Nernst-Planck-Navier-Stokes Systems Far From Equilibrium

TL;DR

This paper proves the existence of global smooth solutions for the Nernst-Planck-Navier-Stokes system modeling ionic electrodiffusion in fluids, under broad conditions including multiple ionic species and various boundary conditions.

Contribution

It establishes the first rigorous proof of global smooth solutions for this complex coupled system in three dimensions with arbitrary data.

Findings

Global smooth solutions exist for two ionic species with arbitrary positive initial and boundary conditions.

The result extends to Navier-Stokes coupling if the velocity field is regular.

The proof applies to multiple ionic species with identical diffusivities.

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.