Stabilization of gravity water waves

Abstract

Abstract. This paper is devoted to the stabilization of the incompressible Euler equation with free surface. We study the damping of two-dimensional gravity waves by an absorbing beach where the water-wave energy is dissipated by using the variations of the external pressure.

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.…”

“…Consequently, if the solution exists on time intervals of size nT 0 with T 0 ≥ 2C, then E(nT 0 ) ≤ 2 −n E(0), which is the desired exponential decay. We studied a similar problem in [2] for the case without surface tension. Assuming that κ = 0 and defining P ext by…”

“…where the constant C(N ) depends on the frequency localization 2 of the solution (η, ψ). The fact that this constant must depend on the frequency localization can be easily understood by considering the linearized equations (see [2]). Somewhat surprisingly, it was possible to prove this result under some mild smallness condition on the solution which allows to consider truly nonlinear regimes.…”

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.…”