Derivation and well-posedness of the extended Green-Naghdi equations for flat bottoms with surface tension

Abstract

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.

Mathematical modeling and numerical analysis for the higher order Boussinesq system

“…Moreover, higher order approximations may have similar well-posedness results as classcial ones on relevant time scales due to technical but standard mathematical tools. Based on section 3 and previous works [23,25] this can be concluded at least in the one-dimensional case. However, the advantage is obvious in terms of controlling the convergence precision of the approximation error with respect to Euler equations (see in particular Theorem 3 of section 3).…”

“…A better precision is obtained when the O(µ 2 ) terms are kept in the equations: only O(µ 3 ) terms are dropped. Following the work in a series of papers on the extended Green-Naghdi equations [23,25,32,33], one may write the extended Boussinesq equations by incorporating higher order dispersive effects as follows:…”

“…When the surface elevation is of small amplitude, that is, when an assumption is made on the nonlinearity parameter, the extended Green-Naghdi equations [23,25,32,33] can be greatly simplified. Based on this, the extended Boussinesq with ε ∼ µ reads for one-dimensional small amplitude surfaces:…”

A remark on the well‐posedness of the classical Green–Naghdi system

“…The convergence of solution U n = (𝜁 n , v n ) can be established using estimates from the linear theory, that is, Section 4, in combination with additional standard arguments. We recall that such an iterative scheme is applicable for many similar models in one or two space dimension once the linear analysis is accomplished (see Theorem 1 of previous studies 7,14,15 ). Indeed, taking advantage that the Y s -estimate (18) holds for system (42) and the initial assumption U 0 ∈ Y s , then by induction on n, it holds that U n+1 .…”

“…We will give a brief description of the proof commenced by a sequence of nonlinear problems through the induction relation The convergence of solution can be established using estimates from the linear theory, that is, Section 4, in combination with additional standard arguments. We recall that such an iterative scheme is applicable for many similar models in one or two space dimension once the linear analysis is accomplished (see Theorem 1 of previous studies 7,14,15 ). Indeed, taking advantage that the Y s ‐estimate () holds for system () and the initial assumption U 0 ∈ Y s , then by induction on n , it holds that such that for all .…”