{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:SKQAAVFC7X4G72Q42PBS7QCK55","short_pith_number":"pith:SKQAAVFC","schema_version":"1.0","canonical_sha256":"92a00054a2fdf86fea1cd3c32fc04aef5400074dfb1602135c0a056a4b8b7557","source":{"kind":"arxiv","id":"0806.1180","version":1},"attestation_state":"computed","paper":{"title":"Incompressible flow in porous media with fractional diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. Castro, D. Cordoba, F. Gancedo, R. Orive","submitted_at":"2008-06-06T14:55:50Z","abstract_excerpt":"In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critica"},"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":"0806.1180","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-06-06T14:55:50Z","cross_cats_sorted":[],"title_canon_sha256":"b9a77aaa273a901619ba313e585b7ba2c392ac4874a374366e3d77af8d958ffe","abstract_canon_sha256":"f1552a28531ca4eaf65bf3ee23aca5bdb8d76131782b86001de555459ac19e7e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:51.858055Z","signature_b64":"0gDjyXQSiyA5ZDROccpFRGa1ROo6bZz+uaXZpP/Wr0XNlU8WtIUojuGAw3yAFmJT9rCFDag1FTiOPgAwtyKfAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92a00054a2fdf86fea1cd3c32fc04aef5400074dfb1602135c0a056a4b8b7557","last_reissued_at":"2026-05-18T02:15:51.857621Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:51.857621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Incompressible flow in porous media with fractional diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. Castro, D. Cordoba, F. Gancedo, R. Orive","submitted_at":"2008-06-06T14:55:50Z","abstract_excerpt":"In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critica"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.1180","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0806.1180","created_at":"2026-05-18T02:15:51.857687+00:00"},{"alias_kind":"arxiv_version","alias_value":"0806.1180v1","created_at":"2026-05-18T02:15:51.857687+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.1180","created_at":"2026-05-18T02:15:51.857687+00:00"},{"alias_kind":"pith_short_12","alias_value":"SKQAAVFC7X4G","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"SKQAAVFC7X4G72Q4","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"SKQAAVFC","created_at":"2026-05-18T12:25:58.018023+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/SKQAAVFC7X4G72Q42PBS7QCK55","json":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55.json","graph_json":"https://pith.science/api/pith-number/SKQAAVFC7X4G72Q42PBS7QCK55/graph.json","events_json":"https://pith.science/api/pith-number/SKQAAVFC7X4G72Q42PBS7QCK55/events.json","paper":"https://pith.science/paper/SKQAAVFC"},"agent_actions":{"view_html":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55","download_json":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55.json","view_paper":"https://pith.science/paper/SKQAAVFC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0806.1180&json=true","fetch_graph":"https://pith.science/api/pith-number/SKQAAVFC7X4G72Q42PBS7QCK55/graph.json","fetch_events":"https://pith.science/api/pith-number/SKQAAVFC7X4G72Q42PBS7QCK55/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55/action/storage_attestation","attest_author":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55/action/author_attestation","sign_citation":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55/action/citation_signature","submit_replication":"https://pith.science/pith/SKQAAVFC7X4G72Q42PBS7QCK55/action/replication_record"}},"created_at":"2026-05-18T02:15:51.857687+00:00","updated_at":"2026-05-18T02:15:51.857687+00:00"}