{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NS6HQ2LP2H4UVHFHQ243OOPWNM","short_pith_number":"pith:NS6HQ2LP","canonical_record":{"source":{"id":"1812.09388","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-21T22:01:05Z","cross_cats_sorted":[],"title_canon_sha256":"1cb074394594bc78036d1dd55c0cbf240f78ba49229943cb86ced5dcf0fa1735","abstract_canon_sha256":"906af021e7c6cd8a3a50c2e0a8ce9c6ab9a057eb5a1db35ca3adccac54b008dc"},"schema_version":"1.0"},"canonical_sha256":"6cbc78696fd1f94a9ca786b9b739f66b3df253581e7bd0f99c286bcc107c643e","source":{"kind":"arxiv","id":"1812.09388","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.09388","created_at":"2026-05-17T23:44:29Z"},{"alias_kind":"arxiv_version","alias_value":"1812.09388v2","created_at":"2026-05-17T23:44:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.09388","created_at":"2026-05-17T23:44:29Z"},{"alias_kind":"pith_short_12","alias_value":"NS6HQ2LP2H4U","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NS6HQ2LP2H4UVHFH","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NS6HQ2LP","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NS6HQ2LP2H4UVHFHQ243OOPWNM","target":"record","payload":{"canonical_record":{"source":{"id":"1812.09388","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-21T22:01:05Z","cross_cats_sorted":[],"title_canon_sha256":"1cb074394594bc78036d1dd55c0cbf240f78ba49229943cb86ced5dcf0fa1735","abstract_canon_sha256":"906af021e7c6cd8a3a50c2e0a8ce9c6ab9a057eb5a1db35ca3adccac54b008dc"},"schema_version":"1.0"},"canonical_sha256":"6cbc78696fd1f94a9ca786b9b739f66b3df253581e7bd0f99c286bcc107c643e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:29.687912Z","signature_b64":"Ed+JqVQvUumHTgOo3psOJVOtf+1XjhYww0l7d9nU06KNdhT9LUXvFGPgm0qA6GoGEFCfF+ctjWu0q9CGf8axAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6cbc78696fd1f94a9ca786b9b739f66b3df253581e7bd0f99c286bcc107c643e","last_reissued_at":"2026-05-17T23:44:29.687379Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:29.687379Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.09388","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":"2026-05-17T23:44:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bx5zDTa290svq8W9kT9M9aayvO51SSDxCsFUUsngtfwGVZvAaxDCkpifVp7JXOgG9dAOQLBfCVCPFyUNBu+OCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T15:02:37.147709Z"},"content_sha256":"2acece4f65d95b19cacb5d1d77722376e55440648f354fc7555356b0fd49f39f","schema_version":"1.0","event_id":"sha256:2acece4f65d95b19cacb5d1d77722376e55440648f354fc7555356b0fd49f39f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NS6HQ2LP2H4UVHFHQ243OOPWNM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity of Boltzmann equation with external fields in convex domains of diffuse reflection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yunbai Cao","submitted_at":"2018-12-21T22:01:05Z","abstract_excerpt":"We consider the Boltzmann equation with external fields in strictly convex domains with diffuse reflection boundary condition. As long as the normal derivative of external fields satisfy some sign condition on the boundary (1.8) we construct classical $C^1$ solutions away from the grazing set. As a consequence we construct solutions of Vlasov-Poisson-Boltzmann system having bounded derivatives away from the grazing set (weighted $W^{1,\\infty}$ estimate). In particular this improves the recent regularity estimate of such system in weighted $W^{1,p}$ space for $p<6$ in [1]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.09388","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":"2026-05-17T23:44:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZXHwYspFwJXMxlcV/DHmXF92XOA7mhCj5vXmLapZPcXlpwaqJD86RQVPifevgjwQx6yxlR4h6u1qb77z4YVpCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T15:02:37.148322Z"},"content_sha256":"5db01446c95d7c6c2fff4685c9716e3ff3d43db974a3abe4b6f008cb1f8a8c7f","schema_version":"1.0","event_id":"sha256:5db01446c95d7c6c2fff4685c9716e3ff3d43db974a3abe4b6f008cb1f8a8c7f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NS6HQ2LP2H4UVHFHQ243OOPWNM/bundle.json","state_url":"https://pith.science