Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion

TL;DR

This paper studies a three-dimensional Poisson-Nernst-Planck system with nonlinear diffusion, proving exponential convergence to a unique equilibrium under certain conditions using entropy methods.

Contribution

It establishes exponential convergence to equilibrium for a nonlinear diffusion Poisson-Nernst-Planck system with small electric drift, extending previous existence results.

Findings

Existence of a unique steady state under convex external potentials.

Exponential convergence of solutions to the steady state when electric drift is small.

Application of entropy-dissipation techniques to analyze long-time behavior.

Abstract

We investigate a Poisson-Nernst-Planck type system in three spatial dimensions where the strength of the electric drift depends on a possibly small parameter and the particles are assumed to diffuse quadratically. On grounds of the global existence result proved by Kinderlehrer, Monsaingeon and Xu (2015) using the formal Wasserstein gradient flow structure of the system, we analyse the long-time behaviour of weak solutions. We prove under the assumption of uniform convexity of the external drift potentials that the system possesses a unique steady state. If the strength of the electric drift is sufficiently small, we show convergence of solutions to the respective steady state at an exponential rate using entropy-dissipation methods.