{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:C7NFCJM7HO65LTWA5JFCXBLINV","short_pith_number":"pith:C7NFCJM7","canonical_record":{"source":{"id":"1212.4222","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-18T03:45:41Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"688886e6dc79961f751fe3917f6ebe1e5b8a40ecd80c460e17bb580f7965ecbc","abstract_canon_sha256":"2872d0f483b951ccd1a011bae41475df13ce1867273b160002e402324e075827"},"schema_version":"1.0"},"canonical_sha256":"17da51259f3bbdd5cec0ea4a2b85686d4c7dbc2455840f8fb68b651a46d20c65","source":{"kind":"arxiv","id":"1212.4222","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.4222","created_at":"2026-05-18T02:50:51Z"},{"alias_kind":"arxiv_version","alias_value":"1212.4222v2","created_at":"2026-05-18T02:50:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.4222","created_at":"2026-05-18T02:50:51Z"},{"alias_kind":"pith_short_12","alias_value":"C7NFCJM7HO65","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"C7NFCJM7HO65LTWA","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"C7NFCJM7","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:C7NFCJM7HO65LTWA5JFCXBLINV","target":"record","payload":{"canonical_record":{"source":{"id":"1212.4222","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-18T03:45:41Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"688886e6dc79961f751fe3917f6ebe1e5b8a40ecd80c460e17bb580f7965ecbc","abstract_canon_sha256":"2872d0f483b951ccd1a011bae41475df13ce1867273b160002e402324e075827"},"schema_version":"1.0"},"canonical_sha256":"17da51259f3bbdd5cec0ea4a2b85686d4c7dbc2455840f8fb68b651a46d20c65","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:51.986310Z","signature_b64":"2nd0btUrYsEllRUzaAdX3/ggKDXhWRjERMCMza7LQuP51MVdi9rwkTyivjBj2WmF+7mgzTUFWs2jWcMVJUolCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17da51259f3bbdd5cec0ea4a2b85686d4c7dbc2455840f8fb68b651a46d20c65","last_reissued_at":"2026-05-18T02:50:51.985778Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:51.985778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.4222","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:50:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L0QX8kLkFM9dgZzk9QRCjHjtxkdu3cRhePvPgf6FP37AshigkG0a0An29P5gcIXI+xHl42DnAi1bGoKXgRxUCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T02:10:44.360185Z"},"content_sha256":"1c5eaecd78e7e30cfd30ad7ba1d75e52d8a78de36036d6d4d8ce5e208b8d144e","schema_version":"1.0","event_id":"sha256:1c5eaecd78e7e30cfd30ad7ba1d75e52d8a78de36036d6d4d8ce5e208b8d144e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:C7NFCJM7HO65LTWA5JFCXBLINV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Xiaoxin Zheng","submitted_at":"2012-12-18T03:45:41Z","abstract_excerpt":"In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*}\n  {{array}{ll}\n  (\\partial_{t}+u\\cdot\\nabla)u-\\kappa\\Delta_{h} u+\\nabla \\Pi=\\rho e_{3},\\quad(t,x)\\in\\mathbb{R}^{+}\\times\\mathbb{R}^{3},\n  (\\partial_{t}+u\\cdot\\nabla)\\rho=0,\n  \\text{div}u=0.\n  {array}. {equation*} Under the assumption that the support of the axisymmetric initial data $\\rho_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\\frac\\rho r$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4222","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T02:50:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JXL2aBH9mAb7oyq6LZ1biOzqs8apvHMpQLyIHa3VOHeOMFsCx3L+Np0clUlaq8v2xS/7lvxaAWzOeiMOx/7sCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T02:10:44.360816Z"},"content_sha256":"253c34557ffae2daa4a82c753ac2fb8d1bc4356f6a99226b163dcc6672a8c3a0","schema_version":"1.0","event_id":"sha256:253c34557ffae2daa4a82c753ac2fb8d1bc4356f6a99226b163dcc6672a8c3a0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C7NFCJM7HO65LTWA5JFCXBLINV/bundle.json","state_url":"https://pith.science/pith/C7NFCJM7HO65LTWA5JFCXBLINV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