Mathematical Analysis of a Diffuse Interface Model for Multi-Phase Flows of Incompressible Viscous Fluids with Different Densities

TL;DR

This paper provides a rigorous mathematical analysis of a diffuse interface model for multi-phase flows involving incompressible, viscous fluids with different densities, establishing existence, regularity, and long-term behavior of solutions.

Contribution

It proves existence of weak solutions in 2D and 3D, global solutions in 2D, local strong solutions in 3D, and convergence to stationary states, advancing theoretical understanding of multi-phase flow models.

Findings

Existence of weak solutions in 2D and 3D.

Global existence of solutions in 2D.

Convergence to stationary solutions over time.

Abstract

We analyze a diffuse interface model for multi-phase flows of $N$ incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space dimensions and a singular free energy density is shown. Moreover, in two space dimensions global existence for sufficiently regular initial data is proven. In three space dimension, existence of strong solutions locally in time is shown as well as regularization for large times in the absence of exterior forces. Moreover, in both two and three dimensions, convergence to stationary solutions as time tends to infinity is proved.