Diffuse interface model for two-phase flows on evolving surfaces with different densities: local well-posedness

A Cahn–Hilliard–Navier–Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn–Hilliard energy with a singular (logarithmic) potential, short-time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal L² L 2 -regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal Lᵖ L p -regularity for the linearized Cahn–Hilliard system.