A Dispersive Estimate for the Linearized Water-Waves Equations in Finite Depth

We prove a dispersive estimate for the solutions of the linearized Water-Waves equations in dimension d = 1 and d = 2 in presence of a flat bottom. Adapting the proof from Aynur (An optimal decay estimate for the linearized water wave equation in 2d. arXiv:1411.0963, 2014) in the case of infinite depth, we prove a decay with respect to time t of order vertical bar t vertical bar(-d/3) for solutions with initial data phi such that vertical bar phi vertical bar(1)(H), vertical bar x phi vertical bar(1)(H) are bounded. We also give variants to this result with different decays for a more convenient use of the dispersive estimate. We then give an existence result for the full Water-Waves equations in weighted spaces for practical uses of the proven dispersive estimates.