We establish a weak-strong uniqueness principle for the flow of two immiscible, incompressible and viscous fluids with surface tension under the assumption of identical viscosities and densities. As long as there exists a strong solution to the system, every varifold solution originating from the same initial condition has to coincide with it. The global-in-time existence of varifold solutions was established by H. Abels (Interfaces Free Bound. 9, 2007). The key ingredient of our result is the construction of a relative entropy functional which is capable of controlling the interface error.
This is joint work with Julian Fischer.