Long time well-posedness and full justification of a Whitham-Green-Naghdi system

We establish the full justification of a "Whitham-Green-Naghdi" system modeling the propagation of surface gravity waves with bathymetry in the shallow water regime. It is an asymptotic model of the water waves equations with the same dispersion relation. The model under study is a nonlocal quasilinear symmetrizable hyperbolic system without surface tension. We prove the consistency of the general water waves equations with our system at the order of precision $O(\mu^2 (\varepsilon + \beta))$, where $\mu$ is the shallow water parameter, $\varepsilon$ the nonlinearity parameter, and $\beta$ the topography parameter. Then we prove the long time well-posedness on a time scale $O(\frac{1}{\max\{\varepsilon,\beta\}})$. Lastly, we show the convergence of the solutions of the Whitham-Green-Naghdi system to the ones of the water waves equations on the later time scale.