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In this paper we study the dynamics of an incompressible viscous fluidevolving in an open-top container in two dimensions. The fluid mechanics aredictated by the Navier-Stokes equations. The upper boundary of the fluid isfree and evolves within the container. The fluid is acted upon by a uniformgravitational field, and capillary forces are accounted for along the freeboundary. The triple-phase interfaces where the fluid, air above the vessel,and solid vessel wall come in contact are called contact points, and the anglesformed at the contact point are called contact angles. The model that weconsider integrates boundary conditions that allow for full motion of thecontact points and angles. Equilibrium configurations consist of quiescentfluid within a domain whose upper boundary is given as the graph of a functionminimizing a gravity-capillary energy functional, subject to a fixed massconstraint. The equilibrium contact angles can take on any values between $0$and $\pi$ depending on the choice of capillary parameters. The main thrust ofthe paper is the development of a scheme of a priori estimates that show thatsolutions emanating from data sufficiently close to the equilibrium existglobally in time and decay to equilibrium at an exponential rate.

Stability of contact lines in fluids: 2D Navier-Stokes flow