Long-time behavior of free energy in the nonlinear Fokker-Planck equation

TL;DR

This paper investigates the long-term behavior of free energy in nonlinear Fokker-Planck equations with spatially inhomogeneous nonlinear diffusion, extending entropy dissipation methods to analyze asymptotic properties.

Contribution

It introduces a novel approach to analyze the asymptotic behavior of inhomogeneous nonlinear diffusion in Fokker-Planck equations using energy dissipation laws.

Findings

Long-time decay of dissipation function for large diffusion coefficients

Extension of entropy dissipation method to inhomogeneous diffusion

Characterization of free energy behavior in nonlinear Fokker-Planck equations

Abstract

We study the asymptotic behavior of Fokker-Planck equations with spatially inhomogeneous nonlinear diffusion, based on the energy dissipation law. First, we consider the Fokker-Planck equation with porous-medium-type nonlinear diffusion that satisfies the energy dissipation law by introducing spatial inhomogeneity into the free energy. We obtain a result on the long-time behavior of the dissipation function for sufficiently large diffusion coefficients by extending the entropy dissipation method to the case of inhomogeneous diffusion.