A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility

TL;DR

This paper develops a mathematical model for two-phase incompressible flows incorporating nonlocal interactions and variable mobility, proving existence of solutions and attractors in different scenarios.

Contribution

It introduces a diffuse interface model with nonlocal interactions and non-constant mobility, extending existence results to degenerate cases and establishing global attractors.

Findings

Existence of global weak solutions for non-degenerate mobilities.

Existence of global attractors in two and three dimensions.

Extension to degenerate mobilities and singular potentials.

Abstract

We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.