{"paper":{"title":"Kinetic Fokker-Planck Equations with Nonlinear Diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Xicheng Zhang, Zhengyan Wu, Zimo Hao","submitted_at":"2026-07-01T05:25:29Z","abstract_excerpt":"We study existence, regularity, and uniqueness for the nonlinear kinetic Fokker--Planck equation $$\n  \\partial_t f=\\Delta_v\\Psi(f)-v\\cdot\\nabla_x f,\n  \\qquad f|_{t=0}=f_0, $$ on $\\mathbb R^{2d}$. In the model case $\\Psi(r)=r^s$, this equation couples nonlinear fast-diffusion/porous-medium type diffusion with kinetic transport. A distinctive feature is that the diffusion acts only in the velocity variable $v$, so that compactness in the spatial variable $x$ cannot be obtained from standard elliptic estimates and must instead be recovered through the hypoelliptic structure.\n  Under general struc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00458","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/2607.00458/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"}