Communications in Mathematical Sciences

Volume 17 (2019)

Number 8

Existence and local uniqueness for the Stokes-Nernst-Planck-drift-diffusion-Poisson system modeling nanopore and nanowire sensors

Pages: 2089 – 2112



Leila Taghizadeh (Institute for Analysis and Scientific Computing, Technical University Vienna, Austria)

Clemens Heitzinger (Institute for Analysis and Scientific Computing, Technical University Vienna, Austria)


This work gives analytical results for a system of transport equations which is the underlying mathematical model for nanopore sensors and for all types of affinity-based nanowire sensors. This model consists of the Poisson equation for the electrostatic potential ensuring self-consistency and including interface conditions stemming from a homogenized boundary layer, the drift-diffusion equations describing the transport of charge carriers in the sensor, the Nernst–Planck equations describing the transport of ions, and the Stokes equations describing the flow of the background medium water. We present existence and local uniqueness theorems for this stationary, nonlinear, and fully coupled system. The existence proof is based on the Schauder fixed-point theorem and local uniqueness around equilibrium is obtained from the implicit-function theorem. The maximum principle is used to obtain a priori estimates for the solution. Due to the multiscale problem inherent in affinity-based field-effect sensors, a homogenized equation for the potential with interface conditions at a surface is used.


Stokes-Nernst-Planck-drift-diffusion-Poisson system, nanowire sensors, nanopore sensors, existence, local uniqueness

2010 Mathematics Subject Classification

35Q20, 62P30, 76R50, 82D37, 82D80

The authors acknowledge support from the FWF (Austrian Science Fund) START project no. Y660, “PDE Models for Nanotechnology”.

Received 6 March 2017

Accepted 1 July 2019

Published 3 February 2020