Existence theory for a Poisson-Nernst-Planck model of electrophoresis

TL;DR

This paper establishes a local in time existence result for weak solutions of a coupled Poisson-Nernst-Planck and fluid dynamics model describing electrophoretic motion of charged macromolecules in an ionized fluid.

Contribution

It provides the first rigorous existence proof for a complex electrohydrodynamic system combining Nernst-Planck, Poisson, Navier-Stokes, and particle dynamics.

Findings

Proves local existence of weak solutions.

Extends previous methods to a coupled electrofluid system.

Lays groundwork for future global existence and stability analysis.

Abstract

A system modeling the electrophoretic motion of a charged rigid macromolecule immersed in a incompressible ionized fluid is considered. The ionic concentration is governing by the Nernst-Planck equation coupled with the Poisson equation for the electrostatic potential, Navier-Stokes and Newtonian equations for the fluid and the macromolecule dynamics, respectively. A local in time existence result for suitable weak solutions is established, following the approach of Desjardins and Esteban [Comm. Partial Diff. Eq., 25 (2000), 1399--1414].