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# Communications in Mathematical Sciences

## Volume 19 (2021)

### Number 3

### Stability of equilibria to the model for non-isothermal electrokinetics

Pages: 687 – 720

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n3.a6

#### Authors

#### Abstract

Recently, energetic variational approach was employed to derive models for nonisothermal electrokinetics by Liu *et al.* [P. Liu, S. Wu, and C. Liu, *Commun. Math. Sci.*, 16(5):1451–1463, 2018]. 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 that, we prove the stability of these equilibria, thus the global well-posedness of PNPF near them.

#### Keywords

Poisson–Nernst–Planck–Fourier system, linearized dissipative law, stability of equilibria

#### 2010 Mathematics Subject Classification

35Q35, 35Q79, 76A02, 80A20

Received 11 December 2019

Accepted 16 October 2020

Published 5 May 2021