Analysis of a degenerate parabolic cross-diffusion system for ion transport
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.
Keywords
Cite
@article{arxiv.1706.07261,
title = {Analysis of a degenerate parabolic cross-diffusion system for ion transport},
author = {Anita Gerstenmayer and Ansgar Jüngel},
journal= {arXiv preprint arXiv:1706.07261},
year = {2017}
}