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.

Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

“…Assumption (A1) is supposed for simplicity only. Mixed Dirichlet-Neumann boundary conditions can be included in the analysis (see, e.g., [6]), but the proofs become even more technical. Mixed boundary conditions are chosen in the numerical experiments; therefore, the numerical scheme is defined for that case.…”

“…On the discrete level, we replace u 0 w Φ by an upwind approximation. This allows us to apply the discrete maximum principle showing that u 0 ≥ 0 and hence u = (u 1 , … , u n ) ∈  with  defined in (6). Finally, Assumption (A3) is needed to derive a discrete version of the entropy inequality.…”

“…Our aim is to design a numerical approximation of (1) which preserves the structural properties of the continuous equations under simplifying assumptions (the coefficients D i are the same and the drift term vanishes). This suggests to use the entropy variables as the unknowns, as it was done in our previous work [6] with simulations in one space dimension. Unfortunately, we have not been able to perform a numerical convergence analysis with these variables.…”

“…If additionally the drift part vanishes, the solution is unique. The existence proof uses a topological degree argument in finite space dimensions, while the uniqueness proof is based on the entropy method of Gajewski [7], recently extended to cross-diffusion systems [6,8]. • If D i = D for all i, the scheme preserves the nonnegativity and upper bound for the concentrations.…”

Corrector estimates in homogenization of a nonlinear transmission problem for diffusion equations in connected domains

“…The corresponding PDE system obeys a structure of the gradient flow; see, eg, other works. [17][18][19] The paper 20 considers the homogenization over a two-phase domain for static PNP equations and homogeneous interface conditions. In Kovtunenko and Zubkova, 21 residual error estimates for the averaged monodomain solution with first-order correctors were justified under the simplifying assumption that the flux across the interface is of order O( 2 ).…”