Asymptotic gradient flow structures of a nonlinear Fokker-Planck equation

TL;DR

This paper investigates a nonlinear Fokker-Planck equation modeling particle diffusion amidst obstacles, revealing its asymptotic gradient flow structure, analyzing its dynamics, and establishing existence and long-term behavior of solutions.

Contribution

It introduces the concept of asymptotic gradient flow for a nonlinear Fokker-Planck equation and explores its properties and dynamics as a simplified test case.

Findings

The equation can be interpreted as an asymptotic gradient flow.

Numerical simulations illustrate the dynamics of the system.

Global existence and long-time behavior of solutions are established.

Abstract

In this paper we consider a nonlinear Fokker-Planck equation with asymptotically small parameters. It describes the diffusion of finite-size particles in the presence of a fixed distribution of obstacles in the limit of low-volume fraction. The equation does not have a gradient flow structure, but can be interpreted as an asymptotic gradient flow, that is, as a gradient flow up to a certain asymptotic order. We use this scalar equation as a simple testbed model for more complicated systems of this kind. We discuss several possible entropy-mobility pairs, illustrate their dynamics with numerical simulations, present global in time existence results and study the long time behavior of solutions.