{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:6TCVMGUVB3PGRGVCRXCRR6TDPS","short_pith_number":"pith:6TCVMGUV","canonical_record":{"source":{"id":"1112.4724","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-12-20T15:24:29Z","cross_cats_sorted":[],"title_canon_sha256":"159542e8cb3629f188f8bc9cf3eaae1de547a5db763aa250dfd24bcb61216c0e","abstract_canon_sha256":"254c20c6be7b07dc48976e1d3ae2b3a75de8062615c507ae02b785eb3c1bb182"},"schema_version":"1.0"},"canonical_sha256":"f4c5561a950ede689aa28dc518fa637cb57a4012cbc3e0dd192c482462e76db5","source":{"kind":"arxiv","id":"1112.4724","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4724","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4724v2","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4724","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"pith_short_12","alias_value":"6TCVMGUVB3PG","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6TCVMGUVB3PGRGVC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6TCVMGUV","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:6TCVMGUVB3PGRGVCRXCRR6TDPS","target":"record","payload":{"canonical_record":{"source":{"id":"1112.4724","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-12-20T15:24:29Z","cross_cats_sorted":[],"title_canon_sha256":"159542e8cb3629f188f8bc9cf3eaae1de547a5db763aa250dfd24bcb61216c0e","abstract_canon_sha256":"254c20c6be7b07dc48976e1d3ae2b3a75de8062615c507ae02b785eb3c1bb182"},"schema_version":"1.0"},"canonical_sha256":"f4c5561a950ede689aa28dc518fa637cb57a4012cbc3e0dd192c482462e76db5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:59.764795Z","signature_b64":"h0ikdpfS/mqyTYQlwOjainkVHPMg+zU02424NN3lzLJ+b3qdcCKkkJgT0opyFWAjgV25BCFeVSGGAJkB5e2lDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4c5561a950ede689aa28dc518fa637cb57a4012cbc3e0dd192c482462e76db5","last_reissued_at":"2026-05-18T03:49:59.764371Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:59.764371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.4724","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":"2026-05-18T03:49:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QnfImjG8uTmkPJzDXxSlC5A/JaP4uK1/1l17vRlGqM2wALeIKa0NS6kuqlflkZpOEpwqw2mZ80L8wxRt45wBBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T10:47:52.169837Z"},"content_sha256":"ff70f004c3d5b49c16403681be84bd6161bfab2a77cd3ca767fa4c68ae75208e","schema_version":"1.0","event_id":"sha256:ff70f004c3d5b49c16403681be84bd6161bfab2a77cd3ca767fa4c68ae75208e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:6TCVMGUVB3PGRGVCRXCRR6TDPS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the BBM-Burgers Equation: Well-posedness, Ill-posedness and Long Period Limit","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Carlos Banquet Brango","submitted_at":"2011-12-20T15:24:29Z","abstract_excerpt":"In this work we study a dispersive equation with a dissipative term, the Benjamin-Bona-Mahony-Burgers equation. First we prove that the initial value problem for this equation is well-posed in $H^s(\\mathbb{R}),$ for $s\\geq 0$ and ill-posed if $s< 0.$ The ill-posedness is in the sense that the flow-map cannot be continuous at the origin from $H^s(\\mathbb{R})$ to even $\\mathcal{D}'(\\mathbb{R}).$ Additionally, we establish an exact theory of convergence of the periodic solutions to the continuous one, in Sobolev spaces, as the period goes to infinity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4724","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":"2026-05-18T03:49:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QieOvDv16CHfYXx1oNO66uHZkZxm74RQdYR1L6nKRcyeVFb/SWgtNAqvpZRW7U5pCyl4FJ+6jU2Y4ofWProTAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T10:47:52.170182Z"},"content_sha256":"1c7e150e310adbccfaa8ae55e27a459ca99a68285b05dc016ae4331469bc4159","schema_version":"1.0","event_id":"sha256:1c7e150e310adbccfaa8ae55e27a459ca99a68285b05dc016ae4331469bc4159"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6TCVMGUVB3PGRGVCRXCRR6TDPS/bundle.json","state_url":"https://pith.