Appears in collection : 2026 - T2 - WS3 - Idealised mathematical models for geophysical flows

We will talk about a weighted a priori energy estimate for the two dimensional water-waves problem with contact points in the absence of gravity and surface tension and some related topics. 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.