In this talk, I present a new diffuse interface model for incompressible, viscous fluid mixtures with bulk-surface interaction. This system consists of a Navier-Stokes-Cahn-Hilliard model in the bulk that is coupled to a surface Navier-Stokes-Cahn-Hilliard model on the boundary. Compared with previous models in the literature, the inclusion of an additional surface Navier-Stokes equation is motivated, for example, by biological applications. We prove the existence of weak solutions by means of a semi-Galerkin scheme combined with a fixed-point argument. To discretize the Navier-Stokes subsystem, we analyze a novel bulk-surface Stokes system and its corresponding bulk-surface Stokes operator, whose eigenfunctions serve as a natural basis to approximate the velocity fields. Lastly, if time permits, I will mention ongoing work concerning the existence of strong solutions and their uniqueness. > The results are partly based on joint work with Patrik Knopf (University of Regensburg).
