Holder Continuity for a Family of Nonlocal Hypoelliptic Kinetic\n Equations

2018/12/21 by Logan F. Stokols, Stokols, Logan F.
Engineering · Mathematics · Physics and Astronomy · #35B65 #35H10 #35Q84 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.1812.09006

openalex publication_date 2018/12/21 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

In this work, Holder continuity is obtained for solutions to the nonlocal\nkinetic Fokker-Planck Equation, and to a family of related equations with\ngeneral integro-differential operators. These equations can be seen as a\ngeneralization of the Fokker-Planck Equation, or as a linearization of\nnon-cutoff Boltzmann. Difficulties arise because our equations are\nhypoelliptic, so we utilize the theory of averaging lemmas. Regularity is\nobtained using De Giorgi's method, so it does not depend on the regularity of\ninitial conditions or coefficients. This work assumes stronger constraints on\nthe nonlocal operator than in the work of Imbert and Silvestre [22], but allows\nunbounded source terms.\n

Related

• H "older regularity for hypoelliptic kinetic equations with rough\n diffusion coefficients
• Hölder continuity of solutions to hypoelliptic equations with bounded measurable coefficients
• De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions
• Holder estimates for kinetic Fokker-Planck equations up to the boundary
• Quantitative De Giorgi methods in kinetic theory for non-local operators