On two-phase flows of viscous imcompressible fluids

We consider the flow of two immiscible, incompressible viscous fluids of same density but different viscosities. There are many results on short-time existence of (strong) solutions for this and similar free boundary value problems in fluid mechanics. Also the case of global in time existence of strong solutions for initial data near an equilibrium was studied by many mathematicians. But there are only few results on large time existence of (weak) solutions to such kind of free boundary value problems for arbitrary initial data.

In this talk we will discuss some results on large time existence of suitably defined weak solutions to the two phase flow described above. Neglecting the effect of surface tension, weak solutions are known to exist for arbitrary large data and time; but there is almost no information on the interface between the two fluids in this case. Taking surface tension into account gives an a priori bound of the area of the interface, which can be used in the construction of weak solutions, which have a countably rectifiable interface of finite area. We present a conditional result showing the existence of weak solutions in the case of surface tension if no area or energy is lost during the approximation.