{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:F4NDQXEU6BCI2PZKJAAQ3IHDGR","short_pith_number":"pith:F4NDQXEU","canonical_record":{"source":{"id":"1109.5328","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-25T06:54:17Z","cross_cats_sorted":[],"title_canon_sha256":"213f2cee0e4324a1fdf5482fe938ec4527cade26ca6cf9712028c4acecdba8c9","abstract_canon_sha256":"68bcef0dc7d4085232a56753d7ba7efb6f7c5f30d219ccbd6676d49d18c2fb76"},"schema_version":"1.0"},"canonical_sha256":"2f1a385c94f0448d3f2a48010da0e334560f36dee14f6674fd89ab90fce55144","source":{"kind":"arxiv","id":"1109.5328","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5328","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5328v2","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5328","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"pith_short_12","alias_value":"F4NDQXEU6BCI","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"F4NDQXEU6BCI2PZK","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"F4NDQXEU","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:F4NDQXEU6BCI2PZKJAAQ3IHDGR","target":"record","payload":{"canonical_record":{"source":{"id":"1109.5328","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-25T06:54:17Z","cross_cats_sorted":[],"title_canon_sha256":"213f2cee0e4324a1fdf5482fe938ec4527cade26ca6cf9712028c4acecdba8c9","abstract_canon_sha256":"68bcef0dc7d4085232a56753d7ba7efb6f7c5f30d219ccbd6676d49d18c2fb76"},"schema_version":"1.0"},"canonical_sha256":"2f1a385c94f0448d3f2a48010da0e334560f36dee14f6674fd89ab90fce55144","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:38.801891Z","signature_b64":"1TUAK0glxeoo4adDl9ANlKg44OTVcXq6TDpUq1gE/g/RKgjgpHb7oqFmnZQfMltkxbsrpQjuf2Yi0feVL7ufDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f1a385c94f0448d3f2a48010da0e334560f36dee14f6674fd89ab90fce55144","last_reissued_at":"2026-05-18T03:57:38.801028Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:38.801028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.5328","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":"2026-05-18T03:57:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QwBzvMfxlnZF3eDurTg9KGbq4e7Hyre69QfSp5BKDQ4gKK8dPfRp6KSIH5DIFnMKOQK18NJ12B61qjPSYSW0Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:47:08.085124Z"},"content_sha256":"17dcaf097afe6d65b942d36cef7d5994c6cad69d67ce6c43982b5d400dc107e3","schema_version":"1.0","event_id":"sha256:17dcaf097afe6d65b942d36cef7d5994c6cad69d67ce6c43982b5d400dc107e3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:F4NDQXEU6BCI2PZKJAAQ3IHDGR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changjiang Zhu, Huanyao Wen","submitted_at":"2011-09-25T06:54:17Z","abstract_excerpt":"First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate {\\it a priori} estimates globally in time which depend on the assumption $\\kappa=O(1+\\theta^q)$ where $q>r$ ($r$ can be zero), which relaxes the condition $q\\ge2+2r$ in [14,29,42]. Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5328","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":"2026-05-18T03:57:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BRJCwodwH7qjx12uvxuyo42lk1LMI6/NRgx2eWyrCVauCP6tvsPYzzviYrISGf45rqGE6aHqA+Kahaxis/kNCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:47:08.085754Z"},"content_sha256":"4e9fc410aa8343e5f85af7a311a46699db5ca4f0fbcac8bbafc8063426ade74a","schema_version":"1.0","event_id":"sha256:4e9fc410aa8343e5f85af7a311a46699db5ca4f0fbcac8bbafc8063426ade74a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F4NDQXEU6BCI2PZKJAAQ3IHDGR/bundle.json","state_url":"https://pith.science/pith/F4NDQXEU6BCI2PZKJAAQ3IHDGR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F4NDQXEU6BCI2PZKJAAQ3IHDGR/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":"2026-05-27T14:47:08Z","links":{"resolver":"https://pith.science/pith/F4NDQXEU6BCI2PZKJAAQ3IHDGR","bundle":"https://pith.science/pith/F4NDQXEU6BCI2PZKJAAQ3IHDGR/bundle.json","state":"https://pith.science/pith/F4NDQXEU6BCI2PZKJAAQ3IHDGR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F4NDQXEU6BCI2PZKJAAQ3IHDGR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:F4NDQXEU6BCI2PZKJAAQ3IHDGR","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":"68bcef0dc7d4085232a56753d7ba7efb6f7c5f30d219ccbd6676d49d18c2fb76","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-25T06:54:17Z","title_canon_sha256":"213f2cee0e4324a1fdf5482fe938ec4527cade26ca6cf9712028c4acecdba8c9"},"schema_version":"1.0","source":{"id":"1109.5328","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5328","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5328v2","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5328","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"pith_short_12","alias_value":"F4NDQXEU6BCI","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"F4NDQXEU6BCI2PZK","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"F4NDQXEU","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:4e9fc410aa8343e5f85af7a311a46699db5ca4f0fbcac8bbafc8063426ade74a","target":"graph","created_at":"2026-05-18T03:57:38Z","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":"First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate {\\it a priori} estimates globally in time which depend on the assumption $\\kappa=O(1+\\theta^q)$ where $q>r$ ($r$ can be zero), which relaxes the condition $q\\ge2+2r$ in [14,29,42]. Th","authors_text":"Changjiang Zhu, Huanyao Wen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-25T06:54:17Z","title":"Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5328","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":"sha256:17dcaf097afe6d65b942d36cef7d5994c6cad69d67ce6c43982b5d400dc107e3","target":"record","created_at":"2026-05-18T03:57:38Z","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":"68bcef0dc7d4085232a56753d7ba7efb6f7c5f30d219ccbd6676d49d18c2fb76","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-25T06:54:17Z","title_canon_sha256":"213f2cee0e4324a1fdf5482fe938ec4527cade26ca6cf9712028c4acecdba8c9"},"schema_version":"1.0","source":{"id":"1109.5328","kind":"arxiv","version":2}},"canonical_sha256":"2f1a385c94f0448d3f2a48010da0e334560f36dee14f6674fd89ab90fce55144","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f1a385c94f0448d3f2a48010da0e334560f36dee14f6674fd89ab90fce55144","first_computed_at":"2026-05-18T03:57:38.801028Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:38.801028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1TUAK0glxeoo4adDl9ANlKg44OTVcXq6TDpUq1gE/g/RKgjgpHb7oqFmnZQfMltkxbsrpQjuf2Yi0feVL7ufDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:38.801891Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.5328","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17dcaf097afe6d65b942d36cef7d5994c6cad69d67ce6c43982b5d400dc107e3","sha256:4e9fc410aa8343e5f85af7a311a46699db5ca4f0fbcac8bbafc8063426ade74a"],"state_sha256":"95454595c1c60dbe5752d68db8b51c65835069415da896bfee668eba667c5148"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mAHlRtO0ZG98w0/BdqkWHTiP8xHQZLimDH+Si3E7Hq569COuryEnI4EAPxyPMn02vKp4bcS6SKkeBy2exeKFAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:47:08.089349Z","bundle_sha256":"0bafcc3355f7a4183578882e1179d9346847fa63266836bc30baa9abfb3d864d"}}