# Analysis of a degenerate parabolic cross-diffusion system for ion transport

## TL;DR

This paper analyzes a complex cross-diffusion system modeling ion transport through biological membranes, establishing existence, uniqueness, and numerical insights into long-term behavior despite mathematical challenges.

## Contribution

It extends the boundedness-by-entropy method to nonhomogeneous boundary conditions and proves global existence and uniqueness for a degenerate, coupled ion transport system.

## Key findings

- Existence of bounded weak solutions is proven.
- Uniqueness of solutions is established under certain regularity.
- Numerical simulations show slow equilibration rates.

## Abstract

A cross-diffusion system describing ion transport through biological membranes or nanopores in a bounded domain with mixed Dirichlet-Neumann boundary conditions is analyzed. The ion concentrations solve strongly coupled diffusion equations with a drift term involving the electric potential which is coupled to the concentrations through a Poisson equation. The global-in-time existence of bounded weak solutions and the uniqueness of weak solutions under moderate regularity assumptions are shown. The main difficulties of the analysis are the cross-diffusion terms and the degeneracy of the diffusion matrix, preventing the use of standard tools. The proofs are based on the boundedness-by-entropy method, extended to nonhomogeneous boundary conditions, and the uniqueness technique of Gajewski. A finite-volume discretization in one space dimension illustrates the large-time behavior of the numerical solutions and shows that the equilibration rates may be very small.