  Based on the global a priori estimates in [Guo-Tice, J. Eur. Math. Soc.
(2024)], we establish the well-posedness of a viscous fluid model satisfying
the dynamic law for the contact line \begin{equation*}
\mathscr{W}(\p_t\zeta(\pm\ell,t))=[\![\gamma]\!]\mp\sigma\frac{\p_1\zeta}{(1+|\p_1\zeta|^2)^{1/2}}(\pm\ell,t)
\end{equation*} in 2D domain, where $\zeta(x_1,t)$ is a free surface with two
contact points $\zeta(\pm\ell,t)$, $[\![\gamma]\!]$ and $\sigma$ are constants
characterizing the solid-fluid-gas free energy, and the increasing
$\mathscr{W}$ is the contact point velocity response function. Motivated by the
energy-dissipation structure, our construction relies on the construction of a
pressureless weak solution for the coupled velocity and free interface for the
linearized problems, via a Galerkin approximation with a time-dependent basis
and an artificial regularization for the capillary operator.
