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We consider some variational inequality formulations related to densitydependent 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.

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We show, using the spectral Galerkin method together with compactness arguments, existence and uniqueness of the periodic strong solutions for the magnetohydrodynamics's type equations with inhomogeneous boundary conditions. Also, we study the asymptotic stability for time periodic solution for this system. In particular, when the magnetic field h(x, t) is zero, we obtain existence, uniqueness and asymptotic behavior of the strong solutions to the Navier-Stokes equations with inhomogeneous boundary conditions.

We prove new regularity criteria of the Prodi-Serrin type with weak Lebesgue integrability in both space and time for a viscous active chemical fluid in a bounded domain.