For 3D gravity water waves with general vorticity, the generalized Dirichlet-Neumann operator is analytic in the surface profile and admits explicit Taylor and paralinearization formulas.
On the Cauchy problem for gravity water waves
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Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity
For 3D gravity water waves with general vorticity, the generalized Dirichlet-Neumann operator is analytic in the surface profile and admits explicit Taylor and paralinearization formulas.