On a Multispecies Electrodiffusion Model in Higher-Dimensional Porous Media

Abstract

We consider the Nernst--Planck--Darcy system describing the electrodiffusion of $N$ ionic species with arbitrary valences and different diffusivities, transported by a Darcy flow driven by the electric force, in the whole space ${\mathbb{R}}^d$, $d\ge 3$. We prove the existence of global solutions for initial ionic concentrations that are small in the critical Lebesgue space $L^{d/2}({\mathbb{R}}^d)$. Moreover, we establish uniqueness for initial data in $L^{d+\epsilon }({\mathbb{R}}^d)$ when $3\le d\le 4$ and in $L^{2d-1}({\mathbb{R}}^d)$ when $d\ge 5$, improving on results in the literature, which require Sobolev regularity of the data. The proofs rely on energy estimates exploiting cancellations of the highest-order nonlinear terms, and on the resulting dissipative structures, which yield an instantaneous gain of regularity used to control the doubly nonlinear velocity term in the uniqueness argument.