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Local well-posedness of the contact line problem in 2-D Stokes flow

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 $ε$--perturbed linear Stokes problem in moving domains, the contraction mapping principle for the $ε$--perturbed nonlinear full contact line problem, and a continuity argument for uniform energy estimates.