In this paper, we study the existence and long-time behavior of global strong solutions to a system describing the mixture of two viscous incompressible Newtonian fluids of the same density. The system consists of a coupling of Navier-Stokes and Cahn-Hilliard equations. We first show the global existence of strong solutions in several cases. Then we prove that the global strong solution of our system will converge to a steady state as time goes to infinity. We also provide an estimate on the convergence rate.
"Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids." Commun. Math. Sci. 7 (4) 939 - 962, December 2009.