Interior Electroneutrality in Nernst–Planck–Navier–Stokes Systems

We consider the limit of vanishing Debye length for ionic diffusion in fluids, described by the Nernst–Planck–Navier–Stokes system. In the asymptotically stable cases of blocking (vanishing normal flux) and uniform selective (special Dirichlet) boundary conditions for the ionic concentrations, we prove that the ionic charge density ρ converges in time to zero in the interior of the domain, in the limit of vanishing Debye length (ϵ→ 0). The order of the limits is important; we take first the long time limit and then the limit of vanishing Debye length. Thus, our main new results in these time asymptotically stable cases concern the vanishing of the ionic charge densities in steady states, as ϵ→ 0. The infinite time behaviors follow as corollaries of our previous works [5, 6]. For the unstable regime of Dirichlet boundary conditions for the ionic concentrations, we prove bounds that are uniform in time and ϵ. We also consider electroneutral boundary conditions, for which we prove that electroneutrality ρ→ 0 is achieved at any fixed ϵ> 0 , exponentially fast in time in L^p, for all 1 ≦ p< ∞. The results hold for two oppositely charged ionic species with arbitrary ionic diffusivities, in bounded domains with smooth boundaries.