{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:JBPH7QTD4SOEBWD3NE65YKP2ME","short_pith_number":"pith:JBPH7QTD","canonical_record":{"source":{"id":"1809.02803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-08T14:10:27Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"db783d35e5449eac031eb63fdfe32bc991435fb3b528d3af0b01a30c12da2e1b","abstract_canon_sha256":"c8b19863a148bc440cfe52779574538be72caaf7fe3662e0d19e9826b9831acd"},"schema_version":"1.0"},"canonical_sha256":"485e7fc263e49c40d87b693ddc29fa61015ee59dec1f0cf7bbb5c244a63ab109","source":{"kind":"arxiv","id":"1809.02803","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.02803","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"arxiv_version","alias_value":"1809.02803v1","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.02803","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"pith_short_12","alias_value":"JBPH7QTD4SOE","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"JBPH7QTD4SOEBWD3","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"JBPH7QTD","created_at":"2026-05-18T12:32:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:JBPH7QTD4SOEBWD3NE65YKP2ME","target":"record","payload":{"canonical_record":{"source":{"id":"1809.02803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-08T14:10:27Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"db783d35e5449eac031eb63fdfe32bc991435fb3b528d3af0b01a30c12da2e1b","abstract_canon_sha256":"c8b19863a148bc440cfe52779574538be72caaf7fe3662e0d19e9826b9831acd"},"schema_version":"1.0"},"canonical_sha256":"485e7fc263e49c40d87b693ddc29fa61015ee59dec1f0cf7bbb5c244a63ab109","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:15.238698Z","signature_b64":"RuegWeOQYGpac2nWoXCQU8BDw0zZQNfF+1jOXjqwY3HvesTBRs4mWd0zzZ7jsCKKpVhHK+qg1OZhw5VyUcLLCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"485e7fc263e49c40d87b693ddc29fa61015ee59dec1f0cf7bbb5c244a63ab109","last_reissued_at":"2026-05-18T00:06:15.238131Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:15.238131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.02803","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":"2026-05-18T00:06:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ukFq58gw7ubtYhVJ4ouIAV1FsEqKZ3TNDsVBjpcwx4Y4xSkEWSP2G73tiPyZtxXd1F9zQKo2zaFKMJFDvB/cCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:42:23.300491Z"},"content_sha256":"d1d1a37bef032f4837d15caf6427c2816bfc2fe18c7d793664341e8ea8098204","schema_version":"1.0","event_id":"sha256:d1d1a37bef032f4837d15caf6427c2816bfc2fe18c7d793664341e8ea8098204"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:JBPH7QTD4SOEBWD3NE65YKP2ME","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Determinstic and stochastic 2d Navier-Stokes equations with anisotropic viscosity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Ping Zhang, Rongchan Zhu, Siyu Liang","submitted_at":"2018-09-08T14:10:27Z","abstract_excerpt":"In this paper, we investigate both deterministic and stochastic 2D Navier Stokes equations with anisotropic viscosity. For the deterministic case, we prove the global well-posedness of the system with initial data in the anisotropic Sobolev space $\\tilde{H}^{0,1}.$ For the stochastic case, we obtain existence of the martingale solutions and pathwise uniqueness of the solutions, which imply existence of the probabilistically strong solution to this system by the Yamada-Watanabe Theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02803","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":""},"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":"2026-05-18T00:06:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aKfp8n1laWTVgj2ULc07yjg+Ap4T86MPnx60r4utVj10F692J1kA+6bgK8iiXUsqriCsyxvIdqnq+35Hr62ECA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:42:23.300838Z"},"content_sha256":"46ee223fc3ab2399166269a6fdf9deee23a29a2123605cd625b75c5fc54dc24d","schema_version":"1.0","event_id":"sha256:46ee223fc3ab2399166269a6fdf9deee23a29a2123605cd625b75c5fc54dc24d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JBPH7QTD4SOEBWD3