{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:P4KMN74BTC5R2OL57NW5PANJE6","short_pith_number":"pith:P4KMN74B","schema_version":"1.0","canonical_sha256":"7f14c6ff8198bb1d397dfb6dd781a92786d40e8f125b7a4ac6813e2ccf4eb069","source":{"kind":"arxiv","id":"2406.07984","version":1},"attestation_state":"computed","paper":{"title":"On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongyi Wei, Feng Shao, Tiantian Hao, Zhifei Zhang","submitted_at":"2024-06-12T08:10:05Z","abstract_excerpt":"In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\\it for the density patch data $\\rho_0=1_{D}$ with"},"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":"2406.07984","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-06-12T08:10:05Z","cross_cats_sorted":[],"title_canon_sha256":"0e58ad3d1c82d575791e0ad90d0c87ae61c4e6649dfeba23b098d6c5cd186897","abstract_canon_sha256":"63ba229eae59eb78e9a833c229de14b8c72757d0433e79755e83ed4c7e4081b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:30:41.834010Z","signature_b64":"dJK2ohUEBvEdbyBDFmRiCNy85o9q/uT86rSYM8l7kjCkgE943nKdHxVyOYzq8w434VhZwzbF1Cp8O4e7/ipdDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f14c6ff8198bb1d397dfb6dd781a92786d40e8f125b7a4ac6813e2ccf4eb069","last_reissued_at":"2026-07-05T08:30:41.833516Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:30:41.833516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongyi Wei, Feng Shao, Tiantian Hao, Zhifei Zhang","submitted_at":"2024-06-12T08:10:05Z","abstract_excerpt":"In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\\it for the density patch data $\\rho_0=1_{D}$ with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.07984","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/2406.07984/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":"2406.07984","created_at":"2026-07-05T08:30:41.833576+00:00"},{"alias_kind":"arxiv_version","alias_value":"2406.07984v1","created_at":"2026-07-05T08:30:41.833576+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.07984","created_at":"2026-07-05T08:30:41.833576+00:00"},{"alias_kind":"pith_short_12","alias_value":"P4KMN74BTC5R","created_at":"2026-07-05T08:30:41.833576+00:00"},{"alias_kind":"pith_short_16","alias_value":"P4KMN74BTC5R2OL5","created_at":"2026-07-05T08:30:41.833576+00:00"},{"alias_kind":"pith_short_8","alias_value":"P4KMN74B","created_at":"2026-07-05T08:30:41.833576+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":25,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6","json":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6.json","graph_json":"https://pith.science/api/pith-number/P4KMN74BTC5R2OL57NW5PANJE6/graph.json","events_json":"https://pith.science/api/pith-number/P4KMN74BTC5R2OL57NW5PANJE6/events.json","paper":"https://pith.science/paper/P4KMN74B"},"agent_actions":{"view_html":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6","download_json":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6.json","view_paper":"https://pith.science/paper/P4KMN74B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2406.07984&json=true","fetch_graph":"https://pith.science/api/pith-number/P4KMN74BTC5R2OL57NW5PANJE6/graph.json","fetch_events":"https://pith.science/api/pith-number/P4KMN74BTC5R2OL57NW5PANJE6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6/action/storage_attestation","attest_author":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6/action/author_attestation","sign_citation":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6/action/citation_signature","submit_replication":"https://pith.science/pith/P4KMN74BTC5R2OL57NW5PANJE6/action/replication_record"}},"created_at":"2026-07-05T08:30:41.833576+00:00","updated_at":"2026-07-05T08:30:41.833576+00:00"}