On the variational inequalities related to viscous density-dependent incompressible fluids

We consider some variational inequality formulations related to density-dependent incompressible fluids. Firstly, we state the density-dependent micropolar model, which let us to introduce a generic (vectorial) differential inequality formulation. Then, two relaxations of this differential inequality will be considered, driving to concepts of weak and generalized solutions (observing that the weak solutions are generalized solutions but the contrary is not clear). Afterwards, under similar conditions imposed to prove the existence of generalized solutions for density-dependent Navier-Stokes equations (see Salvi, Riv Mat Parma 4:453-466, 1982), we prove the existence of weak solutions for this generic problem, which involves several variational inequality problems for viscous density-dependent incompressible fluids.