science/pith/6TCVMGUVB3PGRGVCRXCRR6TDPS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6TCVMGUVB3PGRGVCRXCRR6TDPS/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":"2026-06-05T10:47:52Z","links":{"resolver":"https://pith.science/pith/6TCVMGUVB3PGRGVCRXCRR6TDPS","bundle":"https://pith.science/pith/6TCVMGUVB3PGRGVCRXCRR6TDPS/bundle.json","state":"https://pith.science/pith/6TCVMGUVB3PGRGVCRXCRR6TDPS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6TCVMGUVB3PGRGVCRXCRR6TDPS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:6TCVMGUVB3PGRGVCRXCRR6TDPS","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":"254c20c6be7b07dc48976e1d3ae2b3a75de8062615c507ae02b785eb3c1bb182","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-12-20T15:24:29Z","title_canon_sha256":"159542e8cb3629f188f8bc9cf3eaae1de547a5db763aa250dfd24bcb61216c0e"},"schema_version":"1.0","source":{"id":"1112.4724","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4724","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4724v2","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4724","created_at":"2026-05-18T03:49:59Z"},{"alias_kind":"pith_short_12","alias_value":"6TCVMGUVB3PG","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6TCVMGUVB3PGRGVC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6TCVMGUV","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:1c7e150e310adbccfaa8ae55e27a459ca99a68285b05dc016ae4331469bc4159","target":"graph","created_at":"2026-05-18T03:49:59Z","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 work we study a dispersive equation with a dissipative term, the Benjamin-Bona-Mahony-Burgers equation. First we prove that the initial value problem for this equation is well-posed in $H^s(\\mathbb{R}),$ for $s\\geq 0$ and ill-posed if $s< 0.$ The ill-posedness is in the sense that the flow-map cannot be continuous at the origin from $H^s(\\mathbb{R})$ to even $\\mathcal{D}'(\\mathbb{R}).$ Additionally, we establish an exact theory of convergence of the periodic solutions to the continuous one, in Sobolev spaces, as the period goes to infinity.","authors_text":"Carlos Banquet Brango","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-12-20T15:24:29Z","title":"On the BBM-Burgers Equation: Well-posedness, Ill-posedness and Long Period Limit"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4724","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:ff70f004c3d5b49c16403681be84bd6161bfab2a77cd3ca767fa4c68ae75208e","target":"record","created_at":"2026-05-18T03:49:59Z","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":"254c20c6be7b07dc48976e1d3ae2b3a75de8062615c507ae02b785eb3c1bb182","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-12-20T15:24:29Z","title_canon_sha256":"159542e8cb3629f188f8bc9cf3eaae1de547a5db763aa250dfd24bcb61216c0e"},"schema_version":"1.0","source":{"id":"1112.4724","kind":"arxiv","version":2}},"canonical_sha256":"f4c5561a950ede689aa28dc518fa637cb57a4012cbc3e0dd192c482462e76db5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f4c5561a950ede689aa28dc518fa637cb57a4012cbc3e0dd192c482462e76db5","first_computed_at":"2026-05-18T03:49:59.764371Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:59.764371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h0ikdpfS/mqyTYQlwOjainkVHPMg+zU02424NN3lzLJ+b3qdcCKkkJgT0opyFWAjgV25BCFeVSGGAJkB5e2lDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:59.764795Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.4724","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ff70f004c3d5b49c16403681be84bd6161bfab2a77cd3ca767fa4c68ae75208e","sha256:1c7e150e310adbccfaa8ae55e27a459ca99a68285b05dc016ae4331469bc4159"],"state_sha256":"b0124cbb7a33a8dd27596c3df81c93a6c812538cd86b269dd0f5dcff5184339d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ri9S86L2XlatiZL9t7D843pD2Mq8rBAL7ivbqMZfFJiEcO2JrinM3jWuvlZsVC3WQ/YfLEGUBktgF+Xg4QkfCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T10:47:52.172117Z","bundle_sha256":"5908c2cbc1dc86b744980dfaec05a712e1bed477028288fb60149136541e72e3"}}