Derivation and Well-Posedness Analysis of the Higher-Order Benjamin-Bona-Mahony Equation

TL;DR

The authors derive a fifth-order BBM-type equation by applying small-parameter corrections to the $abcd$-system and performing higher-order asymptotics, yielding a model with enhanced dispersion and nonlinear effects for long-time water-wave dynamics. Local well-posedness in $H^s(\mathbb{R})$ for $s\ge1$ is established via a contraction mapping on an integral formulation built from a unitary linear propagator and Fourier-multiplier operators $\phi(\partial_x)$, $\tau(\partial_x)$, and $\psi(\partial_x)$. Global well-posedness is proved under $\gamma=7/48$ with $\gamma_1,\delta_1>0$, using an energy conservation law to obtain a priori bounds in $H^2$ and a decomposition-into-parts strategy to extend results to $s\ge1$; in particular, global well-posedness in $H^s$ for $s\ge2$ follows from energy control, while $1\le s<2$ uses an iterative, interval-by-interval argument. The work provides a rigorous theoretical basis for high-order water-wave models on long time scales and demonstrates how harmonic-analysis tools and energy methods can handle complex nonlinear and dispersive terms arising in higher-order PDEs.

Abstract

This paper studies the derivation and well-posedness of a class of high - order water wave equations, the fifth - order Benjamin - Bona - Mahony (BBM) equation. Low - order models have limitations in describing strong nonlinear and high - frequency dispersion effects. Thus, it is proposed to improve the modeling accuracy of water wave dynamics on long - time scales through high - order correction models. By making small - parameter corrections to the $abcd-$system, then performing approximate estimations, the fifth - order BBM equation is finally derived.For local well - posedness, the equation is first transformed into an equivalent integral equation form. With the help of multilinear estimates and the contraction mapping principle, it is proved that when $s\geq1$, for a given initial value $η_{0}\in H^{s}(\mathbb{R})$, the equation has a local solution $η\in C([0, T];H^{s})$, and the solution depends continuously on the initial value. Meanwhile, the maximum existence time of the solution and its growth restriction are given.For global well - posedness, when $s\geq2$, through energy estimates and local theory, combined with conservation laws, it is proved that the initial - value problem of the equation is globally well - posed in $H^{s}(\mathbb{R})$. When $1\leq s<2$, the initial value is decomposed into a rough small part and a smooth part, and evolution equations are established respectively. It is proved that the corresponding integral equation is locally well - posed in $H^{2}$ and the solution can be extended, thus concluding that the initial - value problem of the equation is globally well - posed in $H^{s}$.

Theorems & Definitions (18)

• Lemma 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 1: Local Well-Posedness
• proof
• Remark 1