Nernst-Planck-Navier-Stokes systems near equilibrium

TL;DR

This paper proves the global existence and stability of solutions near equilibrium for the Nernst-Planck-Navier-Stokes system, modeling ion electrodiffusion in fluids, under certain boundary conditions in three dimensions.

Contribution

It establishes the first global existence results for solutions close to equilibrium for this coupled system in three dimensions.

Findings

Solutions remain close to equilibrium in strong norms.

Decay of relative entropies is proven, aiding stability analysis.

Global existence of strong solutions is achieved for small initial perturbations.

Abstract

The Nernst-Planck-Navier-Stokes system models electrodiffusion of ions in a fluid. We prove global existence of solutions in bounded domains in three dimensions with either blocking (no-flux) or uniform selective (special Dirichlet) boundary conditions for ion concentrations. The global existence of strong solutions is established for initial conditions that are sufficiently small perturbations of steady state solutions. The solutions remain close to equilibrium in strong norms. The main two steps of the proof are (1) the decay of the sum of relative entropies (Kullback-Leibler divergences) and (2) the control of $L^2$ norms of deviations by the sum of relative entropies.