Stabilization of the Water-Wave Equations with Surface Tension

Abstract

Abstract. This paper is devoted to the stabilization of the water-wave equations with surface tension through of an external pressure acting on a small part of the free surface. It is proved that the energy decays to zero exponentially in time, provided that the external pressure is given by the normal component of the velocity at the free surface multiplied by an appropriate cut-off function.

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.…”

A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension

“…For other results on irrotational fluids with surface tension see [2,3,5,11,12,13,17,48,31,41,45,52,79]. Further related results with non-zero surface tension, including the case of rotational fluids, vortex sheets, twophase fluids, and singular limits, are [22,27,32,33,39,51,66,68].…”

A priori estimates for the free-boundary Euler equations with surface tension in three dimensions

“…It has not been until fairly recently, with the works of Lindblad [61] for σ = 0, Coutand and Shkoller [25] for σ ≥ 0, and Shatah and Zeng [72,73], also for σ ≥ 0, and more recently by the first author and Ebin [34] for σ > 0, that existence and uniqueness for (1.4) have been addressed in full generality. Since the early 2000's, research on (1.4) has blossomed, as is illustrated by the sample list [1,3,4,5,6,7,10,8,2,9,12,11,15,16,17,18,19,20,21,22,23,24,26,27,29,30,31,32,33,40,41,42,44,45,46,47,49,50,51,52,54,55,56,…”