Near‐ and far‐field expansions for stationary solutions of Poisson‐Nernst‐Planck equations

This work is concerned with the stationary Poisson‐Nernst‐Planck equation with a large parameter which describes a huge number of ions occupying an electrolytic region. First, we focus on the model with a single specie of positive charges in one‐dimensional bounded domains due to the assumption that these ions are transported in the same direction along a tubular‐like mircodomain. We show that the solution asymptotically blows up in a thin region attached to the boundary and establishes the refined “near‐field” and “far‐field” expansions for the solutions with respect to the parameter. Moreover, we obtain the boundary concentration phenomenon of the net charge density, which mathematically confirms the physical description that the non‐neutral phenomenon occurs near the charged surface. In addition, we revisit a charge‐conserving Poisson‐Boltzmann model for monovalent binary ions and establish a novel comparison for these two models.