  The Nernst-Planck-Navier-Stokes system models electrodiffusion of ions in a
fluid. We prove global existence of solutions in bounded domains in three
dimensions with either blocking (no-flux) or uniform selective (special
Dirichlet) boundary conditions for ion concentrations. The global existence of
strong solutions is established for initial conditions that are sufficiently
small perturbations of steady state solutions. The solutions remain close to
equilibrium in strong norms. The main two steps of the proof are (1) the decay
of the sum of relative entropies (Kullback-Leibler divergences) and (2) the
control of $L^2$ norms of deviations by the sum of relative entropies.
