  Two-phase flow of two Newtonian incompressible viscous fluids with a soluble
surfactant and different densities of the fluids can be modeled within the
diffuse interface approach. We consider a Navier-Stokes/Cahn-Hilliard type
system coupled to non-linear diffusion equations that describe the diffusion of
the surfactant in the bulk phases as well as along the diffuse interface.
Moreover, the surfactant concentration influences the free energy and therefore
the surface tension of the diffuse interface. For this system existence of weak
solutions globally in time for general initial data is proved. To this end a
two-step approximation is used that consists of a regularization of the time
continuous system in the first and a time-discretization in the second step.
