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

Anita Gerstenmayer; Ansgar Jüngel

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arXiv · 1706.07261

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

Anita Gerstenmayer·Ansgar Jüngel

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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.

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Anita Gerstenmayer, Ansgar Jüngel. 2017-06-22. Analysis of a degenerate parabolic cross-diffusion system for ion transport. https://arxiv.org/abs/1706.07261

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