Publication 15-CNA-002

A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations

David Kinderlehrer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
davidk@andrew.cmu.edu

Léonard Monsaingeon
CAMGSD, Instituto Superior Técnico, Portugal
leonard.monsaingeon@ist.utl.pt

Xiang Xu
Department of Mathematics
Purdue University
xu719@purdue.edu

Abstract: The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of global weak solutions in a unified framework for the cases of both linear and nonlinear diffusion. The proof of the main results relies on the derivation of extra estimates based on the flow interchange technique developed by Matthes, McCann, and Savaré in [25].

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