Exponential Decay Estimates for the Stability of Boundary Layer Solutions to Poisson--Nernst--Planck Systems: One Spatial Dimension Case

Abstract

With a small parameter $\varepsilon$ approaching zero, Poisson--Nernst--Planck (PNP) systems over a finite one dimensional (1D) spatial domain have steady state solutions, called 1D boundary layer solutions, and these profiles form boundary layers near boundary points and become flat in the interior domain. 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 $\varepsilon$ 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 $\varepsilon$ 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.