Long Time Dynamics Of The Three-dimensional Nernst-Planck-Darcy Model

TL;DR

This paper studies the long-term behavior of a 3D electrodiffusion model in porous media, proving exponential decay of ionic concentrations without charges and the existence of a finite-dimensional attractor with fixed charges.

Contribution

It establishes global well-posedness and long-time dynamics results for the Nernst-Planck-Darcy model, including decay rates and attractor existence.

Findings

Ionic concentrations decay exponentially without added charges.

Existence of a finite-dimensional global attractor with fixed charges.

Results hold in all Sobolev norms.

Abstract

We consider an electrodiffusion model describing the evolution of $N$ ionic species in a three-dimensional fluid flowing through a porous medium and forced by added body charges. We address the global well-posedness and long-time dynamics of the model. In the absence of added charges, we prove that the ionic concentrations decay exponentially fast in time to their initial spatial averages in all Sobolev norms. When the fluid undergoes the influence of given time-independent charges, we obtain the existence of a finite-dimensional global attractor.