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On a spatially inhomogeneous nonlinear Fokker–Planck equation: Cauchy problem and diffusion asymptotics

Francesca Anceschi and Yuzhe Zhu

Vol. 17 (2024), No. 2, 379–420

We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker–Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global-in-time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy, and barrier function methods.

nonlinear kinetic Fokker–Planck equation, well-posedness, regularity, diffusion asymptotics
Mathematical Subject Classification
Primary: 35A01, 35B40, 35Q35, 35Q84
Received: 25 February 2021
Revised: 25 May 2022
Accepted: 11 July 2022
Published: 6 March 2024
Francesca Anceschi
Dipartimento di Ingegneria Industriale e Scienze Matematiche
Università Politecnica delle Marche
Yuzhe Zhu
Département de mathématiques et applications
École normale supérieure - PSL

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