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**Well-posedness in Sobolev spaces of the full water wave problem in 2D.**
*(English)*
Zbl 0892.76009

Summary: We consider the motion of the interface of two-dimensional irrotational, incompressible, inviscid water wave, with air above water and zero surface tension. We show that the interface is always not accelerating into the water region, normal to itself, as rapidly as the normal acceleration of gravity, and as long as the interface is nonself-intersect. We obtain the existence and uniqueness of solutions of the full water wave problem, locally in time, for any initial interface which is nonself-intersecting, including the case where the interface is of multiple heights.