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

Abstract 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 $$\displaystyle L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mstyle> </mml:math> -regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal $$\displaystyle L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mstyle> </mml:math> -regularity for the linearized Cahn–Hilliard system.