Exponential decay estimates for the stability of boundary layer solutions to Poisson-Nernst-Planck systems: one spatial dimension case

TL;DR

This paper establishes exponential decay estimates for the stability of boundary layer solutions in 1D Poisson-Nernst-Planck systems, showing decay rates are independent of small parameters under certain boundary conditions.

Contribution

It provides the first exponential decay estimates for boundary layer solutions in 1D PNP systems with a novel energy law approach.

Findings

Exponential decay of the solution in the $H^{-1}_x$ norm independent of $\epsilon$

Boundary layer gradients may blow up as $\epsilon$ approaches zero

Transforming the problem yields a useful energy law for stability analysis

Abstract

With a small parameter $\epsilon$, Poisson-Nernst-Planck (PNP) systems over a finite one-dimensional (1D) spatial domain have steady state solutions, called 1D boundary layer solutions, which profiles form boundary layers near boundary points and become at in the interior domain as $\epsilon$ approaches zero. For the stability of 1D boundary layer solutions to (time-dependent) PNP systems, we estimate the solution of the perturbed problem with global electroneutrality. We prove that the $H^{-1}_x$ norm of the solution of the perturbed problem decays exponentially (in time) with exponent independent of $\epsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound. The main difficulty is that the gradients of 1D boundary layer solutions at boundary points may blow up as $\epsilon$ tends to zero. The main idea of our argument is to transform the perturbed problem into another parabolic system with a new and useful energy law for the proof of the exponential decay estimate.