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

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.