  In this paper we consider a layer of incompressible viscous fluid lying above
a flat periodic surface in a uniform gravitational field. The upper boundary of
the fluid is free and evolves in time. We assume that a mass of surfactants
resides on the free surface and evolves in time with the fluid. The surfactants
dynamics couple to the fluid dynamics by adjusting the surface tension
coefficient on the interface and also through tangential Marangoni stresses
caused by gradients in surfactant concentration. We prove that small
perturbations of equilibria give rise to global-in-time solutions in an
appropriate functional space, and we prove that the solutions return to
equilibrium exponentially fast. In particular this proves the asymptotic
stability of equilibria.
