Stability of equilibria to the model for non-isothermal electrokinetics

TL;DR

This paper investigates the stability of certain equilibria in a non-isothermal electrokinetic model derived via energetic variational methods, establishing conditions for stability and global well-posedness of the system.

Contribution

It reformulates the PNPF system into a simpler evolutional form, characterizes a set of stable equilibria, and proves their stability and global well-posedness.

Findings

Stable equilibria are characterized by specific conditions.

Not all positive constant solutions are stable.

Global well-posedness is established near stable equilibria.

Abstract

Recently, energetic variational approach was employed to derive models for non-isothermal electrokinetics by Liu et. al \cite{Liu-Wu-Liu-CMS2018}. In particular, the Poisson-Nernst-Planck-Fourier (PNPF) system for the dynamics of $N$-ionic species in a solvent was derived. In this paper we first reformulate PNPF ($4N+6$ equations) into an evolutional system with $N+1$ equations, and define a new total electrical charge. We then prove the constant states are stable provided that they are such that the perturbed systems around them are dissipative. However, not all positive constant solutions of PNPF are such that the corresponding perturbed systems are dissipative. We characterize a set of equilibria $\mathcal{S}_{eq}$ whose elements satisfy the conditions {(A1)} and {(A2)}, and prove it is nonempty. After then, we prove the stability of these equilibria, thus the global well-posedness of PNPF near them.