{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:BJYFPL2TE6VHU6BJ5KBHKGAFTV","short_pith_number":"pith:BJYFPL2T","canonical_record":{"source":{"id":"2408.06932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-08-13T14:37:14Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"0166a4b2d14dc1f49b89156c9880f22720c5df056bba8eaf187a49b5df791f66","abstract_canon_sha256":"e09e55e32cf999d2c13f88fdb98dedfa634ca154e5177693f6c5bb8d8e7c211a"},"schema_version":"1.0"},"canonical_sha256":"0a7057af5327aa7a7829ea827518059d70e4edddb95786ce65c5b135fd8f14e9","source":{"kind":"arxiv","id":"2408.06932","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.06932","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"arxiv_version","alias_value":"2408.06932v1","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.06932","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"pith_short_12","alias_value":"BJYFPL2TE6VH","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"pith_short_16","alias_value":"BJYFPL2TE6VHU6BJ","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"pith_short_8","alias_value":"BJYFPL2T","created_at":"2026-07-05T08:55:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:BJYFPL2TE6VHU6BJ5KBHKGAFTV","target":"record","payload":{"canonical_record":{"source":{"id":"2408.06932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-08-13T14:37:14Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"0166a4b2d14dc1f49b89156c9880f22720c5df056bba8eaf187a49b5df791f66","abstract_canon_sha256":"e09e55e32cf999d2c13f88fdb98dedfa634ca154e5177693f6c5bb8d8e7c211a"},"schema_version":"1.0"},"canonical_sha256":"0a7057af5327aa7a7829ea827518059d70e4edddb95786ce65c5b135fd8f14e9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:55:03.844218Z","signature_b64":"E9MnDKWr4eXWz1IO7slse/GfVoyCBE2nyTwDL8ynsGfE5+2p2Qd3NmWAEywiy4qiemq8KgqRCOHuTbYd2oXTDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a7057af5327aa7a7829ea827518059d70e4edddb95786ce65c5b135fd8f14e9","last_reissued_at":"2026-07-05T08:55:03.843828Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:55:03.843828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2408.06932","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":"2026-07-05T08:55:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QI+z1mFrpG+kCI/hydp/UFE56ujr+/lmwvT16FhKDlNBcEhaDyWAG1CvKTwE4+O4qMP14HUPY+96C/4WDmzfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T09:42:22.397773Z"},"content_sha256":"2813419dd8c18b5f326fc78ef6dfa4032b6cc2a4e065047b22448d319fe820dc","schema_version":"1.0","event_id":"sha256:2813419dd8c18b5f326fc78ef6dfa4032b6cc2a4e065047b22448d319fe820dc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:BJYFPL2TE6VHU6BJ5KBHKGAFTV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity II: close to $H^1$ initial data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Chongsheng Cao, Dong Wang, Edriss S. Titi, Jinkai Li","submitted_at":"2024-08-13T14:37:14Z","abstract_excerpt":"In this paper, we consider the initial-boundary value problem to the three-dimensional primitive equations for the oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation in the domain $\\Omega=M\\times(-h,h)$, with $M=(0,1)\\times(0,1)$. Global well-posedness of strong solutions is established, for any initial data $(v_0,T_0) \\in H^1(\\Omega)\\cap L^\\infty(\\Omega)$ with $(\\partial_z v_0, \\nabla_H T_0) \\in L^q(\\Omega)$ and $v_0 \\in L_z^1(B^1_{q,2}(M))$, for some $q \\in (2,\\infty)$, by using"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.06932","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/2408.06932/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":"2026-07-05T08:55:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O9zGcDuOQ4VAR3xVgLqI+9wPB1JW0Ptd0gniaC8uwcmVqhSD51Mhcdw7+LsTBWVnay9TLkheYdW/l6oEUYk9Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T09:42:22.398263Z"},"content_sha256":"c0d0aaa5f83fe93a9354b73f2428ee25326283018b52821cd8b4c851c4866616","schema_version":"1.0","event_id":"sha256:c0d0aaa5f83fe93a9354b73f2428ee25326283018b52821cd8b4c851c4866616"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BJYFPL2TE6VHU6BJ5KBHKGAFTV/bundle.json","state_url":"https://pith.