Abstract
We investigate diffuse-interface models that describe the dynamics of incompressible two-phase viscous flows with surfactant. The first model consists of a sixth-order Cahn-Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn-Hilliard equation for the local concentration of the surfactant. Both equations are coupled with a Navier-Stokes system for the averaged fluid velocity. We prove the existence of a global weak solution, which is unique in two dimensions. Under suitable stronger regularity assumptions on the initial data, we show the existence of a unique global (resp. local) strong solution in two (resp. three) dimensions. Then, we present some recent results for a second model that incorporates a different form of interfacial energy accounting for the adsorption of surfactant. In particular, this model addresses the general case of unmatched densities and variable mobility.