Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

Abstract

<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>