“…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.…”