{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:Q2EJ6GWR7RDDLPTVEKZXG2RQNE","short_pith_number":"pith:Q2EJ6GWR","canonical_record":{"source":{"id":"2109.04295","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-09-09T14:16:13Z","cross_cats_sorted":[],"title_canon_sha256":"4c0258067bed5d2a4e65aaa1c80e3f64ea8e6685cde72129f8cc978f391f086b","abstract_canon_sha256":"3c1ac3746c1dd4cbef07c8af9863d8919de20cfc9cb6c15fd3cd65fa7e71665e"},"schema_version":"1.0"},"canonical_sha256":"86889f1ad1fc4635be7522b3736a3069008159af0cfcb0e85c4c77c0512b81db","source":{"kind":"arxiv","id":"2109.04295","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.04295","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"arxiv_version","alias_value":"2109.04295v1","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.04295","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"pith_short_12","alias_value":"Q2EJ6GWR7RDD","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"pith_short_16","alias_value":"Q2EJ6GWR7RDDLPTV","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"pith_short_8","alias_value":"Q2EJ6GWR","created_at":"2026-07-05T03:12:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:Q2EJ6GWR7RDDLPTVEKZXG2RQNE","target":"record","payload":{"canonical_record":{"source":{"id":"2109.04295","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-09-09T14:16:13Z","cross_cats_sorted":[],"title_canon_sha256":"4c0258067bed5d2a4e65aaa1c80e3f64ea8e6685cde72129f8cc978f391f086b","abstract_canon_sha256":"3c1ac3746c1dd4cbef07c8af9863d8919de20cfc9cb6c15fd3cd65fa7e71665e"},"schema_version":"1.0"},"canonical_sha256":"86889f1ad1fc4635be7522b3736a3069008159af0cfcb0e85c4c77c0512b81db","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:12:57.078809Z","signature_b64":"UB7+zatVa458EJiUQbqAXHJUSjC8XJmZrv7IGv084qlVrv0D30No9fd17NI9rqbKkYeVCc5zsu6+xIR/7plHBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86889f1ad1fc4635be7522b3736a3069008159af0cfcb0e85c4c77c0512b81db","last_reissued_at":"2026-07-05T03:12:57.078445Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:12:57.078445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.04295","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":"2026-07-05T03:12:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SSQdF/k5VXOcm/fgsmYEJ8Vj+NNTmq7MXOz73vOW0RvpiBwGFeN30UfKKQBr+3efTOOMU+WkYHmzChN0s7X6Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:44:47.921515Z"},"content_sha256":"2643184cc11c27d2fe655aabd775d20ad5cbd9c47b7363825e53267650ef1bb0","schema_version":"1.0","event_id":"sha256:2643184cc11c27d2fe655aabd775d20ad5cbd9c47b7363825e53267650ef1bb0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:Q2EJ6GWR7RDDLPTVEKZXG2RQNE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Feimin Huang, Qian Yuan","submitted_at":"2021-09-09T14:16:13Z","abstract_excerpt":"The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $ \\mathbb R \\times \\mathbb T^{n-1} (n\\geq2) $, where $\\mathbb T^{n-1}$ is the $ n-1 $-dimensional torus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.04295","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.04295/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:12:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OTDbj9AMSfKziHnw95CpuQp1Ymcjt7zn7nPu7OnChvsa07pMjkNz0F5u0DKHS9EOGhWdyLrbTwfcLEGoJUskBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:44:47.921873Z"},"content_sha256":"fca18dd41d60c9715cdaea0992949b5d7d91b9244821f02986dd3a891f1ac55d","schema_version":"1.0","event_id":"sha256:fca18dd41d60c9715cdaea0992949b5d7d91b9244821f02986dd3a891f1ac55d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q2EJ6GWR7RDDLPTVEKZXG2RQNE/bundle.json","state_url":"https://pith.science/pith/Q2EJ6GWR7RDDLPTVEKZXG2RQNE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q2EJ6GWR7RDDLPTVEKZXG2RQNE/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":"2026-07-07T13:44:47Z","links":{"resolver":"https://pith.science/pith/Q2EJ6GWR7RDDLPTVEKZXG2RQNE","bundle":"https://pith.science/pith/Q2EJ6GWR7RDDLPTVEKZXG2RQNE/bundle.json","state":"https://pith.science/pith/Q2EJ6GWR7RDDLPTVEKZXG2RQNE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q2EJ6GWR7RDDLPTVEKZXG2RQNE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:Q2EJ6GWR7RDDLPTVEKZXG2RQNE","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":"3c1ac3746c1dd4cbef07c8af9863d8919de20cfc9cb6c15fd3cd65fa7e71665e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-09-09T14:16:13Z","title_canon_sha256":"4c0258067bed5d2a4e65aaa1c80e3f64ea8e6685cde72129f8cc978f391f086b"},"schema_version":"1.0","source":{"id":"2109.04295","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.04295","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"arxiv_version","alias_value":"2109.04295v1","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.04295","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"pith_short_12","alias_value":"Q2EJ6GWR7RDD","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"pith_short_16","alias_value":"Q2EJ6GWR7RDDLPTV","created_at":"2026-07-05T03:12:57Z"},{"alias_kind":"pith_short_8","alias_value":"Q2EJ6GWR","created_at":"2026-07-05T03:12:57Z"}],"graph_snapshots":[{"event_id":"sha256:fca18dd41d60c9715cdaea0992949b5d7d91b9244821f02986dd3a891f1ac55d","target":"graph","created_at":"2026-07-05T03:12:57Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2109.04295/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $ \\mathbb R \\times \\mathbb T^{n-1} (n\\geq2) $, where $\\mathbb T^{n-1}$ is the $ n-1 $-dimensional torus.","authors_text":"Feimin Huang, Qian Yuan","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-09-09T14:16:13Z","title":"Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.04295","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":"sha256:2643184cc11c27d2fe655aabd775d20ad5cbd9c47b7363825e53267650ef1bb0","target":"record","created_at":"2026-07-05T03:12:57Z","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":"3c1ac3746c1dd4cbef07c8af9863d8919de20cfc9cb6c15fd3cd65fa7e71665e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-09-09T14:16:13Z","title_canon_sha256":"4c0258067bed5d2a4e65aaa1c80e3f64ea8e6685cde72129f8cc978f391f086b"},"schema_version":"1.0","source":{"id":"2109.04295","kind":"arxiv","version":1}},"canonical_sha256":"86889f1ad1fc4635be7522b3736a3069008159af0cfcb0e85c4c77c0512b81db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"86889f1ad1fc4635be7522b3736a3069008159af0cfcb0e85c4c77c0512b81db","first_computed_at":"2026-07-05T03:12:57.078445Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:12:57.078445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UB7+zatVa458EJiUQbqAXHJUSjC8XJmZrv7IGv084qlVrv0D30No9fd17NI9rqbKkYeVCc5zsu6+xIR/7plHBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:12:57.078809Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.04295","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2643184cc11c27d2fe655aabd775d20ad5cbd9c47b7363825e53267650ef1bb0","sha256:fca18dd41d60c9715cdaea0992949b5d7d91b9244821f02986dd3a891f1ac55d"],"state_sha256":"161802bcebfadc12fcf4500cc7ea329b0f58e86fd3eaf9d92840fcbeb1076a61"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gtDk98MRfk6bmkh2wsSmn23EXYinnyZW/f5FzDMLaXD354OayTFgCZquQ82fwTGvl7xmWxNf4VawpGQlmcxnBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T13:44:47.923674Z","bundle_sha256":"719c8436db539b6fb72e903f50592f2d3a682705853c6649b21e3d02d55041a2"}}