Seminar on partial differential equations

Regularity and Convergence to Equilibrium for a Navier-Stokes-Cahn-Hilliard System with Unmatched Densities

We study the initial-boundary value problem for an incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory  for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as time tends to infinity. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity belonging only to the Leray-Hopf class, the energy dissipation of the system, the separation property for large times, a weak strong uniqueness type result, and the Lojasiewicz-Simon inequality.