MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Existence of Weak solutions for a Diffuse Interface Model for Viscous, Incompressible Fluids with General Densities

Helmut Abels


We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account.

In contrast to previous works, we study the general case that the fluids have different densities. This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard system, where the density of the mixture depends on the concentration, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential.

We prove existence of weak solutions for the non-stationary system in two and three space dimensions.

MSC Codes:
76T99, 76D27, 76D03, 76D05, 76D45, 35Q30, 35Q35
two-phase flow, free boundary value problems, diffuse interface model, mixtures of viscous fluids, Cahn-Hilliard equation, inhomogeneous Navier-Stokes equation

Related publications

2009 Journal Open Access
Helmut Abels

Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities

In: Communications in mathematical physics, 289 (2009) 1, pp. 45-73