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>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>

Symmetric waves are traveling waves of some shallow water scalar equations

“…In previous studies, 5,6 the authors improved the local existence of solution to (1) to the Besov space setting B 2 p,q (R) where p, q ∈ [0, +∞] and s > max({3∕2, (p + 1)∕p} and provides some blow-up criterion. Recently, the well-posedness for space-periodic solutions is established in H s (R∕Z) for s > 3∕2 in Duruk Mutlubas et al 7 Furthermore, Khorbatly 8 addresses maximal time existence and wave breaking in terms of 𝜀 −1 dependence. On the other hand, Geyer and Quirchmayr 9 classify all (weak) traveling wave solutions of (1) in H 1 loc (R).…”

Rigorous estimates on mechanical balance laws in the Boussinesq–Peregrine equations

“…23, or similar to work done in Refs. 9, 13 and 19, 20 to similar models, it is possible to deduce the following local existence result for () on time scales $$T$$ of order $$\frac{1}{\max (\varepsilon,\beta)}$$. In addition, the author in Ref.…”

“…Now, combining (20) with ( 19) and ( 17) yields the desired 𝜀 2 -approximation of 𝜓 𝑥 and ℎ𝜓 𝑥 in terms of 𝜁 and 𝑣:…”