  In an effort to study the stability of contact lines in fluids, we consider
the dynamics of an incompressible viscous Stokes fluid evolving in a
two-dimensional open-top vessel under the influence of gravity. This is a free
boundary problem: the interface between the fluid in the vessel and the air
above (modeled by a trivial fluid) is free to move and experiences capillary
forces. The three-phase interface where the fluid, air, and solid vessel wall
meet is known as a contact point, and the angle formed between the free
interface and the vessel is called the contact angle. We consider a model of
this problem that allows for fully dynamic contact points and angles. We
develop a scheme of a priori estimates for the model, which then allow us to
show that for initial data sufficiently close to equilibrium, the model admits
global solutions that decay to equilibrium exponentially fast.
