{"paper":{"title":"Towards optimal regularity for the fourth-order thin film equation in $\\re^N$: Graveleau-type focusing self-similarity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jonathan D. Evans, Pablo Alvarez-Caudevilla, Victor A. Galaktionov","submitted_at":"2015-05-06T07:37:20Z","abstract_excerpt":"An approach to some \"optimal\" (more precisely, non-improvable) regularity of solutions of the thin film equation u_{t} = -\\nabla \\cdot(|u|^{n} \\nabla \\D u) in \\ren \\times \\re_+, u(x,0)=u_0(x) in \\re^N, where n in (0,2) is a fixed exponent, with smooth compactly supported initial data u_0(x), in dimensions $N \\geq 2$ is discussed. Namely, a precise exponent for the H\\\"older continuity with respect to the spatial radial variable $|x|$ is obtained by construction of a Graveleau-type focusing self-similar solution. As a consequence, optimal regularity of the gradient $\\nabla u$ in certain $L^p$ sp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01267","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"}