Boundary observability of gravity water waves

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.

Stabilization of the Water-Wave Equations with Surface Tension

“…Proof. The proof follows the analysis in [1,2]. The main novelty is that we are now able to take into account surface tension.…”

“…This approach is not new. It was already performed in our previous works [1,2] and was based on several tools: the multiplier technique (with the multiplier m(x)∂ x for some function m to be determined), the Craig-Sulem-Zakharov reduction to an hamiltonian system on the boundary, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations guided by the analysis done by Benjamin and Olver of the conservation laws for water waves (cf [6]). We need here to adapt this analysis to the present context with surface tension, which requires new ideas.…”

“…Somewhat surprisingly, it was possible to prove this result under some mild smallness condition on the solution which allows to consider truly nonlinear regimes. However, the main drawback is that one cannot control the frequency localization of the solutions in terms of the initial data, unless one makes strong smallness assumptions on these initial data (see §5 in [1]). In this paper, we will prove that one can exploit surface tension to improve the stabilization of water waves in three directions: i) the main improvement is that, under mild smallness assumptions on η, one can prove that E(T ) ≤ (C/T )E(0) for some constant C depending only on the physical parameters g, κ, h, L. In particular, C is independent of the frequency localization.…”

“…The first step is to use the multiplier method to obtain an identity. We use the following variant of the multiplier method (as introduced in our previous paper [1]): we set…”

Controllability of quasi-linear Hamiltonian NLS equations

“…Concerning controllability theory for quasi-linear PDEs, most known results deal with first order quasi-linear hyperbolic systems of the form u t + A(u)u x = 0 (see, for example, Coron [23] chapter 6.2 and the many references therein). Recent results for different kinds of quasi-linear PDEs are contained in Alazard, Baldi and Han-Kwan [6] on the internal controllability of gravitycapillary water waves equations, in Alazard [2,3,4] on the boundary observability and stabilization of gravity and gravity-capillary water waves, and in Baldi, Floridia and Haus [14,15] on the internal controllability of quasi-linear perturbations of the Korteweg-de Vries equation.…”