/pith/NS6HQ2LP2H4UVHFHQ243OOPWNM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NS6HQ2LP2H4UVHFHQ243OOPWNM/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":"2026-06-04T15:02:37Z","links":{"resolver":"https://pith.science/pith/NS6HQ2LP2H4UVHFHQ243OOPWNM","bundle":"https://pith.science/pith/NS6HQ2LP2H4UVHFHQ243OOPWNM/bundle.json","state":"https://pith.science/pith/NS6HQ2LP2H4UVHFHQ243OOPWNM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NS6HQ2LP2H4UVHFHQ243OOPWNM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NS6HQ2LP2H4UVHFHQ243OOPWNM","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":"906af021e7c6cd8a3a50c2e0a8ce9c6ab9a057eb5a1db35ca3adccac54b008dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-21T22:01:05Z","title_canon_sha256":"1cb074394594bc78036d1dd55c0cbf240f78ba49229943cb86ced5dcf0fa1735"},"schema_version":"1.0","source":{"id":"1812.09388","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.09388","created_at":"2026-05-17T23:44:29Z"},{"alias_kind":"arxiv_version","alias_value":"1812.09388v2","created_at":"2026-05-17T23:44:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.09388","created_at":"2026-05-17T23:44:29Z"},{"alias_kind":"pith_short_12","alias_value":"NS6HQ2LP2H4U","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NS6HQ2LP2H4UVHFH","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NS6HQ2LP","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:5db01446c95d7c6c2fff4685c9716e3ff3d43db974a3abe4b6f008cb1f8a8c7f","target":"graph","created_at":"2026-05-17T23:44:29Z","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 consider the Boltzmann equation with external fields in strictly convex domains with diffuse reflection boundary condition. As long as the normal derivative of external fields satisfy some sign condition on the boundary (1.8) we construct classical $C^1$ solutions away from the grazing set. As a consequence we construct solutions of Vlasov-Poisson-Boltzmann system having bounded derivatives away from the grazing set (weighted $W^{1,\\infty}$ estimate). In particular this improves the recent regularity estimate of such system in weighted $W^{1,p}$ space for $p<6$ in [1].","authors_text":"Yunbai Cao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-21T22:01:05Z","title":"Regularity of Boltzmann equation with external fields in convex domains of diffuse reflection"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.09388","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":"sha256:2acece4f65d95b19cacb5d1d77722376e55440648f354fc7555356b0fd49f39f","target":"record","created_at":"2026-05-17T23:44:29Z","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":"906af021e7c6cd8a3a50c2e0a8ce9c6ab9a057eb5a1db35ca3adccac54b008dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-21T22:01:05Z","title_canon_sha256":"1cb074394594bc78036d1dd55c0cbf240f78ba49229943cb86ced5dcf0fa1735"},"schema_version":"1.0","source":{"id":"1812.09388","kind":"arxiv","version":2}},"canonical_sha256":"6cbc78696fd1f94a9ca786b9b739f66b3df253581e7bd0f99c286bcc107c643e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6cbc78696fd1f94a9ca786b9b739f66b3df253581e7bd0f99c286bcc107c643e","first_computed_at":"2026-05-17T23:44:29.687379Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:29.687379Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ed+JqVQvUumHTgOo3psOJVOtf+1XjhYww0l7d9nU06KNdhT9LUXvFGPgm0qA6GoGEFCfF+ctjWu0q9CGf8axAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:29.687912Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.09388","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2acece4f65d95b19cacb5d1d77722376e55440648f354fc7555356b0fd49f39f","sha256:5db01446c95d7c6c2fff4685c9716e3ff3d43db974a3abe4b6f008cb1f8a8c7f"],"state_sha256":"593368c37edc3f85aea412cd097c74f89fd3b38ffb9be58c881b012ab00c4783"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+no6HEpE6mmIDb+wQ9BY+Cz74rwFf10j8VReGM4khBargg1Pgp7UXNJzFcvS+8rzz4I+T85d25Oh82Ms3KsTAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T15:02:37.150513Z","bundle_sha256":"86bf2a4e097c8e76aad718d7253302ede8a7394312d04eb93d2385deed868026"}}