{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:42TA2BCLJ4G4N2O5R4RKHRJLTX","short_pith_number":"pith:42TA2BCL","canonical_record":{"source":{"id":"1204.3938","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-17T23:12:11Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"5b2e1fb68f51fa93f736aa4ace4785beb9976a9ca5aad0fe494d4438c201f15f","abstract_canon_sha256":"25486deb5d22c6f35ef38b1e83ef06ec3ab614f3798f083ebfe5cb755924295c"},"schema_version":"1.0"},"canonical_sha256":"e6a60d044b4f0dc6e9dd8f22a3c52b9dfaa3030a2b13855f4afa9d7f8199c521","source":{"kind":"arxiv","id":"1204.3938","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.3938","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"arxiv_version","alias_value":"1204.3938v1","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.3938","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"pith_short_12","alias_value":"42TA2BCLJ4G4","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"42TA2BCLJ4G4N2O5","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"42TA2BCL","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:42TA2BCLJ4G4N2O5R4RKHRJLTX","target":"record","payload":{"canonical_record":{"source":{"id":"1204.3938","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-17T23:12:11Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"5b2e1fb68f51fa93f736aa4ace4785beb9976a9ca5aad0fe494d4438c201f15f","abstract_canon_sha256":"25486deb5d22c6f35ef38b1e83ef06ec3ab614f3798f083ebfe5cb755924295c"},"schema_version":"1.0"},"canonical_sha256":"e6a60d044b4f0dc6e9dd8f22a3c52b9dfaa3030a2b13855f4afa9d7f8199c521","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:38.229554Z","signature_b64":"BqSANtDtyV7EVtL9zv1ElDql4eQmAVQikJHidk8Qpi26rdC/ZvRNjQ44PLXNOYaW4yFiwxaLy+w1awJp35K+Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6a60d044b4f0dc6e9dd8f22a3c52b9dfaa3030a2b13855f4afa9d7f8199c521","last_reissued_at":"2026-05-18T03:57:38.228875Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:38.228875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.3938","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":"2026-05-18T03:57:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5q+gggtCARJHFMuXO1s20hyw1Oohlo3V6npYQOilDH2CjL5JM3QxQZcfCpnHGZ5z/VnNxWrgX5fD732pXpq3BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T01:45:32.321602Z"},"content_sha256":"5f001ab318d5e353a0645c8699272a93d4958ade91501aeb5913365b501d7b5d","schema_version":"1.0","event_id":"sha256:5f001ab318d5e353a0645c8699272a93d4958ade91501aeb5913365b501d7b5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:42TA2BCLJ4G4N2O5R4RKHRJLTX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An aggregation equation with degenerate diffusion: qualitative property of solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Inwon Kim, Lincoln Chayes, Yao Yao","submitted_at":"2012-04-17T23:12:11Z","abstract_excerpt":"We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKean-Vlasov equation where here the \"diffusive\" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on $m\\in(1,2]$ with particular emphasis on $m = 2$. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For $m=2$, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3938","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":"2026-05-18T03:57:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2uHWb/WbgrwMfUNOozOKAKyjdOe55xsiPIv16yIx/N0EfEf1rRTtjLvE/jvWiSVA3UqmX9F1yANd4wl5nW08AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T01:45:32.322257Z"},"content_sha256":"38d84b50596b6e39854d0409047d6a278ab9895aa9df0547eeec284a7f57eba8","schema_version":"1.0","event_id":"sha256:38d84b50596b6e39854d0409047d6a278ab9895aa9df0547eeec284a7f57eba8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/42TA2BCLJ4G4N2O5R4RKHRJLTX/bundle.json","state_url":"https://pith.science/pith/42TA2BCLJ4G4N2O5R4RKHRJLTX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/42TA2BCLJ4G4N2O5R4RKHRJLTX/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":"2026-05-26T01:45:32Z","links":{"resolver":"https://pith.science/pith/42TA2BCLJ4G4N2O5R4RKHRJLTX","bundle":"https://pith.science/pith/42TA2BCLJ4G4N2O5R4RKHRJLTX/bundle.json","state":"https://pith.science/pith/42TA2BCLJ4G4N2O5R4RKHRJLTX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/42TA2BCLJ4G4N2O5R4RKHRJLTX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:42TA2BCLJ4G4N2O5R4RKHRJLTX","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":"25486deb5d22c6f35ef38b1e83ef06ec3ab614f3798f083ebfe5cb755924295c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-17T23:12:11Z","title_canon_sha256":"5b2e1fb68f51fa93f736aa4ace4785beb9976a9ca5aad0fe494d4438c201f15f"},"schema_version":"1.0","source":{"id":"1204.3938","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.3938","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"arxiv_version","alias_value":"1204.3938v1","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.3938","created_at":"2026-05-18T03:57:38Z"},{"alias_kind":"pith_short_12","alias_value":"42TA2BCLJ4G4","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"42TA2BCLJ4G4N2O5","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"42TA2BCL","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:38d84b50596b6e39854d0409047d6a278ab9895aa9df0547eeec284a7f57eba8","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":"We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKean-Vlasov equation where here the \"diffusive\" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on $m\\in(1,2]$ with particular emphasis on $m = 2$. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For $m=2$, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional ","authors_text":"Inwon Kim, Lincoln Chayes, Yao Yao","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-17T23:12:11Z","title":"An aggregation equation with degenerate diffusion: qualitative property of solutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3938","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":"sha256:5f001ab318d5e353a0645c8699272a93d4958ade91501aeb5913365b501d7b5d","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":"25486deb5d22c6f35ef38b1e83ef06ec3ab614f3798f083ebfe5cb755924295c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-04-17T23:12:11Z","title_canon_sha256":"5b2e1fb68f51fa93f736aa4ace4785beb9976a9ca5aad0fe494d4438c201f15f"},"schema_version":"1.0","source":{"id":"1204.3938","kind":"arxiv","version":1}},"canonical_sha256":"e6a60d044b4f0dc6e9dd8f22a3c52b9dfaa3030a2b13855f4afa9d7f8199c521","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6a60d044b4f0dc6e9dd8f22a3c52b9dfaa3030a2b13855f4afa9d7f8199c521","first_computed_at":"2026-05-18T03:57:38.228875Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:38.228875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BqSANtDtyV7EVtL9zv1ElDql4eQmAVQikJHidk8Qpi26rdC/ZvRNjQ44PLXNOYaW4yFiwxaLy+w1awJp35K+Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:38.229554Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.3938","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5f001ab318d5e353a0645c8699272a93d4958ade91501aeb5913365b501d7b5d","sha256:38d84b50596b6e39854d0409047d6a278ab9895aa9df0547eeec284a7f57eba8"],"state_sha256":"8f2216449417b0c56424b8545ac40e2d14e4e071bc17c1048596876bbcfb8eb6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7RKyURUM9JiO+ezUa0xhEZpo9DKBRfO0sUBItqRBuiw2iBvG4k2qMoyTUSScyBg0pZgeS6pQQ73gYMhA+Kp1DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T01:45:32.326108Z","bundle_sha256":"b5899d3bbfb1db0e853982cd6ac53840c5e4307a9b75a26c063ba67565094f47"}}