A new operator generalizing the Dirichlet-Neumann operator to constant-vorticity flows over arbitrary bottoms is analyzed, and local well-posedness for the resulting water-wave system is proved.
Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity
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abstract
In this paper we consider three-dimensional water waves with vorticity, under the action of gravity. We discuss a generalized Zakharov-Craig-Sulem formulation of the problem introduced by Castro and Lannes, which involves a generalized Dirichlet-Neumann operator. We study this operator in detail, extending some well-known results about the classical Dirichlet-Neumann operator for irrotational water waves, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and a paralinearization result. We stress the fact that no geometric condition on either the velocity field or the vorticity is assumed.
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Two-dimensional water waves with constant vorticity and general bottom topography
A new operator generalizing the Dirichlet-Neumann operator to constant-vorticity flows over arbitrary bottoms is analyzed, and local well-posedness for the resulting water-wave system is proved.