Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

TL;DR

This paper proves the global existence of weak solutions for a nonlocal diffuse interface model describing two incompressible fluids with different densities, using a novel approach independent of previous local models.

Contribution

It establishes the existence of global weak solutions for a nonlocal two-phase flow model with singular potentials, advancing the mathematical understanding of such systems.

Findings

Existence of global dissipative weak solutions proven.

Applicable to nonlocal models with singular double-well potentials.

Method is independent of previous local model approaches.

Abstract

We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Gr\"{u}n and consists in a Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.