{"paper":{"title":"Optimal regularity for kinetic Fokker-Planck equations in domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Marvin Weidner, Xavier Ros-Oton","submitted_at":"2025-05-17T10:10:58Z","abstract_excerpt":"We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $\\Omega\\subset \\mathbb{R}^n$ with specular reflection condition, including Kolmogorov's equation $\\partial_t f +v\\cdot\\nabla_x f-\\Delta_v f=h$. Our main results establish the following:\n  - Solutions are always $C^\\infty$ in $t,v,x$ away from the grazing set $\\{x\\in\\partial\\Omega,\\ v\\cdot n_x=0\\}$.\n  - They are $C^{4,1}_{\\text{kin}}$ up to the grazing set.\n  - This regularity is optimal, i.e. we show that that they are in general not $C^5_{\\text{kin}}$.\n  These results show for the first time that solutio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.11943","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.11943/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}