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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