  We establish the global well-posedness of overdamped dynamic density
functional theory (DDFT): a nonlinear, nonlocal integro-partial differential
equation used in statistical mechanical models of colloidal fluids, and other
applications including nonlinear reaction-diffusion systems and opinion
dynamics. With nonlinear no-flux boundary conditions, we determine the
existence and uniqueness of the weak density and flux, subject to two-body
hydrodynamic interactions (HI). We also show that the density is Lyapunov
stable with respect to the usual (Helmholtz) free energy functional.
Principally, this is done by rewriting the dynamics for the density in an
implicit gradient flow form, resembling the classical Smoluchowski equation but
with spatially inhomogeneous diffusion and advection tensors. We also
rigorously show that the stationary density is independent of the HI tensors,
and prove exponentially fast convergence to equilibrium.
