  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 Savar\'e.
