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

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