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Well-posedness in Sobolev spaces of the full water wave problem in 3D. (English) Zbl 0921.76017
Summary: We consider the motion of the interface of a three-dimensional inviscid, incompressible, irrotational water wave, with air region above water region and zero surface tension. We prove that the motion of the interface of water wave is not subject to Taylor instability, as long as the interface separates the whole three-dimensional space into two simply connected \(C^{2}\) regions. We prove further the existence and uniqueness of solutions of the full three-dimensional water wave problem, locally in time, for any initial interface that separates the whole three-dimensional space into two simply connected regions.

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
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