The water-waves problem is one of the classical free boundary problems arising in the context of incompressible fluid flows. It describes an inviscid, irrotational, incompressible fluid flow in a time-dependent domain governed by Euler equations. The free boundary moves with the fluid, separating fluid region from vacuum region and can be characterized as the set of discontinuities in the fluid density.
We will first discuss how to suitably reformulate the problem as a system of contour equations for boundary velocity and parametrization of the interface. We will then discuss local well-posedness in a class of weighted Sobolev spaces which allow propagation of sharp crests on the interface. Finally, as a corollary of our approach, we will show there exist initial data for which the fluid becomes singular in finite-time.