Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

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

Pages: 911 – 940



Chiun-Chang Lee (Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu, Taiwan)

Hijin Lee (Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea)

Yunkyong Hyon (Division of Mathematical Models, National Institute for Mathematical Sciences, Daejeon, Korea)

Tai-Chia Lin (Department of Mathematics, Center for Advanced Study in Theoretical Sciences (CASTS), National Center for Theoretical Sciences (NCTS), National Taiwan University, Taipei, Taiwan)

Chun Liu (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)


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.


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

2010 Mathematics Subject Classification

65C30, 76A05, 76M99

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