Communications in Mathematical Sciences

Volume 19 (2021)

Number 3

Stability of equilibria to the model for non-isothermal electrokinetics

Pages: 687 – 720



Ning Jiang (School of Mathematics and Statistics, Wuhan University, Wuhan, China)

Yi-Long Luo (School of Mathematics, South China University of Technology, Guangzhou, China)

Xu Zhang (School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China)


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.


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