Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows

We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove the existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher.