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.

中文翻译：

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Poisson-Nernst-Planck 方程平稳解的近场和远场扩展

这项工作涉及具有大参数的平稳泊松-能斯特-普朗克方程，该方程描述了占据电解区的大量离子。首先，我们关注在一维有界域中具有单一种类正电荷的模型，因为假设这些离子沿管状微域以相同方向传输。我们表明，该解在附着在边界上的一个薄区域中渐近膨胀，并为与参数相关的解建立了精细的“近场”和“远场”扩展。此外，我们获得了净电荷密度的边界集中现象，这从数学上证实了非中性现象发生在带电表面附近的物理描述。此外，