C7NFCJM7HO65LTWA5JFCXBLINV/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":"2026-05-27T02:10:44Z","links":{"resolver":"https://pith.science/pith/C7NFCJM7HO65LTWA5JFCXBLINV","bundle":"https://pith.science/pith/C7NFCJM7HO65LTWA5JFCXBLINV/bundle.json","state":"https://pith.science/pith/C7NFCJM7HO65LTWA5JFCXBLINV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C7NFCJM7HO65LTWA5JFCXBLINV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:C7NFCJM7HO65LTWA5JFCXBLINV","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":"2872d0f483b951ccd1a011bae41475df13ce1867273b160002e402324e075827","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-18T03:45:41Z","title_canon_sha256":"688886e6dc79961f751fe3917f6ebe1e5b8a40ecd80c460e17bb580f7965ecbc"},"schema_version":"1.0","source":{"id":"1212.4222","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.4222","created_at":"2026-05-18T02:50:51Z"},{"alias_kind":"arxiv_version","alias_value":"1212.4222v2","created_at":"2026-05-18T02:50:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.4222","created_at":"2026-05-18T02:50:51Z"},{"alias_kind":"pith_short_12","alias_value":"C7NFCJM7HO65","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"C7NFCJM7HO65LTWA","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"C7NFCJM7","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:253c34557ffae2daa4a82c753ac2fb8d1bc4356f6a99226b163dcc6672a8c3a0","target":"graph","created_at":"2026-05-18T02:50:51Z","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"},"paper":{"abstract_excerpt":"In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*}\n  {{array}{ll}\n  (\\partial_{t}+u\\cdot\\nabla)u-\\kappa\\Delta_{h} u+\\nabla \\Pi=\\rho e_{3},\\quad(t,x)\\in\\mathbb{R}^{+}\\times\\mathbb{R}^{3},\n  (\\partial_{t}+u\\cdot\\nabla)\\rho=0,\n  \\text{div}u=0.\n  {array}. {equation*} Under the assumption that the support of the axisymmetric initial data $\\rho_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\\frac\\rho r$","authors_text":"Changxing Miao, Xiaoxin Zheng","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-18T03:45:41Z","title":"Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4222","kind":"arxiv","version":2},"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":"sha256:1c5eaecd78e7e30cfd30ad7ba1d75e52d8a78de36036d6d4d8ce5e208b8d144e","target":"record","created_at":"2026-05-18T02:50:51Z","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":"2872d0f483b951ccd1a011bae41475df13ce1867273b160002e402324e075827","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-18T03:45:41Z","title_canon_sha256":"688886e6dc79961f751fe3917f6ebe1e5b8a40ecd80c460e17bb580f7965ecbc"},"schema_version":"1.0","source":{"id":"1212.4222","kind":"arxiv","version":2}},"canonical_sha256":"17da51259f3bbdd5cec0ea4a2b85686d4c7dbc2455840f8fb68b651a46d20c65","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"17da51259f3bbdd5cec0ea4a2b85686d4c7dbc2455840f8fb68b651a46d20c65","first_computed_at":"2026-05-18T02:50:51.985778Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:51.985778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2nd0btUrYsEllRUzaAdX3/ggKDXhWRjERMCMza7LQuP51MVdi9rwkTyivjBj2WmF+7mgzTUFWs2jWcMVJUolCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:51.986310Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.4222","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c5eaecd78e7e30cfd30ad7ba1d75e52d8a78de36036d6d4d8ce5e208b8d144e","sha256:253c34557ffae2daa4a82c753ac2fb8d1bc4356f6a99226b163dcc6672a8c3a0"],"state_sha256":"ace3acb89fba278c7a83e25e6288b967259100e99d4febe94a09f924c4cb5502"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HBGygzfty6x21oXpR1leZmdVBN+ygDtMNjd1Jh+RRrQ1vBZ3pWPy8pdFGhzkM/yUYL+UyMk5qdmO03Z3n7QXBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T02:10:44.368675Z","bundle_sha256":"fb111193a3cdf779fef1a68fde90c5d701032925569e5ff5746f40134804ab98"}}