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We study an initial-boundary value problem for the incompressibleNavier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels,Garcke and Gr\"{u}n in 2012. This model arises in the diffuse interface theoryfor binary mixtures of viscous incompressible fluids. This system is ageneralization of the well-known model H in the case of fluids with unmatcheddensities. In three dimensions, we prove that any global weak solution (forwhich uniqueness is not known) exhibits a propagation of regularity in time andstabilizes towards an equilibrium state as $t \rightarrow \infty$. Moreprecisely, the concentration function $\phi$ is a strong solution of theCahn-Hilliard equation for (arbitrary) positive times, whereas the velocityfield $\mathbf{u}$ becomes a strong solution of the momentum equation for largetimes. Our analysis hinges upon the following key points: a novel globalregularity result (with explicit bounds) for the Cahn-Hilliard equation withdivergence-free velocity belonging only to $L^2(0,\infty;\mathbf{H}^1_{0,\sigma}(\Omega))$, the energy dissipation of the system, theseparation property for large times, a weak-strong uniqueness type result, andthe Lojasiewicz-Simon inequality. Additionally, in two dimensions, we show theexistence and uniqueness of global strong solutions for the full system.Finally, we discuss the existence of global weak solutions for the case of thedouble obstacle potential.

Global regularity and asymptotic stabilization for the incompressible Navier-Stokes-Cahn-Hilliard model with unmatched densities