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.1. Introduction 379 2. Preliminaries 385 3. Kolmogorov-Fokker-Planck equation 387 4. Well-posedness of the nonlinear model 391 5. Diffusion asymptotics 403 Appendix A. Maximum principle 414 Appendix B. Spreading of positivity 415 Appendix C. Gaining regularity of spatial increment 417