density. Mathematically, the convergence to equilibrium will depend inherently on the spectral properties of the effective diffusion tensor and effective drift vector arising from the HI. What is more, since the full $N$-body Fokker-Planck equation is a PDE in a very high dimensional phase space, well-posed nonlinear, nonlocal PDEs governing the evolution of the one-particle distribution function, valid in the mean field limit, describing the flow of nonhomogeneous fluids are desirable for computational reasons.

The equations studied in this paper are related to the McKean-Vlasov equation [@chayes2010mckean], a nonlinear nonlocal PDE of Fokker-Planck type that arises in the meanfield limit of weakly interacting diffusions. The novelty of the present problem lies in the space dependent diffusion tensor and nonlinear, nonlocal boundary conditions. Additionally, the problem that we study in this paper may in general not be written as a gradient flow, with the exception of the modelling assumption that the off-diagonal elements of the friction tensor $\bm{\Gamma}$ are zero. This choice is equivalent to setting $\bm{Z}_2$ to zero, and would be physically relevant for a diffuse system of particles with a strong hydrodynamic interaction with a wall but weak inter-particle hydrodynamic interactions [@goddard2016dynamical].

Description of the Model.
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In this 