Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

Chia-Yu Hsieh

doi:10.3934/dcdsb.2018269

Article Contents

2019, Volume 24, Issue 6: 2657-2681. Doi: 10.3934/dcdsb.2018269

This issue Previous Article Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study Next Article Nondegenerate multistationarity in small reaction networks

Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

Chia-Yu Hsieh^,

National Center for Theoretical Sciences, National Taiwan University, Taipei City 10617, Taiwan

* Corresponding author: Chia-Yu Hsieh
* Corresponding author: Chia-Yu Hsieh

Received: December 31, 2017

Revised: May 31, 2018

Early access: October 2018

Published: June 2019

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Abstract

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.

Keywords:

Poisson-Nernst-Planck system,
charge conserving Poisson-Boltzmann equation,
variable dielectric,
boundary layer solution,
stability,
exponential decay estimate.

Mathematics Subject Classification: Primary: 35M33; Secondary: 35B25, 35B45.

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