Strong Well-posedness of a Diffuse Interface Model for a Viscous, Quasi-incompressible Two-phase Flow

In this article we discuss the existence of strong solutions locally in time for a model of a binary mixture of viscous incompressible fluids in a bounded domain. The model was derived by Lowengrub and Truskinovski. It is used to describe a diffuse interface model for a two-phase flow of two viscous incompressible Newtonian fluids with different densities. The fluids are macroscopically immiscible but partially mix in a small interfacial region. The model leads to a system of Navier–Stokes/Cahn–Hilliard type. Using a suitable result on maximal $L^2$-regularity for the linearized system, the existence of strong solutions is shown with the aid of the contraction mapping principle. The analysis shows that in the case of different densities the system is coupled in highest order and the principal part of the linearized system is of very different structure compared to the case of same densities. The linear system is solved with the aid of a general result on an abstract damped wave equation by Chen and Triggiani.

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