Communications in Mathematical Sciences

Volume 18 (2020)

Number 6

Global existence for Nernst–Planck–Navier–Stokes system in $\mathbb{R}^n$

Pages: 1743 – 1754



Jian-Guo Liu (Departments of Physics and Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Jinhuan Wang (School of Mathematics, Liaoning University, Shenyang, China)


In this note, we study the Nernst–Planck–Navier–Stokes system for the transport and diffusion of ions in electrolyte solutions. The key feature is to establish three energy-dissipation equalities. As their direct consequence, we obtain global existence for two-ionic species case in $\mathbb{R}^n , n \geq 2$, and multi-ionic species case in $\mathbb{R}^n , n=2,3$.


electrolyte, electro-osmosis, electrochemical transport and diffusion, global weak solution, entropy method

2010 Mathematics Subject Classification

35Q30, 35Q35, 35Q92

Jinhuan Wang is partially supported by the National Natural Science Foundation of China (Grant No. 11926338), and by the Key Project of Education Department of Liaoning Province (Grant No. LZD201701).

The work of Jian-Guo Liu was partially supported by KI-Net NSF RNMS (Grant No. 1107444) and by NSF DMS (Grant No. 1812573).

Accepted 17 April 2020

Published 4 November 2020