Nonlinear inhomogeneous Fokker-Planck models: energetic-variational structures and long time behavior

TL;DR

This paper analyzes a nonlinear inhomogeneous Fokker-Planck model inspired by grain growth, developing long-term behavior analysis through an extended entropy method and supporting results with numerical tests.

Contribution

It introduces a novel analytical framework for nonlinear inhomogeneous Fokker-Planck equations with variable mobility, extending classical entropy methods.

Findings

Established large time asymptotic behavior of the model

Extended entropy method for inhomogeneous diffusion

Numerical tests validating analytical results

Abstract

Inspired by the modeling of grain growth in polycrystalline materials, we consider a nonlinear Fokker-Plank model, with inhomogeneous diffusion and with variable mobility parameters. We develop large time asymptotic analysis of such nonstandard models by reformulating and extending the classical entropy method, under the assumption of periodic boundary condition. In addition, illustrative numerical tests are presented to highlight the essential points of the current analytical results and to motivate future analysis.