Poisson-Nernst-Planck Systems for Narrow Tubular-like Membrane Channels

TL;DR

This paper derives a one-dimensional Poisson-Nernst-Planck model for narrow membrane channels from a three-dimensional system, analyzing its stability and dynamics, especially for large Debye numbers.

Contribution

It introduces a geometry-encoded 1D PNP system as a limit of 3D models and proves its stability and attractor properties.

Findings

Derived the 1D limiting PNP system from 3D models as cross-section radius approaches zero.

Proved the upper semi-continuity of global attractors from 3D to 1D systems.

Analyzed steady-states and stability for large Debye numbers using geometric singular perturbation theory.

Abstract

We study global dynamics of the Poisson-Nernst-Planck (PNP) system for flows of two types of ions through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previous studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous to that of the limiting PNP system. We then examine the dynamics of the one-dimensional limiting PNP system. For large Debye number, the steady-state of the one-dimensional limiting PNP system is completed analyzed using the geometric singular perturbation theory. For a special case, an entropy-type Lyapunov functional is constructed to show the global, asymptotic stability of the steady-state.