{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:DO33O7UV4N6BDZSJ6RGEMUOXSQ","short_pith_number":"pith:DO33O7UV","canonical_record":{"source":{"id":"2107.07688","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-07-16T03:36:58Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"263e7cc009824f3322cbb9a601f3a13e882de0fa9e1e339fcae2e0e82eea9f52","abstract_canon_sha256":"aa6ea205843b5cddf0122270adc6ba018920fc00e03a2912e9d87548730c8371"},"schema_version":"1.0"},"canonical_sha256":"1bb7b77e95e37c11e649f44c4651d794307ecd469a1bf8b27df6cd18b0b5bf81","source":{"kind":"arxiv","id":"2107.07688","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.07688","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"arxiv_version","alias_value":"2107.07688v1","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.07688","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"pith_short_12","alias_value":"DO33O7UV4N6B","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"pith_short_16","alias_value":"DO33O7UV4N6BDZSJ","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"pith_short_8","alias_value":"DO33O7UV","created_at":"2026-07-05T03:57:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:DO33O7UV4N6BDZSJ6RGEMUOXSQ","target":"record","payload":{"canonical_record":{"source":{"id":"2107.07688","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-07-16T03:36:58Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"263e7cc009824f3322cbb9a601f3a13e882de0fa9e1e339fcae2e0e82eea9f52","abstract_canon_sha256":"aa6ea205843b5cddf0122270adc6ba018920fc00e03a2912e9d87548730c8371"},"schema_version":"1.0"},"canonical_sha256":"1bb7b77e95e37c11e649f44c4651d794307ecd469a1bf8b27df6cd18b0b5bf81","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:57:09.631651Z","signature_b64":"BFe1Jq0m2CenhWgDqbeDFnmnuBXTbD9j185/m0R1n8wOZqOx7GSlHFASbxPtjim3D3JfLSloZL99ihwWI7l7Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bb7b77e95e37c11e649f44c4651d794307ecd469a1bf8b27df6cd18b0b5bf81","last_reissued_at":"2026-07-05T03:57:09.631137Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:57:09.631137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2107.07688","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":"2026-07-05T03:57:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C6pFX5tn1VnrOUsUN2+1n99x/F6QYXN06AmybdJDu2RF7XW4BJJC4DsxZj+kpxo8UQglzcWv6DMQjaffv4EqDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T19:56:55.689540Z"},"content_sha256":"852d6747f9743969eeb7622ebaa95c9dc1eec4a00662237a1379e9036612c672","schema_version":"1.0","event_id":"sha256:852d6747f9743969eeb7622ebaa95c9dc1eec4a00662237a1379e9036612c672"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:DO33O7UV4N6BDZSJ6RGEMUOXSQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global well-posedness of $z$-weak solutions to the primitive equations without vertical diffusivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Guozhi Yuan, Jinkai Li","submitted_at":"2021-07-16T03:36:58Z","abstract_excerpt":"In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of $z$-weak solution is established for any such initial datum that itself and its vertical derivative belong to $L^2$. This not only extends the results in \\cite{Cao5} from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be $H^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.07688","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/2107.07688/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":"2026-07-05T03:57:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8rLGveDPbiJP1V7vbclr2Nge6etCUnb+SfPOBJi16aff7byrGsHLAbu/5BlhxZouV5ZkhWqiT5uCqDLNq16HDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T19:56:55.690050Z"},"content_sha256":"73d210e9a35fd9749fb373acb8d72eaa1699351271f76f596c12ebb43547fa74","schema_version":"1.0","event_id":"sha256:73d210e9a35fd9749fb373acb8d72eaa1699351271f76f596c12ebb43547fa74"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DO33O7UV4N6BDZSJ6RGEMUOXSQ/bundle.json","state