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On a spatially
inhomogeneous nonlinear Fokker–Planck equation: Cauchy
problem and diffusion asymptotics
Francesca Anceschi and Yuzhe Zhu |
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Vol. 17 (2024), No. 2, 379–420
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Abstract |
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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. |
Keywords
nonlinear kinetic Fokker–Planck equation, well-posedness,
regularity, diffusion asymptotics
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Mathematical Subject Classification
Primary: 35A01, 35B40, 35Q35, 35Q84
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Milestones
Received: 25 February 2021
Revised: 25 May 2022
Accepted: 11 July 2022
Published: 6 March 2024
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Authors |
| Francesca Anceschi | |
| Dipartimento di Ingegneria
Industriale e Scienze Matematiche Università Politecnica delle Marche Ancona Italy |
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| Yuzhe Zhu | |
| Département de mathématiques et
applications École normale supérieure - PSL Paris France |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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