Chia-Yu Hsieh

In this paper, we consider radial solutions of the Poisson-Nernst-Planck (PNP) system with variable dielectric coefficients $\varepsilon g(x)$ in $N$-dimensional annular domains, $N≥2$. When the parameter $\varepsilon$ tends to zero, the PNP system admits a boundary layer solution as a steady state, which satisfies the charge conserving Poisson-Boltzmann (CCPB) equation. For the stability of the radial boundary layer solutions to the time-dependent radial PNP system, we estimate the radial solution of the perturbed problem with global electroneutrality. We generalize the argument of the one spatial dimension case (cf. [18]) and find a new way to transform the perturbed problem. By choosing a suitable weighted norm, we then derive the associated energy law which can be used to prove that the $H^{-1}_x$ norm of the solution of the perturbed problem decays exponentially in time with the exponent independent of $\varepsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound.

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 ...