The Poisson-Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. In this paper we study a new Poisson-Boltzmann type (PB n) equation with a small dielectric parameter 2 and nonlocal nonlinearity which takes into consideration of the preservation of the total amount of each individual ion. This equation can be derived from the original Poisson-Nernst-Planck (PNP) system. Under Robin type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviors of one dimensional solutions of PB n equations as the parameter approaches to zero. In particular, we show that in case of electro-neutrality, i.e., α = β, solutions of 1-D PB n equations have the similar asymptotic behavior as those of 1-D PB equations. However, as α = β (local non-electroneutrality), solutions of 1-D PB n equations may have blow-up behavior which can not be found in 1-D PB equations. Such a difference between 1-D PB and PB n equations can also be verified by numerical simulations.

Abstract. For multispecies ions, we study boundary layer solutions of charge conserving PoissonBoltzmann (CCPB) equations [50] (with a small parameter ǫ) 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 ǫ 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.

This paper is concerned with the stationary problem of an aero-taxis system with physical boundary conditions proposed by Tuval et al (2005 Proc. Natl Acad. Sci. 102 2277 to describe the boundary layer formation in the air-uid interface in any dimensions. By considering a special case where uid is free, the stationary problem is essentially reduced to a singularly perturbed nonlocal semi-linear elliptic problem. Denoting the diffusion rate of oxygen by ε > 0, we show that the stationary problem admits a unique classical solution of boundarylayer pro le as ε → 0, where the boundary-layer thickness is of order ε. When the domain is a ball, we nd a re ned asymptotic boundary layer pro le up to the rst-order approximation of ε by which we nd that the slope of the layer pro le in the immediate vicinity of the boundary decreases with respect to (w.r.t.) the curvature while the boundary-layer thickness increases w.r.t. the curvature.