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

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 $\log t$.