Analysis of a Poisson-Nernst-Planck cross-diffusion system with steric effects

TL;DR

This paper analyzes a transient Poisson-Nernst-Planck system with steric effects, establishing existence, uniqueness, and long-term behavior of solutions using entropy methods, and illustrates findings with numerical experiments.

Contribution

It introduces a rigorous analysis of a Poisson-Nernst-Planck system with steric effects, proving existence, uniqueness, and decay properties under mixed boundary conditions.

Findings

Proved existence of global weak solutions.

Established weak-strong uniqueness property.

Demonstrated exponential decay to equilibrium in certain cases.

Abstract

A transient Poisson-Nernst-Planck system with steric effects is analyzed in a bounded domain with no-flux boundary conditions for the ion concentrations and mixed Dirichlet-Neumann boundary conditions for the electric potential. The steric repulsion of ions is modeled by a localized Lennard-Jones force, leading to cross-diffusion terms. The existence of global weak solutions, a weak--strong uniqueness property, and, in case of pure Neumann conditions, the exponential decay towards the thermal equilibrium state is proved. The main difficulties are the cross-diffusion terms and the different boundary conditions satisfied by the unknowns. These issues are overcome by exploiting the entropy structure of the equations and carefully taking into account the electric potential term. A numerical experiment illustrates the long-time behavior of the solutions when the potential satisfies mixed boundary conditions.