Abstract
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-regularity for the Stokes part of the linearized system and use maximal L^p-regularity for the linearized Cahn–Hilliard system.
Highlights
Introduction and main resultIn this contribution, we study a thermodynamically consistent, diffuse interface model for two-phase flows of two viscous incompressible system with different densities in a bounded domain in two or three space dimensions
We study a thermodynamically consistent, diffuse interface model for two-phase flows of two viscous incompressible system with different densities in a bounded domain in two or three space dimensions
The model was derived by Garcke and Grün [6] and leads to the following inhomogeneous Navier–Stokes/Cahn– Hilliard system:
Summary
We study a thermodynamically consistent, diffuse interface model for two-phase flows of two viscous incompressible system with different densities in a bounded domain in two or three space dimensions. In the case of non-Newtonian fluids of suitable p-growth, existence of weak solutions was proved by Abels and Breit [3]. For the case of a non-local Cahn–Hilliard equation and Newtonian fluids, the corresponding results was derived by Frigeri [10] and for a model with surfactants by Garcke and the authors in [7]. Giorgini [12] proved existence of local strong solutions in a two-dimensional bounded, sufficiently smooth domain and global existence of strong solutions in the case of a two-dimensional torus. This reformulation will be useful in our analysis.
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