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arXiv · 2105.06626

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

Jhih-Hong Lyu·Chiun-Chang Lee·Tai-Chia Lin
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Abstract

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. Firstly, 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 establish 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 nonlocal Poisson--Boltzmann model for monovalent binary ions and establish a novel comparison for these two models.

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Jhih-Hong Lyu, Chiun-Chang Lee, Tai-Chia Lin. 2021-05-14. Near- and far-field expansions for stationary solutions of Poisson--Nernst--Planck equations. https://doi.org/10.1002/mma.7453

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