On the Cauchy problem for a model of electro-kinetic fluid

In this paper, we consider the Cauchy problem for a model of electro-hydrodynamics in R d ( d = 2 , 3 ) . When d = 2 , we prove that the model has a unique global-in-time smooth solution. When d = 3 , we prove a regularity criterion in terms of the vorticity field only.