Nonlinear Fokker–Planck Equation: Stability, Distance and the Corresponding Extremal Problem in the Spatially Inhomogeneous Case

Abstract

We start with a global Maxwellian M _k , which is a stationary solution, with the constant total density $(\rho(t) \equiv {\tilde{\rho}})$, of the Fokker–Planck equation. The notion of distance between the function M _k and an arbitrary solution f (with the same total density ${\tilde{\rho}}$ at the fixed moment t) of the Fokker–Planck equation is introduced. In this way, we essentially generalize the important Kullback–Leibler distance, which was studied before. Using this generalization, we show local stability of the global Maxwellians in the spatially inhomogeneous case. We compare also the energy and entropy in the classical and quantum cases.

Mathematics Subject Classification (2010). Primary 35Q20, 82B40; Secondary 51K99.