  We prove a weighted a priori energy estimate for the two dimensional
water-waves problem with contact points in the absence of gravity and surface
tension. When the surface graph function and its time derivative have some
decay near the contact points, we show that there is corresponding decay for
the velocity, the pressure and other quantities in a short time interval. As a
result, we have fixed contact points and contact angles. To prove the energy
estimate, a conformal mapping is used to transform the equation for the mean
curvature into an equivalent equation in a flat strip with some weights.
Moreover, the weighted limits at contact points for the velocity, the pressure
etc. are tracked and discussed. Our formulation can be adapted to deal with
more general cases.
