Global weak solutions exist for the quasi-incompressible Navier-Stokes/Cahn-Hilliard model with fractional diffusion, and as the density mismatch alpha tends to zero the solutions converge to Model H at rate alpha on the lifespan of a strong limit solution.
Giorgini, Well-posedness of the two-dimensional Abels-Garcke-Gr¨ un model for two-phase flows with un- matched densities, Calc
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Weak solutions and incompressible limit of a quasi-incompressible Navier--Stokes/Cahn--Hilliard model for viscous two-phase flows
Global weak solutions exist for the quasi-incompressible Navier-Stokes/Cahn-Hilliard model with fractional diffusion, and as the density mismatch alpha tends to zero the solutions converge to Model H at rate alpha on the lifespan of a strong limit solution.