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

A priori energy estimate with decay in weighted norms for the water-waves problem with contact points