{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:ZO6QMS7VMI7FN5GR7RKIBTGABZ","short_pith_number":"pith:ZO6QMS7V","canonical_record":{"source":{"id":"0904.0390","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-02T13:46:07Z","cross_cats_sorted":[],"title_canon_sha256":"d2e17b9e2cc2a2677045a3f9a127a3e7dcb0e6e5cd83d333066cf6af83c9f3dd","abstract_canon_sha256":"3f64e256c1de27012ca0ffb7779f85c3a2b88b428c3f4949f5b80aff26fafe13"},"schema_version":"1.0"},"canonical_sha256":"cbbd064bf5623e56f4d1fc5480ccc00e7b58577fbbe2b1f59ae2bfabd001c9a4","source":{"kind":"arxiv","id":"0904.0390","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.0390","created_at":"2026-05-18T04:37:20Z"},{"alias_kind":"arxiv_version","alias_value":"0904.0390v1","created_at":"2026-05-18T04:37:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.0390","created_at":"2026-05-18T04:37:20Z"},{"alias_kind":"pith_short_12","alias_value":"ZO6QMS7VMI7F","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZO6QMS7VMI7FN5GR","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZO6QMS7V","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:ZO6QMS7VMI7FN5GR7RKIBTGABZ","target":"record","payload":{"canonical_record":{"source":{"id":"0904.0390","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-02T13:46:07Z","cross_cats_sorted":[],"title_canon_sha256":"d2e17b9e2cc2a2677045a3f9a127a3e7dcb0e6e5cd83d333066cf6af83c9f3dd","abstract_canon_sha256":"3f64e256c1de27012ca0ffb7779f85c3a2b88b428c3f4949f5b80aff26fafe13"},"schema_version":"1.0"},"canonical_sha256":"cbbd064bf5623e56f4d1fc5480ccc00e7b58577fbbe2b1f59ae2bfabd001c9a4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:37:20.879472Z","signature_b64":"UY/q72CZkXL3YfS/cD9wCxlCbVnI9TQAudHqfFGxm0yC1K1x8d3tZK+vX/ptjtUTs+bKTaOTMyoFhU+Aq2YPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbbd064bf5623e56f4d1fc5480ccc00e7b58577fbbe2b1f59ae2bfabd001c9a4","last_reissued_at":"2026-05-18T04:37:20.878985Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:37:20.878985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0904.0390","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":"2026-05-18T04:37:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GeLDulaU1V2r9rTWm6a0P8w/k/hBITvT6KWpU06fujd/6UqZbg07YnD/Nr7bbSKsyLrWjDP0rV6GWhswrt20AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:55:59.935333Z"},"content_sha256":"a597a9043216681801f9742e7921fd8e1ef6ca588fcc4d178b9dd6a2bb1b3b08","schema_version":"1.0","event_id":"sha256:a597a9043216681801f9742e7921fd8e1ef6ca588fcc4d178b9dd6a2bb1b3b08"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:ZO6QMS7VMI7FN5GR7RKIBTGABZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Long-time Behavior for Nonlinear Hydrodynamic System Modeling the Nematic Liquid Crystal Flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hao Wu","submitted_at":"2009-04-02T13:46:07Z","abstract_excerpt":"We study a simplified system of the original Ericksen--Leslie equations for the flow of nematic liquid crystals. This is a coupled non-parabolic dissipative dynamic system. We show the convergence of global classical solutions to single steady states as time goes to infinity (uniqueness of asymptotic limit) by using the \\L ojasiewicz--Simon approach. Moreover, we provide an estimate on the convergence rate. Finally, we discuss some possible extensions of the results to certain generalized problems with changing density or free-slip boundary condition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.0390","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":"2026-05-18T04:37:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h9RuJyX/JqWJCwGGdu3wcNXKhQCenCJsUL5sumwI06vc7mJcHrQ44YNNnUJpTMltB/KRUCYO58nT9F/dTclRCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:55:59.935729Z"},"content_sha256":"59b7ebd39d8c3ce3f319146b79259b7f6a9cda0c9c2207f447ad63957de6350e","schema_version":"1.0","event_id":"sha256:59b7ebd39d8c3ce3f319146b79259b7f6a9cda0c9c2207f447ad63957de6350e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZO6QMS7VMI7FN5GR7RKIBTGABZ/bundle.json","state_url":"https://pith.science/