found by classical fluid dynamical formalisms, by taking into account both many body effects and external force fields. This is commonly known as (classical) density functional theory (DFT) [@Mermin:1965lo]. It is able to predict effects driven by the microscale, e.g., the non-smooth droplet profiles which are formed at the gas-liquid-solid trijunction in contact line problems [@berim2009simple] and the coexistence of multiple fluid films at critical values of the chemical potential energy in droplet spreading [@pereira2012equilibrium]. It has been used to resolve the paradox of stress and pressure singularities normally found in classical moving contact line problems [@sibley2013contact]. What is more, DFT agrees well with molecular dynamics simulations; see, e.g., [@Lutsko10] and references therein. These advancements motivate more mathematical analysis, in particular, on the well-posedness of the underlying equations being used and on the number and structure of equilibrium states.

As a non-equilibrium extension to DFT for classical fluids, dynamic DFT (DDFT) has been applied to a wide range of problems: polymeric solutions [@PennaDzubiellaTarazona03], spinodal decomposition [@ArcherEvans04], phase separation [@Archer05], granular dynamics [@MarconiMelchionna07; @MarconiTarazonaCecconi07], nucleation [@vanTeeffelenLikosLowen08], liquid crystals [@WittkowskiLowenBrand10], and evaporating films [@ArcherRobbinsThiele10]. Recently, a stochastic version of DDFT has been derived [@Lutsko12], which allows the study of energy barrier crossings, 