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

TL;DR

This paper studies a nonlinear Fokker-Planck equation with spatial inhomogeneity, establishing well-posedness, diffusion asymptotics, and connections to fast diffusion flows using advanced mathematical techniques.

Contribution

It provides the first global well-posedness and diffusion asymptotics results for a spatially inhomogeneous nonlinear Fokker-Planck model, linking kinetic and diffusive behaviors.

Findings

Proved global well-posedness with smoothness for initial data below Maxwellian.

Derived diffusion asymptotics connecting kinetic model to fast diffusion flow.

Established quantitative convergence to diffusion limit using entropic hypocoercivity.

Abstract

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.