pith/ZO6QMS7VMI7FN5GR7RKIBTGABZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZO6QMS7VMI7FN5GR7RKIBTGABZ/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":"2026-05-25T11:55:59Z","links":{"resolver":"https://pith.science/pith/ZO6QMS7VMI7FN5GR7RKIBTGABZ","bundle":"https://pith.science/pith/ZO6QMS7VMI7FN5GR7RKIBTGABZ/bundle.json","state":"https://pith.science/pith/ZO6QMS7VMI7FN5GR7RKIBTGABZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZO6QMS7VMI7FN5GR7RKIBTGABZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ZO6QMS7VMI7FN5GR7RKIBTGABZ","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":"3f64e256c1de27012ca0ffb7779f85c3a2b88b428c3f4949f5b80aff26fafe13","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-02T13:46:07Z","title_canon_sha256":"d2e17b9e2cc2a2677045a3f9a127a3e7dcb0e6e5cd83d333066cf6af83c9f3dd"},"schema_version":"1.0","source":{"id":"0904.0390","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.0390","created_at":"2026-05-18T04:37:20Z"},{"alias_kind":"arxiv_version","alias_value":"0904.0390v1","created_at":"2026-05-18T04:37:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.0390","created_at":"2026-05-18T04:37:20Z"},{"alias_kind":"pith_short_12","alias_value":"ZO6QMS7VMI7F","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZO6QMS7VMI7FN5GR","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZO6QMS7V","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:59b7ebd39d8c3ce3f319146b79259b7f6a9cda0c9c2207f447ad63957de6350e","target":"graph","created_at":"2026-05-18T04:37:20Z","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 study a simplified system of the original Ericksen--Leslie equations for the flow of nematic liquid crystals. This is a coupled non-parabolic dissipative dynamic system. We show the convergence of global classical solutions to single steady states as time goes to infinity (uniqueness of asymptotic limit) by using the \\L ojasiewicz--Simon approach. Moreover, we provide an estimate on the convergence rate. Finally, we discuss some possible extensions of the results to certain generalized problems with changing density or free-slip boundary condition.","authors_text":"Hao Wu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-02T13:46:07Z","title":"Long-time Behavior for Nonlinear Hydrodynamic System Modeling the Nematic Liquid Crystal Flows"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.0390","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:a597a9043216681801f9742e7921fd8e1ef6ca588fcc4d178b9dd6a2bb1b3b08","target":"record","created_at":"2026-05-18T04:37:20Z","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":"3f64e256c1de27012ca0ffb7779f85c3a2b88b428c3f4949f5b80aff26fafe13","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-02T13:46:07Z","title_canon_sha256":"d2e17b9e2cc2a2677045a3f9a127a3e7dcb0e6e5cd83d333066cf6af83c9f3dd"},"schema_version":"1.0","source":{"id":"0904.0390","kind":"arxiv","version":1}},"canonical_sha256":"cbbd064bf5623e56f4d1fc5480ccc00e7b58577fbbe2b1f59ae2bfabd001c9a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cbbd064bf5623e56f4d1fc5480ccc00e7b58577fbbe2b1f59ae2bfabd001c9a4","first_computed_at":"2026-05-18T04:37:20.878985Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:37:20.878985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UY/q72CZkXL3YfS/cD9wCxlCbVnI9TQAudHqfFGxm0yC1K1x8d3tZK+vX/ptjtUTs+bKTaOTMyoFhU+Aq2YPCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:37:20.879472Z","signed_message":"canonical_sha256_bytes"},"source_id":"0904.0390","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a597a9043216681801f9742e7921fd8e1ef6ca588fcc4d178b9dd6a2bb1b3b08","sha256:59b7ebd39d8c3ce3f319146b79259b7f6a9cda0c9c2207f447ad63957de6350e"],"state_sha256":"36c081eed41f20a9789031f07a8d4189291a7e6537ac7a673981a95e5e22a3fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6oA8i4/s4QC2AM/5F4ByzKqeLYFqIcvYaLYkuS5BTe3hG+R6zwbISxqmoY3B95qGjMlLrJjQj0gRTb8wHS4QBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T11:55:59.938625Z","bundle_sha256":"16b28ced77434fba67b13a29767d63c6d48c1b6b506d501c9a45cf1e0d44cc32"}}