Long time dynamics of the Nernst-Planck-Darcy System on $\mathbb{R}^3$
Elie Abdo, Joe Germany, Mohammad Khalil Hamdan, Kifah Kontar
TL;DR
The paper analyzes the long-time behavior of the Nernst–Planck–Darcy (NPD) system for $N$ ionic species in $\mathbb{R}^3$, detailing how electro-diffusive transport coupled with Darcy flow decays in Sobolev norms and how entropy evolves. Using Fourier splitting and bootstrapping, it proves sharp decay rates for the $L^2$ norm of the $k$-th spatial derivatives of each concentration: $\|\Lambda^k c_i\|_{L^2}^2 \le \tfrac{M_k}{(t+1)^{k+3/2}}$, which corresponds to $\|\Lambda^k c_i\|_{L^2} \lesssim (t+1)^{-(\tfrac{3}{4}+\tfrac{k}{2})}$. It further establishes the sharpness of these rates by comparing to the linear heat flow, showing a matching lower bound $\|\,\Lambda^k c_i\|_{L^2}^2 \gtrsim N_1/(t+1)^{k+3/2}$ for large times. In addition, the relative entropy $\mathcal{E}(t)$ grows like $-\log t$ in $\mathbb{R}^3$ (with precise lower and upper bounds) but decays to zero on bounded domains, illustrating that entropy production is driven by mass spreading rather than core-region dynamics. Overall, the results provide a rigorous description of both diffusion-dominated decay and entropic growth in an unbounded, multi-species electro-diffusion system.
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 $\log t$.