{"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"}