Contributions to Books:
A. Gerstenmayer, A. Jüngel:
"Analysis of a degenerate parabolic cross-diffusion system for ion transport";
in: "ASC Report 10/2017",
Vienna University of Technology,
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.
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.