Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities and Nonlocal Free Energies

TL;DR

This paper proves the existence of weak solutions for a complex two-phase flow model involving nonlocal free energies and different densities, using implicit time discretization in both 2D and 3D.

Contribution

It introduces a novel analysis for a diffuse interface model with singular non-local free energy, extending previous work to more general conditions.

Findings

Existence of weak solutions established for large times.

Model includes singular non-local free energy controlling the volume fraction.

Applicable to both two and three-dimensional flows.

Abstract

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the $H^{\alpha/2}$-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.