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The Nernst-Planck-Navier-Stokes system models electrodiffusion of ions in afluid. We prove global existence of solutions in bounded domains in threedimensions with either blocking (no-flux) or uniform selective (specialDirichlet) boundary conditions for ion concentrations. The global existence ofstrong solutions is established for initial conditions that are sufficientlysmall perturbations of steady state solutions. The solutions remain close toequilibrium in strong norms. The main two steps of the proof are (1) the decayof the sum of relative entropies (Kullback-Leibler divergences) and (2) thecontrol of $L^2$ norms of deviations by the sum of relative entropies.

Nernst-Planck-Navier-Stokes systems near equilibrium