{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:GWZANSHY5F3UKKOASCHRG6OORD","short_pith_number":"pith:GWZANSHY","schema_version":"1.0","canonical_sha256":"35b206c8f8e9774529c0908f1379ce88f107af980c8a982217b6b2f3f76f82a5","source":{"kind":"arxiv","id":"2504.05722","version":4},"attestation_state":"computed","paper":{"title":"Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincar\\'e Inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lukang Sun","submitted_at":"2025-04-08T06:45:24Z","abstract_excerpt":"We study the Cauchy problem for a weighted porous medium equation on $\\R$ associated with a Gibbs probability measure $\\pi=e^{-V}$. Under a Poincar\\'e inequality for $\\pi$ and the convexity assumption on $V$, we prove well-posedness and uniqueness of non-negative weak solutions with initial data in $L^1(\\R,\\pi)$. We also establish an $L^1$--$L^p$ smoothing effect at every positive time. More precisely, for every admissible $p>1$, we show that the logarithm of the ratio between the $L^p(\\R,\\pi)$ norm of the solution and its conserved $L^1(\\R,\\pi)$ mass first decays at a super-exponential rate a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2504.05722","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-04-08T06:45:24Z","cross_cats_sorted":[],"title_canon_sha256":"3fd9188759f52ff8818e6cde8d5da7136cbf2e84dc342250d85ae7fc5c555d31","abstract_canon_sha256":"696bbc69e8e8951383fdfdcec5b86d6352553f636a77b4fc6ed697785a14568f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:39.242682Z","signature_b64":"bobelXfiaB++JZ1IZoM/emWa/LXCMEcfAlKbfJNT5PtU5JkEOmdGq0u859wi5ZhQbcN+jaIpO923622zL3MoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35b206c8f8e9774529c0908f1379ce88f107af980c8a982217b6b2f3f76f82a5","last_reissued_at":"2026-05-18T02:44:39.242138Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:39.242138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincar\\'e Inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lukang Sun","submitted_at":"2025-04-08T06:45:24Z","abstract_excerpt":"We study the Cauchy problem for a weighted porous medium equation on $\\R$ associated with a Gibbs probability measure $\\pi=e^{-V}$. Under a Poincar\\'e inequality for $\\pi$ and the convexity assumption on $V$, we prove well-posedness and uniqueness of non-negative weak solutions with initial data in $L^1(\\R,\\pi)$. We also establish an $L^1$--$L^p$ smoothing effect at every positive time. More precisely, for every admissible $p>1$, we show that the logarithm of the ratio between the $L^p(\\R,\\pi)$ norm of the solution and its conserved $L^1(\\R,\\pi)$ mass first decays at a super-exponential rate a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.05722","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2504.05722","created_at":"2026-05-18T02:44:39.242207+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.05722v4","created_at":"2026-05-18T02:44:39.242207+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.05722","created_at":"2026-05-18T02:44:39.242207+00:00"},{"alias_kind":"pith_short_12","alias_value":"GWZANSHY5F3U","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"GWZANSHY5F3UKKOA","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"GWZANSHY","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD","json":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD.json","graph_json":"https://pith.science/api/pith-number/GWZANSHY5F3UKKOASCHRG6OORD/graph.json","events_json":"https://pith.science/api/pith-number/GWZANSHY5F3UKKOASCHRG6OORD/events.json","paper":"https://pith.science/paper/GWZANSHY"},"agent_actions":{"view_html":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD","download_json":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD.json","view_paper":"https://pith.science/paper/GWZANSHY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.05722&json=true","fetch_graph":"https://pith.science/api/pith-number/GWZANSHY5F3UKKOASCHRG6OORD/graph.json","fetch_events":"https://pith.science/api/pith-number/GWZANSHY5F3UKKOASCHRG6OORD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD/action/storage_attestation","attest_author":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD/action/author_attestation","sign_citation":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD/action/citation_signature","submit_replication":"https://pith.science/pith/GWZANSHY5F3UKKOASCHRG6OORD/action/replication_record"}},"created_at":"2026-05-18T02:44:39.242207+00:00","updated_at":"2026-05-18T02:44:39.242207+00:00"}