_url":"https://pith.science/pith/DO33O7UV4N6BDZSJ6RGEMUOXSQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DO33O7UV4N6BDZSJ6RGEMUOXSQ/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":"2026-08-03T19:56:55Z","links":{"resolver":"https://pith.science/pith/DO33O7UV4N6BDZSJ6RGEMUOXSQ","bundle":"https://pith.science/pith/DO33O7UV4N6BDZSJ6RGEMUOXSQ/bundle.json","state":"https://pith.science/pith/DO33O7UV4N6BDZSJ6RGEMUOXSQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DO33O7UV4N6BDZSJ6RGEMUOXSQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:DO33O7UV4N6BDZSJ6RGEMUOXSQ","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":"aa6ea205843b5cddf0122270adc6ba018920fc00e03a2912e9d87548730c8371","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-07-16T03:36:58Z","title_canon_sha256":"263e7cc009824f3322cbb9a601f3a13e882de0fa9e1e339fcae2e0e82eea9f52"},"schema_version":"1.0","source":{"id":"2107.07688","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.07688","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"arxiv_version","alias_value":"2107.07688v1","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.07688","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"pith_short_12","alias_value":"DO33O7UV4N6B","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"pith_short_16","alias_value":"DO33O7UV4N6BDZSJ","created_at":"2026-07-05T03:57:09Z"},{"alias_kind":"pith_short_8","alias_value":"DO33O7UV","created_at":"2026-07-05T03:57:09Z"}],"graph_snapshots":[{"event_id":"sha256:73d210e9a35fd9749fb373acb8d72eaa1699351271f76f596c12ebb43547fa74","target":"graph","created_at":"2026-07-05T03:57:09Z","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/2107.07688/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of $z$-weak solution is established for any such initial datum that itself and its vertical derivative belong to $L^2$. This not only extends the results in \\cite{Cao5} from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be $H^","authors_text":"Guozhi Yuan, Jinkai Li","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-07-16T03:36:58Z","title":"Global well-posedness of $z$-weak solutions to the primitive equations without vertical diffusivity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.07688","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":"sha256:852d6747f9743969eeb7622ebaa95c9dc1eec4a00662237a1379e9036612c672","target":"record","created_at":"2026-07-05T03:57:09Z","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":"aa6ea205843b5cddf0122270adc6ba018920fc00e03a2912e9d87548730c8371","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-07-16T03:36:58Z","title_canon_sha256":"263e7cc009824f3322cbb9a601f3a13e882de0fa9e1e339fcae2e0e82eea9f52"},"schema_version":"1.0","source":{"id":"2107.07688","kind":"arxiv","version":1}},"canonical_sha256":"1bb7b77e95e37c11e649f44c4651d794307ecd469a1bf8b27df6cd18b0b5bf81","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bb7b77e95e37c11e649f44c4651d794307ecd469a1bf8b27df6cd18b0b5bf81","first_computed_at":"2026-07-05T03:57:09.631137Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:57:09.631137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BFe1Jq0m2CenhWgDqbeDFnmnuBXTbD9j185/m0R1n8wOZqOx7GSlHFASbxPtjim3D3JfLSloZL99ihwWI7l7Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T03:57:09.631651Z","signed_message":"canonical_sha256_bytes"},"source_id":"2107.07688","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:852d6747f9743969eeb7622ebaa95c9dc1eec4a00662237a1379e9036612c672","sha256:73d210e9a35fd9749fb373acb8d72eaa1699351271f76f596c12ebb43547fa74"],"state_sha256":"363413bbff3ef33d7c6ba24fcdd0bd3cd3c84c3d94a91f95df2204d6944cc526"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"owh2LEqor6v5QeBIftz1T+0WyEveZvC26ph5BM4zexD2piIrmu4lhy+1aa7cJVSNxXXWeAEMc7fSUAIpKP8BDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T19:56:55.694984Z","bundle_sha256":"08c07969a9a3dc9b489babcbba66ca3ca753a9fe0ed2faf6d5e9ef797cf2ff9c"}}