Global weak solutions to a diffuse-interface model for quasi-incompressible two-phase flows with unmatched densities and singular potential

TL;DR

This paper proves the existence of global weak solutions for a complex two-phase flow model with unmatched densities, singular potentials, and a non-divergence-free velocity field, advancing mathematical understanding of such systems.

Contribution

It provides the first existence proof for a Navier-Stokes/Cahn-Hilliard system with unmatched densities and mass-averaged velocity without spatial regularization.

Findings

Established global-in-time weak solutions for the model.

Developed a reduction to a Korteweg-type fluid model with two-layer approximation.

Derived tail estimates to handle singular potentials and exclude concentration phenomena.

Abstract

We study a thermodynamically consistent diffuse-interface model that describes the motion of two macroscopically immiscible, incompressible, and viscous Newtonian fluids with unmatched densities. This model is compatible with continuum mixture theory. It adopts a mass-averaged (barycentric) velocity so that the two-phase flow is quasi-incompressible: the velocity is no longer divergence-free, and the pressure enters the equation of the chemical potential. For the initial-boundary value problem in $\mathbb{T}^3$ with a class of physically relevant singular free energy densities, we prove the existence of global-in-time weak solutions. The proof relies on a suitable reduction of the original system to a Korteweg-type fluid model combined with a two-layer approximation, together with delicate estimates for the mass density and the phase-field variable inspired by the celebrated Bresch-Desjardins entropy. A key observation is that capillarity at the free interface provides a damping effect on the density evolution. For the limiting procedure, we derive delicate tail estimates to exclude possible concentrations of the singular potential, since no integrability of the pressure is available \textit{a priori}. This work appears to be the first existence result for the Navier-Stokes/Cahn-Hilliard type system with unmatched densities and mass-averaged velocity without spatial regularization.