Interior Electroneutrality in Nernst–Planck–Navier–Stokes Systems

Abstract

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 $\rho$ converges in time to zero in the interior of the domain, in the limit of vanishing Debye length ($\epsilon\to 0$). For the unstable regime of Dirichlet boundary conditions for the ionic concentrations, we prove bounds that are uniform in time and $\epsilon$. We also consider electroneutral boundary conditions, for which we prove that electroneutrality $\rho\to 0$ is achieved at any fixed $\epsilon> 0$, exponentially fast in time in $L^p$, for all $1\le p<\infty$. The results hold for two oppositely charged ionic species with arbitrary ionic diffusivities, in bounded domains with smooth boundaries.