methods available in the pseudo-spectral collocation scheme 2DChebClass.'
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dynamic density functional theory (DDFT), colloids, overdamped limit, hydrodynamic interactions, nonlocal-differential PDEs, interacting particle systems, McKean-Vlasov equation, phase transitions, bifurcation theory.

60F10; 60J75; 62P10; 92C37

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Introduction {#intro}
============

For suspended particles in a viscous fluid, the Navier-Stokes equations are not sufficient to model flows on a spatial scale comparable with the size of the individual particles. Instead, one requires a computationally tractable model that captures meso/macro-scale dynamics whilst also including physical effects driven by particle-level interactions. Dynamic density functional theories (DDFTs) are excellent candidates for modelling such systems [@marconi1999dynamic; @ArcherEvans04]. They are typically applied in condensed matter physics in the colloidal particle regime with particles of typical diameters 1nm$-$1$\mu$m. Recent advances have allowed the inclusion of inertia [@MarconiTarazonaCecconiMelchionna07; @archer2009dynamical], multiple species [@Archer05; @RothRauscherArcher09; @GNK13; @LichtnerArcherKlapp12], hydrodynamic interactions (HI) [@rex2009dynamical; @Rauscher10; @goddard2012general; @goddard2012unification], background flows [@RauscherDominguezKrugerPenna07], temperature gradients [@WittkowskiLowenBrand12; @anero2013functional], hard spheres [@Rosenfeld89; @RothEvansLangKahl02; @roth2010fundamental; @StopperMaroltRothHansen-Goos15], confined geometries [@goddard2016dynamical; @zimmermann2016flow], arbitrary shaped particles [@WittkowskiLowen11], and active microswimmers [@menzel2016dynamical; @hoell2017dynamical].

For equilibrium fluids, there is a rigorous mathematical framework proving the existence of nontrivial fluid densities, different from those 