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Stabilization of the water-waves equations

Consider a fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity and surface tension. In this talk we explain that one can estimate its energy by looking only at a small localized portion of the free surface. This is used to study the damping of the energy by an absorbing beach where the water-wave energy is dissipated by using only the variations of the external pressure over a localized portion of the free surface. The analysis relies on the multiplier technique, the Craig-Sulem-Zakharov formulation of the water-wave problem, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations inspired by the analysis done by Benjamin and Olver of the conservation laws for water waves.

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