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We consider ionic electrodiffusion in fluids, described by theNernst-Planck-Navier-Stokes system. We prove that the system has global smoothsolutions for arbitrary smooth data: arbitrary positive Dirichlet boundaryconditions for the ionic concentrations, arbitrary Dirichlet boundaryconditions for the potential, arbitrary positive initial concentrations, andarbitrary regular divergence-free initial velocities. The result holds for anypositive diffusivities of ions, in bounded domains with smooth boundary inthree space dimensions, in the case of two ionic species, coupled to Stokesequations for the fluid. The result also holds in the case of Navier-Stokescoupling, if the velocity is regular. The global smoothness of solutions isalso true for arbitrarily many ionic species, if all their diffusivities arethe same.

Nernst-Planck-Navier-Stokes Systems Far From Equilibrium