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.