Charged Fluids in Porous Media

TL;DR

This paper proves the existence and uniqueness of solutions for the Nernst-Planck-Darcy system modeling ionic electrodiffusion in porous media, advancing mathematical understanding of such systems in multiple dimensions.

Contribution

It establishes the global existence and uniqueness of solutions for the Nernst-Planck-Darcy system with two ionic species in periodic domains.

Findings

Global weak solutions exist in $W^{1,p}$ for $p \,\geq\, 2$

Unique smooth solutions exist for large initial data

Results apply to two and three-dimensional periodic domains

Abstract

The Nernst-Planck-Darcy system models ionic electrodiffusion in porous media. We consider the system for two ionic species with opposite valences. We prove that the initial value problem for the Nernst-Planck-Darcy system in periodic domains in two or three dimensions has global weak solutions in $W^{1, p}$ ($p \geq 2$). We obtain furthermore global existence and uniqueness of smooth solutions for arbitrary large data.