Convergence of a Nonlocal to a Local Diffuse Interface Model for Two-Phase Flow with Unmatched Densities

TL;DR

This paper proves that solutions of a nonlocal diffuse interface model for two-phase flow with different densities converge to solutions of a local model, providing a rigorous link between nonlocal and local descriptions.

Contribution

It establishes the convergence of weak solutions from a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities, extending previous results to more complex systems.

Findings

Weak solutions of nonlocal model converge to local model solutions

Convergence proven in smooth bounded domains with specific boundary conditions

Method extends to systems with different densities in two-phase flow

Abstract

We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the corresponding system with a standard "local" Cahn-Hilliard equation. The analysis is done in the case of a sufficiently smooth bounded domain with no-slip boundary condition for the velocity and Neumann boundary conditions for the Cahn-Hilliard equation. The proof is based on the corresponding result in the case of a single Cahn-Hilliard equation and compactness arguments used in the proof of existence of weak solutions for the diffuse interface model.