  We study a dissipative system of nonlinear and nonlocal equations modeling
the flow of electrohydrodynamics. The existence, uniqueness and regularity of
solutions is proven for general $\mathbf{L}^2$ initial data in two space
dimensions and for small data in data in three space dimensions. The existence
in three dimensions is established by studying a linearization of a relative
entropy functional. We also establish the convergence to the stationary
solution with a rate.
