Local well-posedness of a quasi-incompressible two-phase flow

We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier–Stokes/Cahn–Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier–Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end, we show maximal $$L^2$$ L 2 -regularity for the Stokes part of the linearized system and use maximal $$L^p$$ L p -regularity for the linearized Cahn–Hilliard system.