  We consider the evolution of contact lines for viscous fluids in a
two-dimensional open-top vessel. The domain is bounded above by a free moving
boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid
are governed by the incompressible Stokes equations under the influence of
gravity, and the interface between fluid and air is under the effect of
capillary forces. Here we develop a local well-posedness theory of the problem
in the framework of nonlinear energy methods. We utilize several techniques,
including: energy estimates of a geometric formulation of the Stokes equations,
a Galerkin method with a time-dependent basis for an $\epsilon$--perturbed
linear Stokes problem in moving domains, the contraction mapping principle for
the $\epsilon$--perturbed nonlinear full contact line problem, and a continuity
argument for uniform energy estimates.
