Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids

We consider a model describing the behaviour of a mixture of two incompressible fluids with the same density under isothermal conditions. The model consists of three balance equations: a continuity equation, a Navier-Stokes equation for the mean velocity of the mixture and a diffusion equation (Cahn-Hilliard equation). We assume that the chemical potential depends on the velocity of the mixture in such a way that an increase in the velocity improves the miscibility of the mixture. We examine the thermodynamic consistence of the model which leads to the introduction of an additional constitutive force in the motion equation. Then, we prove the existence and uniqueness of the solution of the resulting differential problem