A shallow-water approximation to the full water wave problem

We demonstrate that the system of the Green-Naghdi equations as a two-directional, nonlinearly dispersive wave model is a close approximation to the two-dimensional full water wave problem, Based on the energy estimates and the proof of the well-posedness for the Green-Naghdi equations and the water wave problem, we compare solutions of the two systems, showing that without restrictions on the wave amplitude, any two solutions of the two systems remain close, at least in some finite time within the shallow-water regime, provided that their initial data are close in the Banach space H ^s x H ^s+1 for some s > 3/2. As a consequence, we show that if the depth of the water compared with the wavelength is sufficiently small, the two solutions exist for the same finite time using the uniformly bounded energies defined in the paper.