Stability of Weak Electrokinetic Flow

TL;DR

This paper demonstrates that certain very weak nonequilibrium steady states with nonzero fluid flow in the Nernst-Planck-Stokes system remain globally asymptotically stable despite the breakdown of dissipative structures that typically ensure stability.

Contribution

It establishes the global asymptotic stability of specific weak nonequilibrium steady states with fluid flow, extending stability results beyond equilibrium solutions.

Findings

Weak nonequilibrium steady states are globally stable.

Stability holds despite the loss of dissipative structure.

Nonzero fluid flow states can be stable.

Abstract

We consider the Nernst-Planck-Stokes system on a bounded domain of $\mathbb{R}^d$, $d=2,3$ with general nonequilibrium Dirichlet boundary conditions for the ionic concentrations. It is well known that, in a wide range of cases, equilibrium steady state solutions of the system, characterized by zero fluid flow, are asymptotically stable. In these regimes, the existence of a natural dissipative structure is critical in obtaining stability. This structure, in general, breaks down under nonequilibrium conditions, in which case, in the steady state, the fluid flow may be nontrivial. In this short paper, we show that, nonetheless, certain classes of very weak nonequilibrium steady states, with nonzero fluid flow, remain globally asymptotically stable.