Individual Flux Study via Steady-State Poisson--Nernst--Planck Systems: Effects from Boundary Conditions

We provide a detailed study for ionic flow through ion channels for the case with three ion species, two positively charged having the same valence and one negatively charged, and with zero permanent charge. The work is based on the general geometric theory developed in [W. Liu, J. Differential Equations, 246 (2009), pp. 428--451] for a quasi-one-dimensional steady-state Poisson--Nernst--Planck model. Our focus is on the effects of boundary conditions on the ionic flow. Beyond the existence of solutions of the model problem, we are able to obtain explicit approximations of individual fluxes and the current-voltage relations, from which effects of boundary conditions on ionic flows are examined in a great detail. Critical potentials are identified and their roles in characterizing these effects are studied. Compared to ionic mixtures with two ion species, a number of new features for mixtures of three ion species arise. Numerical simulations are performed, and numerical results are consistent with our analytical ones.