{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:POM3S4NAM6HFNUEUL2J5Y3T4RL","short_pith_number":"pith:POM3S4NA","schema_version":"1.0","canonical_sha256":"7b99b971a0678e56d0945e93dc6e7c8ac7e0a01cd7bb4e969578b00fd5687604","source":{"kind":"arxiv","id":"2404.12858","version":1},"attestation_state":"computed","paper":{"title":"Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Violini, Stefan \\v{S}kondri\\'c, Timoth\\'ee Crin-Barat","submitted_at":"2024-04-19T12:54:17Z","abstract_excerpt":"We present a weak-strong uniqueness result for the inhomogeneous Navier-Stokes (INS) equations in $\\mathbb{R}^d$ ($d=2,3$) for bounded initial densities that are far from vacuum. Given a strong solution within the class employed in Paicu, Zhang and Zhang (2013) and Chen, Zhang and Zhao (2016), and a Leray-Hopf weak solution, we establish that they coincide if the initial data agree. The strategy of our proof is based on the relative energy method and new $W^{-1,p}$-type stability estimates for the density. A key point lies in proving that every Leray-Hopf weak solution originating from initial"},"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":"2404.12858","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-04-19T12:54:17Z","cross_cats_sorted":[],"title_canon_sha256":"890ce5d8ce734fa14464b96bf75df93bac53b845a0898c2c90b06bc6924809a4","abstract_canon_sha256":"d6ac9fecc7fa22c0897eff935444e9d53ab80d31b52dc96f0bc01bdb40c52681"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:09:59.812234Z","signature_b64":"RfBCSRn1Ochh/ZERyJR5oNnQrzN7h9cw/liKje1UaPJdbqncOdQtz1A9RHivTw1rFU1b13ldDbbUm7PnJyKnDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b99b971a0678e56d0945e93dc6e7c8ac7e0a01cd7bb4e969578b00fd5687604","last_reissued_at":"2026-07-05T08:09:59.811860Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:09:59.811860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Violini, Stefan \\v{S}kondri\\'c, Timoth\\'ee Crin-Barat","submitted_at":"2024-04-19T12:54:17Z","abstract_excerpt":"We present a weak-strong uniqueness result for the inhomogeneous Navier-Stokes (INS) equations in $\\mathbb{R}^d$ ($d=2,3$) for bounded initial densities that are far from vacuum. Given a strong solution within the class employed in Paicu, Zhang and Zhang (2013) and Chen, Zhang and Zhao (2016), and a Leray-Hopf weak solution, we establish that they coincide if the initial data agree. The strategy of our proof is based on the relative energy method and new $W^{-1,p}$-type stability estimates for the density. A key point lies in proving that every Leray-Hopf weak solution originating from initial"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.12858","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.12858/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2404.12858","created_at":"2026-07-05T08:09:59.811917+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.12858v1","created_at":"2026-07-05T08:09:59.811917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.12858","created_at":"2026-07-05T08:09:59.811917+00:00"},{"alias_kind":"pith_short_12","alias_value":"POM3S4NAM6HF","created_at":"2026-07-05T08:09:59.811917+00:00"},{"alias_kind":"pith_short_16","alias_value":"POM3S4NAM6HFNUEU","created_at":"2026-07-05T08:09:59.811917+00:00"},{"alias_kind":"pith_short_8","alias_value":"POM3S4NA","created_at":"2026-07-05T08:09:59.811917+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.00390","citing_title":"Global well-posedness and self-similar solution of the inhomogeneous Navier-Stokes system","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL","json":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL.json","graph_json":"https://pith.science/api/pith-number/POM3S4NAM6HFNUEUL2J5Y3T4RL/graph.json","events_json":"https://pith.science/api/pith-number/POM3S4NAM6HFNUEUL2J5Y3T4RL/events.json","paper":"https://pith.science/paper/POM3S4NA"},"agent_actions":{"view_html":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL","download_json":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL.json","view_paper":"https://pith.science/paper/POM3S4NA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.12858&json=true","fetch_graph":"https://pith.science/api/pith-number/POM3S4NAM6HFNUEUL2J5Y3T4RL/graph.json","fetch_events":"https://pith.science/api/pith-number/POM3S4NAM6HFNUEUL2J5Y3T4RL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL/action/storage_attestation","attest_author":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL/action/author_attestation","sign_citation":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL/action/citation_signature","submit_replication":"https://pith.science/pith/POM3S4NAM6HFNUEUL2J5Y3T4RL/action/replication_record"}},"created_at":"2026-07-05T08:09:59.811917+00:00","updated_at":"2026-07-05T08:09:59.811917+00:00"}