{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:TJFY7ZH3HYLWITIFST4RMZ5HH6","short_pith_number":"pith:TJFY7ZH3","schema_version":"1.0","canonical_sha256":"9a4b8fe4fb3e17644d0594f91667a73fb4aa9d054b892111de5a0186c392e8b4","source":{"kind":"arxiv","id":"1403.7491","version":2},"attestation_state":"computed","paper":{"title":"Self-similar solutions for a fractional thin film equation governing hydraulic fractures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Antoine Mellet, Cyril Imbert (LAMA)","submitted_at":"2014-03-28T19:11:31Z","abstract_excerpt":"In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient $K\\geq 0$. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness ($K=0$) and the finite toughness $K\\in(0,\\infty)$ cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all "},"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":"1403.7491","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-03-28T19:11:31Z","cross_cats_sorted":[],"title_canon_sha256":"7e36a71901281cec66822b3b7063d03fe3d7aaa27864dd1b39b25aab4c15c116","abstract_canon_sha256":"548f242731ef059fa3c43afd93b111463a510f0035ac48ee43449bff1fe69fc4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:29.869228Z","signature_b64":"Shbh96FjVf+6nCUXT6fIG33kTYdSTAA2soAHs97dyBcgkB4SwSZl8Ad+dX5h/bBS0coGL8abQCU8Dp1XNQlNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a4b8fe4fb3e17644d0594f91667a73fb4aa9d054b892111de5a0186c392e8b4","last_reissued_at":"2026-05-18T01:23:29.868562Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:29.868562Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self-similar solutions for a fractional thin film equation governing hydraulic fractures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Antoine Mellet, Cyril Imbert (LAMA)","submitted_at":"2014-03-28T19:11:31Z","abstract_excerpt":"In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient $K\\geq 0$. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness ($K=0$) and the finite toughness $K\\in(0,\\infty)$ cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7491","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1403.7491","created_at":"2026-05-18T01:23:29.868676+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.7491v2","created_at":"2026-05-18T01:23:29.868676+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.7491","created_at":"2026-05-18T01:23:29.868676+00:00"},{"alias_kind":"pith_short_12","alias_value":"TJFY7ZH3HYLW","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"TJFY7ZH3HYLWITIF","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"TJFY7ZH3","created_at":"2026-05-18T12:28:49.207871+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/TJFY7ZH3HYLWITIFST4RMZ5HH6","json":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6.json","graph_json":"https://pith.science/api/pith-number/TJFY7ZH3HYLWITIFST4RMZ5HH6/graph.json","events_json":"https://pith.science/api/pith-number/TJFY7ZH3HYLWITIFST4RMZ5HH6/events.json","paper":"https://pith.science/paper/TJFY7ZH3"},"agent_actions":{"view_html":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6","download_json":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6.json","view_paper":"https://pith.science/paper/TJFY7ZH3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.7491&json=true","fetch_graph":"https://pith.science/api/pith-number/TJFY7ZH3HYLWITIFST4RMZ5HH6/graph.json","fetch_events":"https://pith.science/api/pith-number/TJFY7ZH3HYLWITIFST4RMZ5HH6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6/action/storage_attestation","attest_author":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6/action/author_attestation","sign_citation":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6/action/citation_signature","submit_replication":"https://pith.science/pith/TJFY7ZH3HYLWITIFST4RMZ5HH6/action/replication_record"}},"created_at":"2026-05-18T01:23:29.868676+00:00","updated_at":"2026-05-18T01:23:29.868676+00:00"}