{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:5Q43YJFWQPII7JG4OQZ3XKVUG7","short_pith_number":"pith:5Q43YJFW","canonical_record":{"source":{"id":"1412.8267","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-29T06:39:07Z","cross_cats_sorted":[],"title_canon_sha256":"db071853bce48a744b708f3fd2beed7762ce703338caca49692ca107202dc75a","abstract_canon_sha256":"5f1b8e098a76767a3e0e22d749ebfa0c142f04f27aa93ddc3681f4404f864797"},"schema_version":"1.0"},"canonical_sha256":"ec39bc24b683d08fa4dc7433bbaab437f346151b1e6076efce9e568289b09c8c","source":{"kind":"arxiv","id":"1412.8267","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.8267","created_at":"2026-05-18T01:19:06Z"},{"alias_kind":"arxiv_version","alias_value":"1412.8267v1","created_at":"2026-05-18T01:19:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8267","created_at":"2026-05-18T01:19:06Z"},{"alias_kind":"pith_short_12","alias_value":"5Q43YJFWQPII","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5Q43YJFWQPII7JG4","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5Q43YJFW","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:5Q43YJFWQPII7JG4OQZ3XKVUG7","target":"record","payload":{"canonical_record":{"source":{"id":"1412.8267","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-29T06:39:07Z","cross_cats_sorted":[],"title_canon_sha256":"db071853bce48a744b708f3fd2beed7762ce703338caca49692ca107202dc75a","abstract_canon_sha256":"5f1b8e098a76767a3e0e22d749ebfa0c142f04f27aa93ddc3681f4404f864797"},"schema_version":"1.0"},"canonical_sha256":"ec39bc24b683d08fa4dc7433bbaab437f346151b1e6076efce9e568289b09c8c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:06.255163Z","signature_b64":"Sq70OusoAaavNuNGySH9y0VU4LyHTXd9Ijr0yrgd8lveUhwTDNvY/0nFWHtWNVVnEIlS5mKvM9T/MDIyActCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec39bc24b683d08fa4dc7433bbaab437f346151b1e6076efce9e568289b09c8c","last_reissued_at":"2026-05-18T01:19:06.254565Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:06.254565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.8267","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":"2026-05-18T01:19:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I8YIWdREEQQ6t1X83Wm4bBvsOyXkDbfoMEAbVqYs7x75yE/emGy95g+T2kZ9FP2OnCwWhceqW7ZtyS/usn4MBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:27:25.463791Z"},"content_sha256":"449bcc5e943d8216d0fc878193cb70e677aeaeb47c233b6552e687264c8a3c1f","schema_version":"1.0","event_id":"sha256:449bcc5e943d8216d0fc878193cb70e677aeaeb47c233b6552e687264c8a3c1f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:5Q43YJFWQPII7JG4OQZ3XKVUG7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Shangkun Weng","submitted_at":"2014-12-29T06:39:07Z","abstract_excerpt":"In this paper, we first address the space-time decay properties for higher order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions in scaling invariant function spaces. We further investigate the asymptotic profiles and decay properties of these strong solution"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8267","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":"2026-05-18T01:19:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u1WLiV38O5xdKaW3lnkVlk4NCHpwhCagAXe78togP+ANXwPDSLN6mWpKiY1APf1uQEoHP3srxOJD3lb8sYfrCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:27:25.464481Z"},"content_sha256":"483910cfd911dfb144567a1195a338bb7ed473b88288b9cfb12e8cb1d6567d3d","schema_version":"1.0","event_id":"sha256:483910cfd911dfb144567a1195a338bb7ed473b88288b9cfb12e8cb1d6567d3d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5Q43YJFWQPII7JG4OQZ3XKVUG7/bundle.json","state_url":"https://pith.science/pith/5Q43YJFWQPII7JG4OQZ3XKVUG7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5Q43YJFWQPII7JG4OQZ3XKVUG7/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":"2026-05-27T06:27:25Z","links":{"resolver":"https://pith.science/pith/5Q43YJFWQPII7JG4OQZ3XKVUG7","bundle":"https://pith.science/pith/5Q43YJFWQPII7JG4OQZ3XKVUG7/bundle.json","state":"https://pith.science/pith/5Q43YJFWQPII7JG4OQZ3XKVUG7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5Q43YJFWQPII7JG4OQZ3XKVUG7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5Q43YJFWQPII7JG4OQZ3XKVUG7","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":"5f1b8e098a76767a3e0e22d749ebfa0c142f04f27aa93ddc3681f4404f864797","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-29T06:39:07Z","title_canon_sha256":"db071853bce48a744b708f3fd2beed7762ce703338caca49692ca107202dc75a"},"schema_version":"1.0","source":{"id":"1412.8267","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.8267","created_at":"2026-05-18T01:19:06Z"},{"alias_kind":"arxiv_version","alias_value":"1412.8267v1","created_at":"2026-05-18T01:19:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8267","created_at":"2026-05-18T01:19:06Z"},{"alias_kind":"pith_short_12","alias_value":"5Q43YJFWQPII","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5Q43YJFWQPII7JG4","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5Q43YJFW","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:483910cfd911dfb144567a1195a338bb7ed473b88288b9cfb12e8cb1d6567d3d","target":"graph","created_at":"2026-05-18T01:19:06Z","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 first address the space-time decay properties for higher order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions in scaling invariant function spaces. We further investigate the asymptotic profiles and decay properties of these strong solution","authors_text":"Shangkun Weng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-29T06:39:07Z","title":"Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8267","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":"sha256:449bcc5e943d8216d0fc878193cb70e677aeaeb47c233b6552e687264c8a3c1f","target":"record","created_at":"2026-05-18T01:19:06Z","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":"5f1b8e098a76767a3e0e22d749ebfa0c142f04f27aa93ddc3681f4404f864797","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-29T06:39:07Z","title_canon_sha256":"db071853bce48a744b708f3fd2beed7762ce703338caca49692ca107202dc75a"},"schema_version":"1.0","source":{"id":"1412.8267","kind":"arxiv","version":1}},"canonical_sha256":"ec39bc24b683d08fa4dc7433bbaab437f346151b1e6076efce9e568289b09c8c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec39bc24b683d08fa4dc7433bbaab437f346151b1e6076efce9e568289b09c8c","first_computed_at":"2026-05-18T01:19:06.254565Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:06.254565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sq70OusoAaavNuNGySH9y0VU4LyHTXd9Ijr0yrgd8lveUhwTDNvY/0nFWHtWNVVnEIlS5mKvM9T/MDIyActCCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:06.255163Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.8267","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:449bcc5e943d8216d0fc878193cb70e677aeaeb47c233b6552e687264c8a3c1f","sha256:483910cfd911dfb144567a1195a338bb7ed473b88288b9cfb12e8cb1d6567d3d"],"state_sha256":"ea50d760d5922fa3c9a6fc1ee4f6079c674a3cfecbf2ff6deaf9a0d7527192c0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3/z2XWQwKBUTw2phFdZiGl33TmblRwMLultFx0ck4md6lrDOql5C6RPiLi0vBE7Pu+6DICS50We4AkbiVNxzBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T06:27:25.468454Z","bundle_sha256":"58752ae5f46ee2698555479b1889bb85a71dadf79772fa3fdc7cac10a5f6653a"}}