Optimal regularity for kinetic Fokker-Planck equations in domains

2025/05/17 by Xavier Ros‐Oton, Marvin Weidner, Ros-Oton, Xavier +1 · 3 citations
Mathematics · #35B65 #35Q84 #82C40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.11943

openalex publication_date 2025/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains Ω⊂ ℝⁿ with specular reflection condition, including Kolmogorov's equation ∂_t f +v⋅∇_x f-Δ_v f=h. Our main results establish the following: - Solutions are always C^∞ in t,v,x away from the grazing set \x∈∂Ω, v⋅ n_x=0\. - They are C^4,1_kin up to the grazing set. - This regularity is optimal, i.e. we show that that they are in general not C⁵_kin. These results show for the first time that solutions are classical up to boundary, i.e. C¹_t,x and C²_v.

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