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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 ( $$\epsilon \rightarrow 0$$ ). 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 $$\epsilon \rightarrow 0$$ . 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 $$\epsilon $$ . We also consider electroneutral boundary conditions, for which we prove that electroneutrality $$\rho \rightarrow 0$$ is achieved at any fixed $$\epsilon > 0$$ , exponentially fast in time in $$L^p$$ , for all $$1\leqq p < \infty $$ . The results hold for two oppositely charged ionic species with arbitrary ionic diffusivities, in bounded domains with smooth boundaries.

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 in bounded domains with a smooth boundary in three space dimensions, in the following situations. We consider: a 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. Global regularity holds for any positive, possibly different diffusivities of the ions, 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 same global smoothness of solutions is proved to hold for arbitrarily many ionic species as well, but in that case we require all their diffusivities to be the same.

We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is inspired by the recent result (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]).

We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system in bounded domains, in two dimensions, with Dirichlet boundary conditions for the Navier-Stokes and Poisson equations, and blocking (vanishing normal flux) or selective (Dirichlet) boundary conditions for the ionic concentrations. We prove global existence and stability results for large data.