Boundary observability of gravity water waves

TL;DR

This paper demonstrates that the energy of a three-dimensional gravity water wave system can be estimated solely from boundary measurements at the contact points, using advanced mathematical techniques.

Contribution

It establishes a boundary observability result for gravity water waves, linking interior energy to boundary measurements through novel analytical methods.

Findings

Energy can be estimated from boundary contact points.

Utilizes multiplier technique and Craig-Sulem-Zakharov formulation.

Provides new insights into water wave conservation laws.

Abstract

Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. We prove that one can estimate its energy by looking only at the motion of the points of contact between the free surface and the vertical walls. The proof 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.