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The Cahn-Hilliard-Boussinesq system with singular potential

We consider a Cahn-Hilliard-Boussinesq system with positive heat diffusivity and singular potential on a two-dimensional bounded domain with suitable boundary conditions. For the corresponding initial and boundary value problem we prove the existence of strong solutions and the well-posedness for weak solutions. Then we set the diffusivity equal to zero. In this case, the model can be viewed as an approximation of the two-dimensional compressible Navier-Stokes-Cahn-Hilliard system proposed in [J. Lowengrub, L. Truskinovsky, Proc. R. Soc. Lond. A., 454:2617–2654, 1998]. In particular, the heat equation turns out to be the continuity equation for the fluid density. In the case of zero diffusivity, existence and uniqueness of weak and strong solutions are established. In addition, we show that the solution to the diffusive problem does converge to the solution to the diffusionless when the diffusion coefficient goes to zero. In particular, we provide an error estimate for strong solutions. The validity of the uniform separation property from the pure states is finally proven for both the cases.