Well-posedness for a diffuse interface model of non-Newtonian two-phase flows

TL;DR

This paper proves the global existence of weak solutions for a complex non-Newtonian two-phase flow model governed by Navier-Stokes-Cahn-Hilliard equations, addressing initial density zeros and concentration-dependent viscosity.

Contribution

It establishes the first global existence results for weak solutions in a 3D setting with zero initial density and Landau potential, using semi-Galerkin and monotonicity methods.

Findings

Global weak solutions exist for the model with zero initial density.

Local strong solutions are shown to exist in 3D periodic domains.

The proof employs semi-Galerkin scheme and monotonicity techniques.

Abstract

The evolution of two partially miscible, nonhomogeneous, incompressible viscous fluids of non-Newtonian type, can be governed by the Navier-Stokes-Cahn-Hilliard system. In the present work, we prove the global existence of weak solutions for the case of initial density containing zero and the concentration depending viscosity with free energy potential equal to the Landau potential in a bounded domain of three dimensions. Furthermore, we show that a strong solutions exist locally in time in the case of three dimensions periodic domain ${\mathbb T}^3.$ The proof relies on a suitable semi-Galerkin scheme and the monotonicity method.