2025/03/12 by Zeng, Jie
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.09088
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≥1, for a given initial value η0∈ Hs(ℝ), the equation has a local solution η∈ C([0, T];Hs), 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≥2, 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 Hs(ℝ). When 1≤ 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 H2 and the solution can be extended, thus concluding that the initial - value problem of the equation is globally well - posed in Hs.