Existence, Uniqueness, Regularity and Long-term Behavior for Dissipative Systems Modeling Electrohydrodynamics

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.