Long time dynamics of the Nernst-Planck-Darcy System on $\mathbb{R}^3$

TL;DR

This paper analyzes the long-term behavior of ionic concentrations and fluid flow in a porous medium modeled by the Nernst-Planck equations in three-dimensional space, revealing decay rates and entropy growth over time.

Contribution

It provides rigorous decay estimates for ionic concentration derivatives and characterizes the blow-up of relative entropy in the 3D Nernst-Planck-Darcy system.

Findings

Spatial derivatives decay at rate t^{-3/4 - k/2} in L^2

Relative entropy grows logarithmically with time

Long-time dynamics are characterized in unbounded 3D space

Abstract

We study ionic electrodiffusion modeled by the Nernst--Planck equations describing the evolution of $N$ ionic species in a three-dimensional incompressible fluid flowing through a porous medium. We address the long-time dynamics of the resulting system in the three-dimensional whole space $\mathbb{R}^3$. We prove that the $k$-th spatial derivatives of each ionic concentration decays to zero in $L^2$ with a sharp rate of order $t^{-\frac{3}{4}-\frac{k}{2}}$. Moreover, we investigate the behavior of the relative entropy associated with the model and show that it blows up in time with a sharp growth rate of order $\log t$.