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A. Gerstenmayer, A. Jüngel:
"Analysis of a degenerate parabolic cross-diffusion system for ion transport";
in: "ASC Report 10/2017", Vienna University of Technology, Wien, 2017, ISBN: 978-3-902627-10-0, 1 - 24.

A cross-diffusion system describing ion transport through biological mem-branes or nanopores in a bounded domain with mixed Dirichlet-Neumann boundary con-ditions 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 di-mension illustrates the large-time behavior of the numerical solutions and shows that the equilibration rates may be very small.

Ion transport, existence of weak solutions, free energy, entropy method, unique-ness of weak solutions, finite-volume approximation.

http://www.asc.tuwien.ac.at/preprint/2017/asc10x2017.pdf