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Stability of Weak Electrokinetic Flow

Fizay-Noah Lee

TL;DR

The paper analyzes the Nernst-Planck-Stokes system with Dirichlet boundary data to show that extremely weak nonequilibrium steady states, possibly with nonzero fluid flow, are globally asymptotically stable. Using an energy-method framework, it proves that for sufficiently small boundary data there exists a unique steady state $(c_1^*,c_2^*,u^*)$ and that all time-dependent solutions converge to this state in $L^2$ as $t\to\infty$, with an exponential decay controlled by an energy functional $\mathcal{E}(t)$. It also identifies a condition on boundary data under which the steady-state flow is nontrivial ($u^*\neq0$). This work extends stability results beyond equilibrium boundary conditions, bridging to nonequilibrium regimes where electrokinetic instabilities can arise while still guaranteeing global stability in the small-data limit.

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.

Stability of Weak Electrokinetic Flow

TL;DR

The paper analyzes the Nernst-Planck-Stokes system with Dirichlet boundary data to show that extremely weak nonequilibrium steady states, possibly with nonzero fluid flow, are globally asymptotically stable. Using an energy-method framework, it proves that for sufficiently small boundary data there exists a unique steady state and that all time-dependent solutions converge to this state in as , with an exponential decay controlled by an energy functional . It also identifies a condition on boundary data under which the steady-state flow is nontrivial (). This work extends stability results beyond equilibrium boundary conditions, bridging to nonequilibrium regimes where electrokinetic instabilities can arise while still guaranteeing global stability in the small-data limit.

Abstract

We consider the Nernst-Planck-Stokes system on a bounded domain of , 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.
Paper Structure (4 sections, 5 theorems, 33 equations)

This paper contains 4 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Stability of Weak Electrokinetic Flow

Key Result

Theorem 1

There exists a smooth solution of the steady state Nernst-Planck-Stokes system on a smooth, connected, bounded domain $\Omega\subset\mathbb{R}^d$ ($d=2,3)$ together with boundary conditions with $\gamma_i,W\in C^\infty(\partial\Omega)$.

Theorems & Definitions (8)

  • Theorem 1
  • Proposition 1
  • proof
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • proof