NE65YKP2ME/bundle.json","state_url":"https://pith.science/pith/JBPH7QTD4SOEBWD3NE65YKP2ME/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JBPH7QTD4SOEBWD3NE65YKP2ME/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":"2026-05-31T05:42:23Z","links":{"resolver":"https://pith.science/pith/JBPH7QTD4SOEBWD3NE65YKP2ME","bundle":"https://pith.science/pith/JBPH7QTD4SOEBWD3NE65YKP2ME/bundle.json","state":"https://pith.science/pith/JBPH7QTD4SOEBWD3NE65YKP2ME/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JBPH7QTD4SOEBWD3NE65YKP2ME/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:JBPH7QTD4SOEBWD3NE65YKP2ME","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":"c8b19863a148bc440cfe52779574538be72caaf7fe3662e0d19e9826b9831acd","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-08T14:10:27Z","title_canon_sha256":"db783d35e5449eac031eb63fdfe32bc991435fb3b528d3af0b01a30c12da2e1b"},"schema_version":"1.0","source":{"id":"1809.02803","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.02803","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"arxiv_version","alias_value":"1809.02803v1","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.02803","created_at":"2026-05-18T00:06:15Z"},{"alias_kind":"pith_short_12","alias_value":"JBPH7QTD4SOE","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"JBPH7QTD4SOEBWD3","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"JBPH7QTD","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:46ee223fc3ab2399166269a6fdf9deee23a29a2123605cd625b75c5fc54dc24d","target":"graph","created_at":"2026-05-18T00:06:15Z","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 investigate both deterministic and stochastic 2D Navier Stokes equations with anisotropic viscosity. For the deterministic case, we prove the global well-posedness of the system with initial data in the anisotropic Sobolev space $\\tilde{H}^{0,1}.$ For the stochastic case, we obtain existence of the martingale solutions and pathwise uniqueness of the solutions, which imply existence of the probabilistically strong solution to this system by the Yamada-Watanabe Theorem.","authors_text":"Ping Zhang, Rongchan Zhu, Siyu Liang","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-08T14:10:27Z","title":"Determinstic and stochastic 2d Navier-Stokes equations with anisotropic viscosity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02803","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:d1d1a37bef032f4837d15caf6427c2816bfc2fe18c7d793664341e8ea8098204","target":"record","created_at":"2026-05-18T00:06:15Z","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":"c8b19863a148bc440cfe52779574538be72caaf7fe3662e0d19e9826b9831acd","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-08T14:10:27Z","title_canon_sha256":"db783d35e5449eac031eb63fdfe32bc991435fb3b528d3af0b01a30c12da2e1b"},"schema_version":"1.0","source":{"id":"1809.02803","kind":"arxiv","version":1}},"canonical_sha256":"485e7fc263e49c40d87b693ddc29fa61015ee59dec1f0cf7bbb5c244a63ab109","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"485e7fc263e49c40d87b693ddc29fa61015ee59dec1f0cf7bbb5c244a63ab109","first_computed_at":"2026-05-18T00:06:15.238131Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:15.238131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RuegWeOQYGpac2nWoXCQU8BDw0zZQNfF+1jOXjqwY3HvesTBRs4mWd0zzZ7jsCKKpVhHK+qg1OZhw5VyUcLLCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:15.238698Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.02803","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d1d1a37bef032f4837d15caf6427c2816bfc2fe18c7d793664341e8ea8098204","sha256:46ee223fc3ab2399166269a6fdf9deee23a29a2123605cd625b75c5fc54dc24d"],"state_sha256":"b6569d9a1718680400ff6ff27f895646ecd00157b506cdaf0fd884b34d5f3560"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"74O4Q08bA4T0pfBvp1atojIPuPGsjd+ux1wnHedtfP3Lf2lBKYIBQE7f1ileTBW8TGLHq9gYLfPxemKSznsiCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T05:42:23.302978Z","bundle_sha256":"6d2cd8f9663e7371b31687c81bacf1ff3264d1b5917a4f846e01167f849b63d2"}}