In this paper, we will derive the two-dimensional extended Green-Naghdi system {see Matsuno [Proc. R. Soc. A 472, 20160127 (2016)] for determination in a various way} for flat bottoms of order three with respect to the shallowness parameter µ. Then we consider the 1D extended Green-Naghdi equations taking into account the effect of small surface tension. We show that the construction of solution with a standard Picard iterative scheme can be accomplished in which the well-posedness in X s = H s+2 (R) × H s+2 (R) for some s > 3 2 of the new extended 1D system for a finite large time existence t = O(1 ε) is demonstrated.

The aim of this paper is to give an alternative proof for the derivation of a prior energy estimate. Consequently, this allows to define a natural energy norm of the long-term well-posedness result established by Israwi (Derivation and analysis of a new 2D Green-Naghdi system, Nonlinearity, 2010, 23, 2889-2904) but for the original system, in which the partial operator curl is not involved.

<p style='text-indent:20px;'>In the mathematical theory of water waves, this paper focuses on the hierarchy of higher order asymptotic models. The well-posedness of the medium amplitude extended Green-Naghdi model, as well as higher-ordered Boussinesq-Peregrine and Boussinesq models, is first demonstrated. Introducing a regularization term and various physical topography variations, we show that these models admit unique solutions by a standard energy estimate method in the "hyperbolic" space <inline-formula><tex-math id="M1">\begin{document}$ H^{s+2}( \mathbb R)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s&gt;3/2 $\end{document}</tex-math></inline-formula>, but on short/intermediate time scales with respect to amplitude and topography parameters of order <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon^{-1/4} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \max(\sqrt{ \varepsilon}, \beta)^{-1} = \beta^{-1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon^{-1/2} $\end{document}</tex-math></inline-formula> respectively. Furthermore, we show that the extended Green-Naghdi system's long-term well-posedness is reachable on time scales of order <inline-formula><tex-math id="M6">\begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}</tex-math></inline-formula>. The above three specified models, in particular, admit longer time existence of order <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon^{-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ \varepsilon^{-1} $\end{document}</tex-math></inline-formula>, respectively.</p>