Nernst-Planck-Navier-Stokes Systems Far From Equilibrium

Abstract: We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system. We prove that the system has global smooth solutions for arbitrary smooth data: arbitrary positive Dirichlet boundary conditions for the ionic concentrations, arbitrary Dirichlet boundary conditions for the potential, arbitrary positive initial concentrations, and arbitrary regular divergence-free initial velocities. The result holds for any positive diffusivities of ions, in bounded domains with smooth boundary in three space dimensions, in the case of two ionic species, coupled to Stokes equations for the fluid. The result also holds in the case of Navier-Stokes coupling, if the velocity is regular. The global smoothness of solutions is also true for arbitrarily many ionic species, if all their diffusivities are the same.