Poisson–Nernst–Planck Systems for Narrow Tubular-Like Membrane Channels
We study global asymptotic behavior of Poisson–Nernst–Planck (PNP) systems for flow of two ion species 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 previously 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 as the radius of the channel
tends to zero.