGNR26BC7L3DK6X/graph.json","fetch_events":"https://pith.science/api/pith-number/KA3L3L5UBKDIGNR26BC7L3DK6X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X/action/storage_attestation","attest_author":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X/action/author_attestation","sign_citation":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X/action/citation_signature","submit_replication":"https://pith.science/pith/KA3L3L5UBKDIGNR26BC7L3DK6X/action/replication_record"}},"created_at":"2026-05-18T03:45:23.646432+00:00","updated_at":"2026-05-18T03:45:23.646432+00:00"}