In this paper we prove the uniqueness of weak solutions to a model of electro-kinetic fluid which consists of a momentum equation together with transport equations of charges. Our result is new in that it holds even when the momentum vanishes. The existence of weak solutions has been proved in P.Biler, W. Hebisch and T. Nadzieja, Nonlinear Anal. TMA, 23, 1189-1209, 1994, R. Ryham, C. Liu and Z-Q. Wang, Preprint, 2005, R. Ryham, arXiv: 0810.2064 v1 (math.AP) 12, Oct 2008.
"Uniqueness of weak solutions to a model of electro-kinetic fluid." Commun. Math. Sci. 7 (2) 411 - 421, June 2009.