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 ℝd, d ≥ 3. We prove the existence of global solutions for initial ionic concentrations that are small in the critical Lebesgue space Ld/2(ℝd). Moreover, we establish uniqueness for initial data in Ld+ε(ℝd) when 3 ≤ d ≤ 4 and in L2d−1(ℝd) when d ≥ 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.
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BibTeX
@misc{abdo2026multispecies,
title = {{On a Multispecies Electrodiffusion Model in Higher-Dimensional Porous Media}},
author = {Elie Abdo and Christofer Ghazale},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14659},
eprint = {2609.14659},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14659},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a Multispecies Electrodiffusion Model in Higher-Dimensional Porous Media
AU - Elie Abdo
AU - Christofer Ghazale
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14659
UR - https://arxiv.org/abs/2609.14659
AB - 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 ℝ^d, d ≥ 3. We prove the existence of global solutions for initial ionic concentrations that are small in the critical Lebesgue space L^d/2(ℝ^d). Moreover, we establish uniqueness for initial data in L^d+ε(ℝ^d) when 3 ≤ d ≤ 4 and in L^2d−1(ℝ^d) when d ≥ 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.
ER -