Global Regularity for Nernst-Planck-Navier-Stokes Systems with Mixed Boundary Conditions

Abstract: We consider electrodiffusion of ions in fluids, described by the Nernst-Planck-Navier-Stokes system, in three dimensional bounded domains, with mixed blocking (no-flux) and selective (Dirichlet) boundary conditions for the ionic concentrations and Robin boundary conditions for the electric potential, representing the presence of an electrical double layer. We prove global existence of strong solutions for large initial data in the case of two oppositely charged ionic species. The result hold unconditionally in the case where fluid flow is described by the Stokes equations. In the case of Navier-Stokes coupling, the result holds conditionally on Navier-Stokes regularity. We use a simplified argument to also establish global regularity for the case of purely blocking boundary conditions for the ionic concentrations for two oppositely charged ionic species and also for more than two species if the diffusivities are equal and the magnitudes of the valences are also equal.