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We establish the global well-posedness of overdamped dynamic densityfunctional theory (DDFT): a nonlinear, nonlocal integro-partial differentialequation used in statistical mechanical models of colloidal fluids, and otherapplications including nonlinear reaction-diffusion systems and opiniondynamics. With nonlinear no-flux boundary conditions, we determine theexistence and uniqueness of the weak density and flux, subject to two-bodyhydrodynamic interactions (HI). We also show that the density is Lyapunovstable with respect to the usual (Helmholtz) free energy functional.Principally, this is done by rewriting the dynamics for the density in animplicit gradient flow form, resembling the classical Smoluchowski equation butwith spatially inhomogeneous diffusion and advection tensors. We alsorigorously show that the stationary density is independent of the HI tensors,and prove exponentially fast convergence to equilibrium.

Well-Posedness and Equilibrium Behaviour of Overdamped Dynamic Density Functional Theory