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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 4

### Boundary layer solutions of charge conserving Poisson–Boltzmann equations: One-dimensional case

Pages: 911 – 940

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a2

#### Authors

#### Abstract

For multispecies ions, we study boundary layer solutions of charge conserving Poisson–Boltzmann (CCPB) equations [L. Wan, S. Xu, M. Liao, C. Liu, and P. Sheng, *Phys. Rev.* X 4, 011042, 2014] (with a small parameter $\epsilon$) over a finite one-dimensional (1D) spatial domain, subjected to Robin-type boundary conditions with variable coefficients. Hereafter, 1D boundary layer solutions mean that as $\epsilon$ approaches zero, the profiles of solutions form boundary layers near boundary points and become flat in the interior domain. These solutions are related to electric double layers with many applications in biology and physics. We rigorously prove the asymptotic behaviors of 1D boundary layer solutions at interior and boundary points. The asymptotic limits of the solution values (electric potentials) at interior and boundary points with a potential gap (related to zeta potential) are uniquely determined by explicit nonlinear formulas (cannot be found in classical Poisson–Boltzmann equations) which are solvable by numerical computations.

#### Keywords

charge conserving Poisson–Boltzmann equations, boundary layer, multispecies ions

#### 2010 Mathematics Subject Classification

65C30, 76A05, 76M99

Published 6 April 2016