Navier–Stokes–Cahn–Hilliard equations on evolving surfaces

We derive a system of equations which can be seen as an evolving surface version of the diffuse interface “Model H” of Hohenberg and Halperin (1977). We then consider the well-posedness for the corresponding (tangential) system when one prescribes the evolution of the surface. Well-posedness is proved for smooth potentials in the Cahn–Hilliard equation with polynomial growth, and also for a thermodynamically relevant singular potential.