Well-posedness and long-time behavior of the Hele-Shaw-Cahn-Hilliard system

The Hele-Shaw-Cahn-Hilliard model is one of the most popular system describing two-phase flows in porous media or Hele-Shaw cell using the phase-field approach. We will discuss the well-posedness and long-time behavior of the incompressible Hele-Shaw-Cahn-Hilliard system in two and three spatial dimensions. We show the convergence of global weak/strong solution to equilibrium as time goes to infinity with a rate via the Lojasiewicw-Simon technique. Stability of the energy minimizers is also discussed.