{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:JRD7C6IPNNZPMJFC7B5LSY3FZM","short_pith_number":"pith:JRD7C6IP","canonical_record":{"source":{"id":"1009.4538","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-23T08:13:28Z","cross_cats_sorted":[],"title_canon_sha256":"e92127d313b300e96859a2e98932d203692ace363da3ff1c144643614a238439","abstract_canon_sha256":"3d6abb82688c9a240585ee818b2b94169fef1f922fe8a219f90f72c1796e9b68"},"schema_version":"1.0"},"canonical_sha256":"4c47f1790f6b72f624a2f87ab96365cb343ed87a9d1e3105f3c22cb57a3a44d9","source":{"kind":"arxiv","id":"1009.4538","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4538","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4538v1","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4538","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"pith_short_12","alias_value":"JRD7C6IPNNZP","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JRD7C6IPNNZPMJFC","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JRD7C6IP","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:JRD7C6IPNNZPMJFC7B5LSY3FZM","target":"record","payload":{"canonical_record":{"source":{"id":"1009.4538","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-23T08:13:28Z","cross_cats_sorted":[],"title_canon_sha256":"e92127d313b300e96859a2e98932d203692ace363da3ff1c144643614a238439","abstract_canon_sha256":"3d6abb82688c9a240585ee818b2b94169fef1f922fe8a219f90f72c1796e9b68"},"schema_version":"1.0"},"canonical_sha256":"4c47f1790f6b72f624a2f87ab96365cb343ed87a9d1e3105f3c22cb57a3a44d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:30.199232Z","signature_b64":"rycUKBWOd0w5qRplzGOUehsI+e3Qd+RG2mlnobqgbD7nZXfXyo9qkyHFy3wLAqoAw/CVA42x8Dkc9oN1zIIRAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c47f1790f6b72f624a2f87ab96365cb343ed87a9d1e3105f3c22cb57a3a44d9","last_reissued_at":"2026-05-18T04:40:30.198563Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:30.198563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.4538","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":"2026-05-18T04:40:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0oQeeP+q9TrxuwzWUwXyLCS5jrp49FyizqMDDBgKG2ZuMXFg/Px/S/Lp9pDTnyY0wvcHpKsLpfEPyGYexbKlAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:28:51.745233Z"},"content_sha256":"550cee02d38b34e2ed5b9a1ffac18ffa5502b00c79f18fd40cb4387269518ac4","schema_version":"1.0","event_id":"sha256:550cee02d38b34e2ed5b9a1ffac18ffa5502b00c79f18fd40cb4387269518ac4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:JRD7C6IPNNZPMJFC7B5LSY3FZM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Navier-Stokes equations on the $\\beta$-plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Djoko Wirosoetisno, Mustafa Al-Jaboori","submitted_at":"2010-09-23T08:13:28Z","abstract_excerpt":"We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $\\beta$-plane (i.e.\\ with the Coriolis force varying as $f_0+\\beta y$) will become nearly zonal: with the vorticity $\\omega(x,y,t)=\\wb(y,t)+\\wt(x,y,t)$, one has $|\\wt|_{H^s}^2\\le\\beta^{-1} M_s(\\...)$ as $t\\to\\infty$. We use this show that, for sufficiently large $\\beta$, the global attractor of this system reduces to a point."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4538","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":"2026-05-18T04:40:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rRVfj6FLqbzSdf01xCAsciYUmIPfFDeQjUfJHxYGauJSJfhUwjCF/pY60pgOsAelZuJg0edfUXkRz2kwrSYUBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:28:51.745615Z"},"content_sha256":"45bf79ac513f9b91855b90f2e69286adc09f504299cbb85c8e19da0aa4875fd0","schema_version":"1.0","event_id":"sha256:45bf79ac513f9b91855b90f2e69286adc09f504299cbb85c8e19da0aa4875fd0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JRD7C6IPNNZPMJFC7B5LSY3FZM/bundle.json","state_url":"https://pith.science/