Local Well-Posedness of a Quasi-Incompressible Two-Phase Flow

TL;DR

This paper proves local-in-time well-posedness for a diffuse interface model describing two viscous incompressible fluids with different densities, using advanced regularity techniques for coupled Navier-Stokes and Cahn-Hilliard systems.

Contribution

It establishes local existence of strong solutions for a complex two-phase flow model with variable density, employing maximal regularity and contraction mapping methods.

Findings

Existence of strong solutions locally in time.

Maximal regularity results for Stokes and Cahn-Hilliard systems.

Application of contraction mapping for nonlinear problem.

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.