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Poisson-Nernst-Planck Systems for Narrow Tubular-like Membrane Channels

We study global dynamics of the Poisson-Nernst-Planck (PNP) system for flows of two types of ions through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previous studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous to that of the limiting PNP system. We then examine the dynamics of the one-dimensional limiting PNP system. For large Debye number, the steady-state of the one-dimensional limiting PNP system is completed analyzed using the geometric singular perturbation theory. For a special case, an entropy-type Lyapunov functional is constructed to show the global, asymptotic stability of the steady-state.