pith/JRD7C6IPNNZPMJFC7B5LSY3FZM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JRD7C6IPNNZPMJFC7B5LSY3FZM/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":"2026-06-01T04:28:51Z","links":{"resolver":"https://pith.science/pith/JRD7C6IPNNZPMJFC7B5LSY3FZM","bundle":"https://pith.science/pith/JRD7C6IPNNZPMJFC7B5LSY3FZM/bundle.json","state":"https://pith.science/pith/JRD7C6IPNNZPMJFC7B5LSY3FZM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JRD7C6IPNNZPMJFC7B5LSY3FZM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:JRD7C6IPNNZPMJFC7B5LSY3FZM","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":"3d6abb82688c9a240585ee818b2b94169fef1f922fe8a219f90f72c1796e9b68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-23T08:13:28Z","title_canon_sha256":"e92127d313b300e96859a2e98932d203692ace363da3ff1c144643614a238439"},"schema_version":"1.0","source":{"id":"1009.4538","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4538","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4538v1","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4538","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"pith_short_12","alias_value":"JRD7C6IPNNZP","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JRD7C6IPNNZPMJFC","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JRD7C6IP","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:45bf79ac513f9b91855b90f2e69286adc09f504299cbb85c8e19da0aa4875fd0","target":"graph","created_at":"2026-05-18T04:40:30Z","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":"We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $\\beta$-plane (i.e.\\ with the Coriolis force varying as $f_0+\\beta y$) will become nearly zonal: with the vorticity $\\omega(x,y,t)=\\wb(y,t)+\\wt(x,y,t)$, one has $|\\wt|_{H^s}^2\\le\\beta^{-1} M_s(\\...)$ as $t\\to\\infty$. We use this show that, for sufficiently large $\\beta$, the global attractor of this system reduces to a point.","authors_text":"Djoko Wirosoetisno, Mustafa Al-Jaboori","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-23T08:13:28Z","title":"Navier-Stokes equations on the $\\beta$-plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4538","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":"sha256:550cee02d38b34e2ed5b9a1ffac18ffa5502b00c79f18fd40cb4387269518ac4","target":"record","created_at":"2026-05-18T04:40:30Z","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":"3d6abb82688c9a240585ee818b2b94169fef1f922fe8a219f90f72c1796e9b68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-23T08:13:28Z","title_canon_sha256":"e92127d313b300e96859a2e98932d203692ace363da3ff1c144643614a238439"},"schema_version":"1.0","source":{"id":"1009.4538","kind":"arxiv","version":1}},"canonical_sha256":"4c47f1790f6b72f624a2f87ab96365cb343ed87a9d1e3105f3c22cb57a3a44d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c47f1790f6b72f624a2f87ab96365cb343ed87a9d1e3105f3c22cb57a3a44d9","first_computed_at":"2026-05-18T04:40:30.198563Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:30.198563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rycUKBWOd0w5qRplzGOUehsI+e3Qd+RG2mlnobqgbD7nZXfXyo9qkyHFy3wLAqoAw/CVA42x8Dkc9oN1zIIRAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:30.199232Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.4538","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:550cee02d38b34e2ed5b9a1ffac18ffa5502b00c79f18fd40cb4387269518ac4","sha256:45bf79ac513f9b91855b90f2e69286adc09f504299cbb85c8e19da0aa4875fd0"],"state_sha256":"fb93ea3831029df8a51c474312c20be396ccc8c9a22ece2d9fa20a7d9b18191d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w09AxfjWjvigG8MmkDNR3sQdtwzHxHDdsyQGcUqT0Hq1NZhlwtIkpTnFvjPsjArmR76sNer9UJiuKbjZCvhtBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T04:28:51.747807Z","bundle_sha256":"67c43382c3277933699d8a94c1fcc963abedd5cee1d06199d5680a68b488a5f7"}}