Existence and Stability of Nonequilibrium Steady States of Nernst-Planck-Navier-Stokes Systems

TL;DR

This paper proves the existence and stability of nonequilibrium steady states in the Nernst-Planck-Navier-Stokes system, providing conditions for nonzero fluid velocity and demonstrating boundedness and stability of solutions.

Contribution

It establishes the existence of smooth steady states and their stability under certain conditions, advancing understanding of nonequilibrium behaviors in coupled fluid and ionic systems.

Findings

Existence of smooth steady state solutions.

Conditions for nonzero fluid velocity based on boundary data.

Global nonlinear stability of steady states with weak currents.

Abstract

We consider the Nernst-Planck-Navier-Stokes system in a bounded domain of ${\mathbb {R}}^d$, $d=2,3$ with general nonequilibrium Dirichlet boundary conditions for the ionic concentrations. We prove the existence of smooth steady state solutions and present a sufficient condition in terms of only the boundary data that guarantees that these solutions have nonzero fluid velocity. We show that time evolving solutions are ultimately bounded uniformly, independently of their initial size. In addition, we consider one dimensional steady states with steady nonzero currents and show that they are globally nonlinearly stable as solutions in a three dimensional periodic strip, if the currents are sufficiently weak.