Linear dispersive decay estimates for the water-wave system with fractional boundary operators

In this paper, we try to obtain the decay rate for the free waves Tα(t)f≔eit|∇|+|∇|αf in two dimension for α > 0. We will characterize the impact of α on the decay estimate. In fact, we shall show the decay rate of Tα(t)f is t−1 if α ≤ 2, while the decay rate of Tα(t)f becomes t−5/6 if α > 2. It is well known it is associated with the linearized gravity-capillary water waves equation when α = 3 and gravity-hydroelastic waves equation when α = 5. Hence, our results are very helpful to establish the global well-posedness of a series of water wave models such as hydroelastic waves equation.