{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:GL2ED6HBBACYRLU55QE4F4GH5I","short_pith_number":"pith:GL2ED6HB","canonical_record":{"source":{"id":"2411.05438","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-08T09:37:01Z","cross_cats_sorted":[],"title_canon_sha256":"481431b1f84f95ad09b68eba75813f04a59d5a3b7dec0f55b644c4e91b064e39","abstract_canon_sha256":"71fba57eb2c51337b2b52ac8df0ccf9b101042e7d2998575ebd9bb4a0bed30b7"},"schema_version":"1.0"},"canonical_sha256":"32f441f8e1080588ae9dec09c2f0c7ea3d60a7470f92dee10d55a38e01012ffb","source":{"kind":"arxiv","id":"2411.05438","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.05438","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"arxiv_version","alias_value":"2411.05438v1","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.05438","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"pith_short_12","alias_value":"GL2ED6HBBACY","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"GL2ED6HBBACYRLU5","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"GL2ED6HB","created_at":"2026-07-05T09:33:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:GL2ED6HBBACYRLU55QE4F4GH5I","target":"record","payload":{"canonical_record":{"source":{"id":"2411.05438","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-08T09:37:01Z","cross_cats_sorted":[],"title_canon_sha256":"481431b1f84f95ad09b68eba75813f04a59d5a3b7dec0f55b644c4e91b064e39","abstract_canon_sha256":"71fba57eb2c51337b2b52ac8df0ccf9b101042e7d2998575ebd9bb4a0bed30b7"},"schema_version":"1.0"},"canonical_sha256":"32f441f8e1080588ae9dec09c2f0c7ea3d60a7470f92dee10d55a38e01012ffb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:33:01.025840Z","signature_b64":"MH/TNchHl7VMdkWiIbuoyfIvWXb6KKM5jysH3fqU4HuiGNi59qtpnMO6r9tQ068PjcanjI94ynAoce2PP9+gAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32f441f8e1080588ae9dec09c2f0c7ea3d60a7470f92dee10d55a38e01012ffb","last_reissued_at":"2026-07-05T09:33:01.025280Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:33:01.025280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.05438","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:33:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uvPU/i/3dys7n/gcdxnWPsUZ4IsjwyWZiKioTs/TA3aGAnFcU2JAetMSNmis30LQiC9QyfkZMJ3hYNHR1DrYBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T21:43:52.152524Z"},"content_sha256":"e1e5cf8e3ef6e74d7cbe5dd77a224a079ffa1ce3d2283aabf515dca013f93f6e","schema_version":"1.0","event_id":"sha256:e1e5cf8e3ef6e74d7cbe5dd77a224a079ffa1ce3d2283aabf515dca013f93f6e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:GL2ED6HBBACYRLU55QE4F4GH5I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inhomogenous Navier--Stokes equations with unbounded density","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean-Paul Adogbo, Maja Szlenk, Piotr B. Mucha","submitted_at":"2024-11-08T09:37:01Z","abstract_excerpt":"In the current state of the art regarding the Navier--Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper.\n  Our main result demonstrates the global existence of a unique solution for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.05438","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/2411.05438/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:33:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hbHwR6J0FjTeD0eGKCpdfUxcGqJQ/cYyJtxpI/D73DAgnVqZd4NuCPnjsmDGGEznunuHqm1T8GfUNEUyHnDQDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T21:43:52.152901Z"},"content_sha256":"20c2d5b30214af98df69f2a4884f2c06a851a5e2b949dc8538e61f8e7f575793","schema_version":"1.0","event_id":"sha256:20c2d5b30214af98df69f2a4884f2c06a851a5e2b949dc8538e61f8e7f575793"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GL2ED6HBBACYRLU55QE4F4GH5I/bundle.json","state_url":"https://pith.science/pith/GL2ED6HBBACYRLU55QE4F4GH5I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GL2ED6HBBACYRLU55QE4F4GH5I/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-17T21:43:52Z","links":{"resolver":"https://pith.science/pith/GL2ED6HBBACYRLU55QE4F4GH5I","bundle":"https://pith.science/pith/GL2ED6HBBACYRLU55QE4F4GH5I/bundle.json","state":"https://pith.science/pith/GL2ED6HBBACYRLU55QE4F4GH5I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GL2ED6HBBACYRLU55QE4F4GH5I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:GL2ED6HBBACYRLU55QE4F4GH5I","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"71fba57eb2c51337b2b52ac8df0ccf9b101042e7d2998575ebd9bb4a0bed30b7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-08T09:37:01Z","title_canon_sha256":"481431b1f84f95ad09b68eba75813f04a59d5a3b7dec0f55b644c4e91b064e39"},"schema_version":"1.0","source":{"id":"2411.05438","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.05438","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"arxiv_version","alias_value":"2411.05438v1","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.05438","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"pith_short_12","alias_value":"GL2ED6HBBACY","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"GL2ED6HBBACYRLU5","created_at":"2026-07-05T09:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"GL2ED6HB","created_at":"2026-07-05T09:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:20c2d5b30214af98df69f2a4884f2c06a851a5e2b949dc8538e61f8e7f575793","target":"graph","created_at":"2026-07-05T09:33:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.05438/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In the current state of the art regarding the Navier--Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper.\n  Our main result demonstrates the global existence of a unique solution for","authors_text":"Jean-Paul Adogbo, Maja Szlenk, Piotr B. Mucha","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-08T09:37:01Z","title":"Inhomogenous Navier--Stokes equations with unbounded density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.05438","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e1e5cf8e3ef6e74d7cbe5dd77a224a079ffa1ce3d2283aabf515dca013f93f6e","target":"record","created_at":"2026-07-05T09:33:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"71fba57eb2c51337b2b52ac8df0ccf9b101042e7d2998575ebd9bb4a0bed30b7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-08T09:37:01Z","title_canon_sha256":"481431b1f84f95ad09b68eba75813f04a59d5a3b7dec0f55b644c4e91b064e39"},"schema_version":"1.0","source":{"id":"2411.05438","kind":"arxiv","version":1}},"canonical_sha256":"32f441f8e1080588ae9dec09c2f0c7ea3d60a7470f92dee10d55a38e01012ffb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32f441f8e1080588ae9dec09c2f0c7ea3d60a7470f92dee10d55a38e01012ffb","first_computed_at":"2026-07-05T09:33:01.025280Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:33:01.025280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MH/TNchHl7VMdkWiIbuoyfIvWXb6KKM5jysH3fqU4HuiGNi59qtpnMO6r9tQ068PjcanjI94ynAoce2PP9+gAg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:33:01.025840Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.05438","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e1e5cf8e3ef6e74d7cbe5dd77a224a079ffa1ce3d2283aabf515dca013f93f6e","sha256:20c2d5b30214af98df69f2a4884f2c06a851a5e2b949dc8538e61f8e7f575793"],"state_sha256":"41833bda44d6f0b00b879e84adcf90e577cb12982ce4a5a87e3b9410c45f5ef7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N/W1L9v+FJJLsLNDotUWurj8zJbS/Tw1Tyct5SyVkr4WCG9w0tEz6lEcKt12LR4iUQZjNziP3z5MCt+3YKnJDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T21:43:52.155668Z","bundle_sha256":"2c640abfce25411c1715166805c30699ac5944d796900fe91a016c936eac678b"}}