science/pith/BJYFPL2TE6VHU6BJ5KBHKGAFTV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BJYFPL2TE6VHU6BJ5KBHKGAFTV/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":"2026-08-15T09:42:22Z","links":{"resolver":"https://pith.science/pith/BJYFPL2TE6VHU6BJ5KBHKGAFTV","bundle":"https://pith.science/pith/BJYFPL2TE6VHU6BJ5KBHKGAFTV/bundle.json","state":"https://pith.science/pith/BJYFPL2TE6VHU6BJ5KBHKGAFTV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BJYFPL2TE6VHU6BJ5KBHKGAFTV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:BJYFPL2TE6VHU6BJ5KBHKGAFTV","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":"e09e55e32cf999d2c13f88fdb98dedfa634ca154e5177693f6c5bb8d8e7c211a","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-08-13T14:37:14Z","title_canon_sha256":"0166a4b2d14dc1f49b89156c9880f22720c5df056bba8eaf187a49b5df791f66"},"schema_version":"1.0","source":{"id":"2408.06932","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.06932","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"arxiv_version","alias_value":"2408.06932v1","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.06932","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"pith_short_12","alias_value":"BJYFPL2TE6VH","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"pith_short_16","alias_value":"BJYFPL2TE6VHU6BJ","created_at":"2026-07-05T08:55:03Z"},{"alias_kind":"pith_short_8","alias_value":"BJYFPL2T","created_at":"2026-07-05T08:55:03Z"}],"graph_snapshots":[{"event_id":"sha256:c0d0aaa5f83fe93a9354b73f2428ee25326283018b52821cd8b4c851c4866616","target":"graph","created_at":"2026-07-05T08:55:03Z","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/2408.06932/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 to the three-dimensional primitive equations for the oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation in the domain $\\Omega=M\\times(-h,h)$, with $M=(0,1)\\times(0,1)$. Global well-posedness of strong solutions is established, for any initial data $(v_0,T_0) \\in H^1(\\Omega)\\cap L^\\infty(\\Omega)$ with $(\\partial_z v_0, \\nabla_H T_0) \\in L^q(\\Omega)$ and $v_0 \\in L_z^1(B^1_{q,2}(M))$, for some $q \\in (2,\\infty)$, by using","authors_text":"Chongsheng Cao, Dong Wang, Edriss S. Titi, Jinkai Li","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-08-13T14:37:14Z","title":"Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity II: close to $H^1$ initial data"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.06932","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":"sha256:2813419dd8c18b5f326fc78ef6dfa4032b6cc2a4e065047b22448d319fe820dc","target":"record","created_at":"2026-07-05T08:55:03Z","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":"e09e55e32cf999d2c13f88fdb98dedfa634ca154e5177693f6c5bb8d8e7c211a","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-08-13T14:37:14Z","title_canon_sha256":"0166a4b2d14dc1f49b89156c9880f22720c5df056bba8eaf187a49b5df791f66"},"schema_version":"1.0","source":{"id":"2408.06932","kind":"arxiv","version":1}},"canonical_sha256":"0a7057af5327aa7a7829ea827518059d70e4edddb95786ce65c5b135fd8f14e9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a7057af5327aa7a7829ea827518059d70e4edddb95786ce65c5b135fd8f14e9","first_computed_at":"2026-07-05T08:55:03.843828Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:55:03.843828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E9MnDKWr4eXWz1IO7slse/GfVoyCBE2nyTwDL8ynsGfE5+2p2Qd3NmWAEywiy4qiemq8KgqRCOHuTbYd2oXTDg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:55:03.844218Z","signed_message":"canonical_sha256_bytes"},"source_id":"2408.06932","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2813419dd8c18b5f326fc78ef6dfa4032b6cc2a4e065047b22448d319fe820dc","sha256:c0d0aaa5f83fe93a9354b73f2428ee25326283018b52821cd8b4c851c4866616"],"state_sha256":"7234246fa2d6d3dae5ba179d6ef7b9291cc088331d9917af0f2b9330b2d9067d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1FtvcyMmSKKZGRp4L0LC7KCVXcCsA6aN98gSVteN6hPpVsrKtWeQ0k9a2C0net4jCxZxhZW/XCgJDHsyv0oKAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T09:42:22.402480Z","bundle_sha256":"fe663dcb5a8ee7a55cf9ba215803fbf4bbb946ad2a0c6b3ac9e6b039e7ddb2cc"}}