Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The onedimensional steady-state Poisson-Nernst-Planck (PNP) system is a useful representation of these devices but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for two types of ions with three regions of ...

Abstract For ? &lt; 1 3 , we consider ${C}^{\beta }\left({\mathbb{T}}^{3}\times [0,T]\right)$?> C <mml:m...