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

On Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids

Helmut Abels


We discuss the existence of generalized solutions of the flow of two immiscible, incompressible, viscous Newtonian and Non-Newtonian fluids with and without surface tension. In the case without surface tension, the existence of weak solutions can be shown, but little is known about the interface between both fluids. If surface tension is present, the energy estimates gives an a priori bound on the $(d-1)$-dimensional Hausdorff measure of the interface, but the existence of weak solutions is open. This might be due to possible oscillation and concentrations effect of the interface related to instabilities of the interface as f.e. fingering, emulsification or just cancellation of area, when two parts of the interface meet. Nevertheless one can show the existence of so-called measure-valued varifold solutions, which are weak solution if an energy equality is satisfied.

MSC Codes:
35Q30, 35Q35, 76D27, 76D45, 76T99
two-phase flow, free boundary value problems, varifold solutions, measure-valued solutions, surface tension

Related publications

2007 Journal Open Access
Helmut Abels and Mathias Wilke

On generalized solutions of two-phase flows for viscous incompressible fluids

In: Interfaces and free boundaries, 9 (2007) 1, pp. 31-65